Asymptotically Safe QuantumGravity:
Bimetric Actions, Boundary Terms, & aC-Function
Dissertation
zur Erlangung des Grades »Doktor der Naturwissenschaften«
am Fachbereich Physik, Mathematik und Informatik
der Johannes Gutenberg-Universität in Mainz
Daniel Becker
geboren in Hachenburg (Ww.)
Mainz, 17. August 2015
Daniel Becker
Quantum Gravity - THEP
Institute of Physics
Staudinger Weg 7
55128 Mainz (Germany)
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D77 / Dissertation der Johannes Gutenberg-Universität Mainz
1. Gutachter:
2. Gutachter:
Datum der mündlichen Prüfung: 01.Februar 2016
dedicated to
my family
“ Da steh ich nun, ich armer Tor! Und bin so klug als wie zuvor.(And here, poor fool, I stand once more, No wiser than I was before.)J. W. v. Goethe, Faust I (transl. W. Arndt)

ABSTRACT
Abstract
Today our understanding of Nature is based on two pillars, the classical theory of General Rel-
ativity and the Standard Model of particle physics, each remarkably successful and predictive.
However, the coalescence of a classical spacetime with the quantum nature of matter leads to
severe inconsistencies which are at the heart of many open problems in modern physics. In
constructing a quantum theory of gravity which cures these deficiencies one not only faces the
problem that General Relativity is perturbatively non-renormalizable, but one also has to incor-
porate the fundamental requirement of Background Independence, expressing that no particular
spacetime (such as Minkowski space) is singled out a priori.
In this thesis we employ a non-perturbative, universal method which has played an impor-
tant role in the past decades: the Functional Renormalization Group Equation (FRGE). In the
Effective Average Action (EAA) approach the search for a quantum field theory of gravity is
guided by the Asymptotic Safety conjecture. Proposed by Steven Weinberg, it employs a gen-
eralized notion of renormalizability which goes beyond the standard perturbative ones. The key
requirement is the existence of a non-trivial ultraviolet (UV) fixed point of the Renormalization
Group (RG) flow which has a finite-dimensional UV-critical hypersurface to ensure predictivity.
While classical General Relativity is recovered as an effective description in the infrared (IR),
the bare (or ‘classical’) action emerges from the fixed point condition and is thus a prediction
rather than an input.
Since the original works on Asymptotic Safety in the 90’s, based on the Einstein-Hilbert
truncation, various extensions and generalizations thereof have been studied in the past decades,
all of which provide strong evidence for the existence of a suitable non-trivial fixed point with
a finite number of relevant directions. Thus, it seems very likely that Quantum Gravity is non-
perturbatively renormalizable.
On the other hand, the status of Background Independence in the context of Asymptotic
Safety remained largely unclear. It is a requirement on the global properties of the RG flow:
there must exist RG trajectories which emanate from the fixed point in the UV and restore the
broken split-symmetry (relating background to quantum fields) in the IR.
Employing a ‘bi-metric’ ansatz for the EAA, we demonstrate for the first time in a non-
trivial setting that the two key requirements of Background Independence and Asymptotic
Safety can be satisfied simultaneously. Taken together they are even found to lead to an in-
creased predictivity. A new powerful calculational scheme (‘deformed α = 1 gauge’) is in-
troduced and applied to derive the beta-functions for the bi-metric Einstein-Hilbert truncation
including a Gibbons-Hawking-York term.
Exploring further the global properties of the RG flow, a generalized notion of aC-function
is proposed and successfully tested for bi-metric RG trajectories consistent with Background
Independence. As a byproduct, we also develop a new, completely general testing device to
judge the reliability of truncated computations in the FRGE scheme.
Furthermore, we investigate the occurrence of propagating graviton modes in the Asymp-
totic Safety scenario. The relevant properties of the graviton propagator depend on the sign of
the anomalous dimension, η , of Newton’s coupling. While asymptotically safe RG trajectories
possess a negative η in the UV, we observe that it turns positive in the semi-classical regime.
This feature, not observed in the older single-metric truncations, is found relevant for the scale
dependent (non-)occurrence of gravitational waves. The implications for the generation of pri-
mordial density perturbations in the early Universe are discussed.
The second main focus of this thesis is the inclusion of boundary terms into the EAA. We
present a detailed analysis of the Gibbons-Hawking-York functional and of its RG evolution
induced by the bi-metric bulk invariants. This generalization to spacetime topologies with a
non-empty boundary is particularly important in black hole thermodynamics. We observe a
stabilization of their thermodynamical properties near the Planck scale.
Kurzfassung
Unser heutiges Verständnis der fundamentalen Wechselwirkungen beruht auf zwei sehr erfol-
greichen Theorien großer Vorhersagekraft: der Allgemeinen Relativitätstheorie (ART) und dem
Standardmodell der Teilchenphysik. Beim Versuch die klassische Raumzeit mit der Quanten-
natur der Materie zu verbinden, treten grundlegende konzeptionelle Probleme auf, die eng mit
zentralen offenen Fragen der modernen Physik zusammenhängen. Eine Lösung dieser Probleme
scheint erst in einer Quantentheorie der Gravitation selbst möglich zu sein. In der Konstruktion
einer solchen Theorie treten jedoch neben der störungstheoretischen Nichtrenormierbarkeit der
ART auch konzeptionelle Schwierigkeiten auf, die von der unabdingbaren Forderung nach Hin-
tergrundunabhängigkeit herrühren.
In dieser Arbeit wenden wir eine universelle, nicht-störungstheoretische Methode an, die
in den letzten Jahrzehnten zu weitreichenden Erkenntnissen in den verschiedensten Bereichen
der Physik geführt hat: die Funktionale Renormierungsgruppengleichung (FRGE). Basierend
auf der “Effective Average Action” (EAA) ist die Suche nach einer geeigneten Quantenfeldthe-
orie der Gravitation durch die von Steven Weinberg postulierte Forderung nach asymptotis-
cher Sicherheit bestimmt. Hierbei zeichnet sich eine (nicht-)störungstheoretisch renormierbare
Theorie dadurch aus, dass ihr UV-Verhalten vollständig durch einen (nicht-)trivialen Fixpunkt
(NGFP) des RG-Flusses kontrolliert wird, dessen UV-kritische Hyperfläche endlich-dimensional
ist. Dadurch ergibt sich ein klassisches Regime auf makroskopischen Skalen, wohingegen die
fundamentale (nackte) Theorie aus der Fixpunktbedingung folgt und damit eine Vorhersage
dieses Zugangs ist.
Seit Ende der 90er Jahre die ersten Anzeichen für Asymptotische Sicherheit auf Basis einer
Einstein-Hilbert Rechnung gefunden wurden, hat sich in zahlreichen Erweiterungen und Mod-
ifikationen der Hinweis auf die Existenz eines nicht-trivialen Fixpunktes der Gravitation stark
verdichtet. Demnach scheint die Quantengravitation tatsächlich im nicht-störungstheoretischen
Sinn renormierbar zu sein.
Andererseits ist der Status der Hintergrundunabhängigkeit im Kontext der Asymptotischen
Sicherheit noch weitgehend unklar, was maßgeblich damit zusammenhängt, dass diese Frage
mit globalen Eigenschaften des RG-Flusses zusammenhängt: Asymptotisch sichere RG-Trajek-
torien die im UV an einem NGFP beginnen, müssen zusätzlich die sogenannte Split-Symmetrie
im IR erfüllen.
Um dieser Fragestellung nachzugehen, betrachten wir in dieser Arbeit einen bi-metrischen
Ansatz für die EAA und zeigen erstmalig, dass die Forderungen nach Asymptotischer Sicher-
heit und Hintergrundunabhängigkeit gleichzeitig erfüllt werden können. Es zeigt sich, dass
dadurch die Vorhersagekraft des Zugangs sogar noch größer wird. Wir entwickeln eine neue,
leistungsstarke Methode, die auf einer ‘deformierten’ α = 1 Eichung basiert und es erlaubt, die
RG-Gleichungen der bi-metrischen Einstein-Hilbert Trunkierung, erweitert um einen Gibbons-
Hawking-York Term, abzuleiten und zu studieren.
Im zweiten Teil der Arbeit untersuchen wir allgemeine globale Eigenschaften des RG-
Flusses über den Rahmen der Quantengravitation hinaus und postulieren eine verallgemeinerte
C-Funktion, welche das globale Verhalten der RG-Trajektorien bestimmt. Wir testen diesen
Vorschlag erfolgreich anhand der obigen RG-Trajektorien für die Klasse der physikalisch rele-
vanten Theorien mit Split-Symmetrie. Als Nebenprodukt erhalten wir eine allgemeine Methode
zur Beurteilung der Güte von beliebigen Näherungslösungen der FRGE.
Anschließend wenden wir die Resultate der obigen bi-metrischen Rechnung auf die Frage
nach der Existenz propagierender Graviton-Moden im Rahmen der asymptotischen Sicherheit
an. Die Eigenschaften des Gravitonpropagators werden wesentlich durch das Vorzeichen der
anomalen Dimension η der Newton-Kopplung bestimmt. Während asymptotisch sichere RG-
Trajektorien im UV ein negatives η besitzen, zeigen unsere Untersuchungen, dass η im semik-
lassischen Bereich positiv wird. Diese Eigenschaft war in vorhergehenden ‘single-metric’ Be-
rechnungen nie gefunden worden. Wir untersuchen, auf welchen Skalen das Auftreten von pro-
pagierenden Gravitationswellen zu erwarten ist und diskutieren mögliche Auswirkungen auf
die Erzeugung primordialer Dichtefluktuationen im frühen Universum.
Weiterhin betrachten wir Raumzeiten mit Rand, die Trunkierungsansätze für die EAA mit
Oberflächentermen erfordert. Wir stellen eine detaillierte Analyse des Gibbons-Hawking-York
Funktionals vor und untersuchen sein RG-Verhalten, welches durch die bi-metrischen Volumen-
Terme induziert wird. Diese Verallgemeinerung des Formalismus ist von zentraler Bedeutung
für die Thermodynamik schwarzer Löcher. Wir beobachten eine thermodynamische Stabil-
isierung schwarzer Löcher, deren Masse sich durch Hawking-Evaporation der Planck-Masse
annähert.

PREFACE
“ Brevity is the soul of wit.This thesis is divided into three main parts, each addressing a conceptually different aspects of
the Asymptotic Safety program to Quantum Gravity. We start from scratch and stepwise intro-
duce the ingredients necessary to construct, at least on a formal level, quantum field theories in
the Functional Renormalization Group (FRG) approach. The purpose of this first, foundational
part goes far beyond a simple introduction. Indeed, it does include standard textbook descrip-
tions of topology, group theory, differential geometry, and measure theory and its relation to
physics, well known since decades. While there are excellent books on each of these topics, our
emphasis lays in providing a consistent and unified language and thereby fixing the notation
and conventions used in this thesis. Our aspiration is a combination of a self-consistent and
a transparent formalization of the employed techniques and thus besides paving the grounds
for the following chapters and derivations the first part of this thesis can be thought of as a
reference resource that can be consulted at any point for clarification. The general structure
of this part considers the dependencies of the various relevant subjects, which starts with the
most fundamental ingredient, spacetime, and culminates in the universal tool of the Functional
Renormalization Group Equation. For the latter we present a coordinate-free version which also
allows to study spacetime constructions with a non-vanishing boundary. Readers interested in
the foundations and structural aspects of the Functional Renormalization Group approach as
well as Asymptotic Safety should focus on this part.
Moving on to the second part, we leave the theoretical description and enter the techni-
cal section of this thesis in which the beta-functions of the so-called bi-metric-bulk – pure-
background-boundarytruncation are derived within the Effective Average Action approach to
Quantum Einstein Gravity. Explicitly, we focus on a bi-metric Einstein-Hilbert ansatz equipped
with a set of boundary invariants, as the Gibbons-Hawking-York term. The computation of the
associated beta-functions is performed in great detail, showing on several stages possible ex-
tensions that can be studied in future work. Therefore, we have kept the notation as general as
possible, in particular to modify the class of boundary conditions in subsequent projects. Again,
we stress the assumptions and the range of validity of the results. The reader who is interested
in a practical calculation and the conceptual difficulties in evaluating the non-linear functional
differential equation, FRGE, is invited to focus on this part.
Finally, the third and concluding part of this thesis sheds light on the implications of the
previously derived beta-functions. We study the Asymptotic Safety conjecture within this trun-
cation and confront this non-perturbative generalization of renormalizability with the funda-
mental requirement of Background Independence within theories of Quantum Gravity. Having
discussed the ultraviolet and infrared properties of the results, we propose a generalized mode
counting device, in certain features reminiscent of Zamolodchikov’s C-function, which would
impose global constraints on the RG evolution. As such, it can also be used to test the re-
liability of truncations and would be very helpful in designing future explorations of the full
infinite dimensional theory space. The final chapters of this part are devoted to the concept of
propagating gravitons and the study of the boundary sector and its implications on black hole
thermodynamics. Readers who are most interested in the results and cosmological applications
are invited to browse the corresponding chapters in this part of the thesis.
At this point it should be clear
that even though this work is struc- Subsection 1.1.5 Section 1.1
tured such that one can start at Spacetime Spacetime
page one and continues to the list
of abbreviations, it is also possi-
ble to sidestep certain aspects at Section 2.3 Section 1.2
first reading and in case of ques-
QFT Symmetries
tions come back or consult the in-
dex. Most of the material is pre- Section 1.3
sented with an emphasize on trans-
Field space
parency and precision, while at the
same time respecting the global Section 2.1
consistency of the thesis. There-
fore, we almost completely aban- Measure theory
doned the idea of appendices in or-
der to retain the line of reasoning Section 3.2 Section 3.1
in the full work and not rip apart
Quantum Gravity General Relativity
conceptually related ideas. For the
reader who is interested in a broad Section 3.3
overview with a brief but consis-
Theories of QG
tent introduction should follow the
road map 1 that provides a journey
through selected topics of this the- Chapter 4 Section 2.2
sis, relevant for understanding in
particular result section. FRGE Effective Action
To further improve the read-
ability of this manuscript, we ap- Figure 1: Road map for the first part of this thesis.
pended an index of important key-
words and keep the list of abbreviations comparatively short. The electronic version is also
equipped with a comprehensive hyperlink ‘net’ that allows to easily transit the thesis. Further-
more, we make use of the following symbols:
R A short remark S A brief summary C Chapter based on self-publication
CONTENTS
I Foundations
1 Physical Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Spacetime 1
1.1.1 Point sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Topological manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.1.5 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.2 Symmetries 14
1.2.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 Group actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.3 Lie groups & Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.2.4 Examples of Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3 Field space 38
1.3.1 Fiber bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.3.2 Field space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
1.4 Operators 66
1.4.1 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1.4.2 Trace of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
1.4.3 Determinant of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
1.5 Functionals 71
1.5.1 Functional derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
1.5.2 Basis invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.1 Measure theory 76
2.1.1 Measurable spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.1.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.1.3 Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.1.4 Relation between measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
2.1.5 Measures for gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1.6 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.2 Effective action 94
2.2.1 Schwinger functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.2.2 Effective action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.2.3 Free theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.3 Quantum field theories 103
2.3.1 Quantum field theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.3.2 Theories, phases, and anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
2.3.3 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
2.3.4 Euclidean quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.1 General Relativity 122
3.2 General aspects of Quantum Gravity 124
3.2.1 A motivation for theories of Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2.2 Field space of Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.2.3 Background Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2.4 Boundaries in Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
3.2.5 Euclidean Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.3 Theories of Quantum Gravity 131
3.3.1 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.3.2 Loop Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.3.3 Causal Dynamical Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3.3.4 Experimental verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4 Functional Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.1 Renormalization group 138
4.1.1 Theory space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.1.2 General aspects of the Renormalization Group . . . . . . . . . . . . . . . . . . . . . . 140
4.2 Infrared-cutoff implementation 142
4.2.1 Eigenmode expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.2 Rescaling relation between Φ̄ and τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.2.3 Smooth cutoff implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
4.3 The Functional Renormalization Group Equation 145
4.3.1 The derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.3.2 Properties of the FRGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4.3.3 The normalization factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
4.4 Renormalizability & Asymptotic Safety 151
4.5 Truncations 153
II Calculations
5 The truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1 Theory space 161
5.1.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1.2 Field space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
5.1.3 Space of functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
5.1.4 Notation and convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2 Preliminary truncation 165
5.2.1 Gravitational functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.2.2 Gauge fixing functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.2.3 Ghost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.3 The final truncation 169
5.3.1 Split-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.3.2 Coefficients and couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.4 Cutoff-operator 173
6 The Hessian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1 Hessian of the cutoff action 175
6.2 Hessian of the gauge fixing functional 176
6.3 Hessian of the ghost functional 177
6.3.1 Hessian contribution of NI(v,w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
6.3.2 Hessian contribution of NI(w,v) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.3.3 Hessian contribution of NII(v,w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
6.3.4 Hessian for the ghost functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
6.4 Hessian of the gravitational functional 183
6.4.1 Hessian for the volume term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.4.2 Hessian for the boundary volume term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.4.3 Hessian for the curvature term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.4.4 Hessian for the boundary curvature term . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.4.5 Hessian for the gravitational functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.4.6 Linear field parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.4.7 Geometric field parametrization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.5 Hessian of the full truncation ansatz 196
6.5.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
6.5.2 The Hessian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.6 First variation formula 198
6.6.1 Stationary point solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.7 Implementation of the gauge fixing condition 200
6.7.1 Hessian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.7.2 First variation formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.8 The level expansion 203
7 Trace evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.1 Inversion of the Hessian operator 208
7.1.1 Inversion of a matrix with non-commuting entries . . . . . . . . . . . . . . . . . . . . . 208
7.2 Decomposition of symmetric 2-tensor fields 213
7.2.1 Hodge theorem for k-forms with ∂M 6= 0/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.2.2 Decomposition theorem for symmetric 2-tensors for ∂M 6= 0/ . . . . . . . . . . . . . 216
7.2.3 The transverse-traceless decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
7.3 Projection techniques for truncations 218
7.3.1 Fixing the ghost sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
7.3.2 Conformal projection technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.3.3 The Hessian in the trace-traceless decomposition . . . . . . . . . . . . . . . . . . . . 229
7.3.4 Projection to maximally symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.4 The ‘Ω deformed’ α = 1 harmonic gauge 235
7.5 Heat kernel 239
7.5.1 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.5.2 Heat kernel expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.5.3 Tensor space traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
8 The beta-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.1 Threshold functions 250
8.1.1 The Mellin transform ofW (∆;ℓ; ū•p k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.1.2 The Mellin transform of ∆Wp(∆;ℓ; ū•k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
8.1.3 Properties and estimates of the threshold functions . . . . . . . . . . . . . . . . . . . 251
8.2 Beta-functions for the coefficients 254
8.2.1 Comparison of coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.2.2 The Einstein-Hilbert invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
8.2.3 The bulk volume invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
8.2.4 The Gibbons-Hawking-York invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.2.5 The boundary volume invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
8.3 Beta-functions for dimensionful couplings 258
8.4 Beta-functions for dimensionless couplings 260
8.4.1 Arbitrary cutoff shape function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
8.4.2 Optimized cutoff shape function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
8.5 The semiclassical approximation 267
8.6 Beta-functions in d = 4 269
8.6.1 The D sector: beta-functions for gD, λD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
8.6.2 The B-sector: beta-functions for gB, λB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.6.3 The level-description: beta-functions for g(p), λ (p) . . . . . . . . . . . . . . . . . . . . . . 271
8.6.4 The boundary sector: beta-functions for g∂ (0), λ ∂ (0) . . . . . . . . . . . . . . . . . . . . 271
8.A Expanded beta-functions in d = 2+ ε 272
8.B Beta-functions in d = 3 274
III Results
9 Investigating Asymptotic Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
9.1 Structure 282
9.1.1 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
9.1.2 The hierarchical structure of the truncated RG equations . . . . . . . . . . . . . . 284
9.1.3 No anti-screening in the semiclassical regime . . . . . . . . . . . . . . . . . . . . . . . 286
9.2 Fixed points 287
9.2.1 The dynamical sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
9.2.2 The level-(0) or B-sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
9.2.3 The boundary level-∂ (0)-sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
9.2.4 Recovering the mechanism of paramagnetic dominance . . . . . . . . . . . . . 291
9.2.5 Impact of the ghost contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
9.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
9.3 Predictivity 295
9.3.1 The classification of trajectories in the linearized region . . . . . . . . . . . . . . . . 295
9.3.2 Critical exponents and scaling fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
9.4 Conclusion 302
9.A Critical exponents of the background-sector 303
10 Background Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10.1 Introduction 306
10.2 Can split-symmetry coexists with Asymptotic Safety? 311
10.2.1 Trajectories of the ‘D’ sub-system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
10.2.2 Solving the non-autonomous ‘B’ system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
10.2.3 The running UV attractor in the B-sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.2.4 Split-symmetry restoration in the physical limit . . . . . . . . . . . . . . . . . . . . . . . . 322
10.2.5 All classes of split-symmetry restoring trajectories . . . . . . . . . . . . . . . . . . . . . 327
10.3 Single-metric vs. bi-metric truncation: a confrontation 331
10.3.1 Collapsed level hierarchies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
10.3.2 Relating single- and bi-metric beta-functions . . . . . . . . . . . . . . . . . . . . . . . . 332
10.3.3 Conditions for the reliability of a single-metric calculation . . . . . . . . . . . . . . 332
10.3.4 The anomalous dimensions: structural differences . . . . . . . . . . . . . . . . . . . . 333
10.3.5 (Un-)Reliable portions of the single-metric theory space . . . . . . . . . . . . . . . 334
10.3.6 The (un-)reliability of the critical exponent calculations . . . . . . . . . . . . . . . . 337
10.3.7 Comparison with the TT-based approach of ref. [172] . . . . . . . . . . . . . . . . . 339
10.4 An application: the running spectral dimension 342
10.5 A brief look at d = 2+ ε and d = 3 344
10.5.1 Near dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
10.5.2 Three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
10.6 Summary and conclusion 349
10.6.1 Summary of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
10.6.2 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
11 A generalized C-theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
11.1 Introduction 353
11.2 The effective average action as a ‘C-function’ 357
11.2.1 Counting field modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.2.2 Running actions and self-consistent backgrounds . . . . . . . . . . . . . . . . . . . . 359
11.2.3 FIDE, FRGE, and WISS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
11.2.4 Pointwise monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
11.2.5 Monotonicity vs. stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
11.2.6 The proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11.2.7 The mode counting property revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.3 Asymptotically safe quantum gravity 368
11.3.1 The single- and bi-metric Einstein-Hilbert truncations . . . . . . . . . . . . . . . . . . 368
11.3.2 Gravitational instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
11.3.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
11.3.4 Crossover trajectories and their mode count . . . . . . . . . . . . . . . . . . . . . . . . 384
11.4 Discussion and outlook 387
11.A The special status of Faddeev-Popov ghosts 391
11.B The trajectory types Ia and IIa 392
12 Propagating modes in Quantum Gravity . . . . . . . . . . . . . . . . . . . . 397
12.1 Introduction 397
12.2 Anomalous dimension in single- and bi-metric truncations 402
12.2.1 Single-metric truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
12.2.2 The (TT-based) bi-metric calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
12.2.3 The (‘Ω-deformed’) bi-metric calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
12.2.4 Significance of the cosmological constant . . . . . . . . . . . . . . . . . . . . . . . . . . 411
12.2.5 From anti-screening to screening and back . . . . . . . . . . . . . . . . . . . . . . . . . 412
12.3 Interpretation and Applications 413
12.3.1 The dark matter interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
12.3.2 Primordial density perturbations from the NGFP regime . . . . . . . . . . . . . . . . 419
12.4 Summary 419
13 The Gibbons-Hawking-York boundary term . . . . . . . . . . . . . . . . . 421
13.1 Introduction 422
13.2 Global properties of the boundary sector 423
13.2.1 The running UV attractor in the ∂ (0)-sector . . . . . . . . . . . . . . . . . . . . . . . . . . 423
13.2.2 The UV behavior of the boundary sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
13.2.3 The IR behavior of the boundary sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
13.2.4 The semiclassical regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
13.3 Variational problems 431
13.3.1 Effective field equations and stationary points . . . . . . . . . . . . . . . . . . . . . . . 432
13.3.2 Variational principle in presence of a boundary . . . . . . . . . . . . . . . . . . . . . . 434
13.4 Counting field modes 437
13.4.1 Split-symmetry and ∂M= 0/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
13.4.2 Euclidean space with non-vanishing boundary . . . . . . . . . . . . . . . . . . . . . . 438
13.4.3 Thermodynamics of the Schwarzschild black hole . . . . . . . . . . . . . . . . . . . . 439
13.5 Conclusion 445
14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
14.1 Summary 447
14.1.1 Asymptotic Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
14.1.2 Background Independence vs. Asymptotic Safety . . . . . . . . . . . . . . . . . . . 448
14.1.3 Towards a C-function in quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
14.1.4 Positivity vs. Asymptotic Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
14.1.5 Asymptotic Safety on manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . 450
14.2 Outlook 450
A Appendix
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Books 455
Articles 457
Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

Foundations
I
1 Physical Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Spacetime
1.2 Symmetries
1.3 Field space
1.4 Operators
1.5 Functionals
2 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.1 Measure theory
2.2 Effective action
2.3 Quantum field theories
3 Quantum Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.1 General Relativity
3.2 General aspects of Quantum Gravity
3.3 Theories of Quantum Gravity
4 Functional Renormalization Group . . . . . . . . . . . . . 137
4.1 Renormalization group
4.2 Infrared-cutoff implementation
4.3 The Functional Renormalization Group Equation
4.4 Renormalizability & Asymptotic Safety
4.5 Truncations
The thesis is divided into three parts, each describing a different aspect of the Functional Renor-
malization Group approach to Quantum Gravity. This first part investigates its theoretical back-
ground in detail, starting from scratch and stepwise develop the necessary machinery to con-
struct this universal tool. While it has an introductory character, presenting some standard
textbook results on the various involved branches of physics and mathematics, we intend giv-
ing a consistent generalization of those concepts for spacetimes augmented with a boundary.
Though more involved, this class of geometries are also well studied in the literature. Rather,
we are aiming at a formulation designed to consistently intertwine boundary contributions with
the standard description of the Renormalization Group. We further present a (formal) measure
theoretic motivation for the Functional Renormalization Group Equation which defines a vector
field on theory space describing the Renormalization Group evolution of theories.
The following chapters present a comprehensive description of the ingredients necessary to
study the Asymptotic Safety conjecture of Quantum Gravity in the presence of boundaries. The
reader who is interested in a condensed introduction may follow the road map in the preface,
fig. 1, and may consult the index if details are required.
This part starts in chapter 1 with the three ingredients theory space and in general a stan-
dard Quantum Field theory is based on: spacetime, symmetry principles, and field space. We
pursue the path of gauge theories and use differential geometry for the definition of the field
content leading to a natural combination of symmetries and interactions. With the introduction
of operators and functionals field space will become more vivid, allowing for transformations
and maps which lead to observables.
In chapter 2 we begin the construction of (probability) theories using functional integration,
which has to be understood on a formal level. After some basics on measure theory, we consider
the general ‘recipe’ to endow field space with a suitable theoretical framework that is consistent
with the symmetries and the underlying spacetime. From this perspective we address the pertur-
bative concept of renormalizability and the general interpretation of perturbation theory along
with its range of validity.
Chapter 3 then provides an example of a classical theory which turns out to be incompatible
with the standard (perturbative) renormalization procedure, General Relativity. We briefly com-
ment on the general aspects and peculiarities of this interaction and then present an incomplete
list of proposals for a quantum theory of gravitation.
The Asymptotic Safety approach to Quantum Gravity is discussed in a separate chapter, 4,
where we first introduce the universal applicable Functional Renormalization Group Equation
and then describe its application to Quantum Gravity. We conclude with a brief summary on
the current status of the AS conjecture.
1.1 Spacetime
Point sets
Topological spaces
Topological manifolds
Smooth manifolds
Spacetime
1.2 Symmetries
Groups
Group actions
Lie groups & Lie algebras
Examples of Groups
1.3 Field space
Fiber bundles
Field space
1.4 Operators
The spectral theorem
Trace of operators
Determinant of operators
1.5 Functionals
Functional derivatives
Basis invariants
1. PHYSICAL CONTENT
This first chapter is devoted to a general introduction of the main mathematical concepts that we
encounter in the FRG approach to quantum field theories. Besides revealing the conventions
and notations used in this thesis, it presents the foundations for the framework on which our
theoretical description is based on.
We will start with a very central object in studying field theories, namely the literally ground
on which the theory is based, spacetime. In our definition, it corresponds to an equivalence class
of smooth manifolds, in general with boundary. The next section presents a summary of group
theory with a special emphasis on Lie groups and -algebras. The special status of symmetries in
today’s understanding of Nature, for instance in the shape of gauge theories, is the reason why
group theoretical aspects take a very important place in the introduction of this thesis. Especially
universality and the phases of theories are quite essential concepts in studying the FRG and
most of the strongest mathematical restrictions are obtained by symmetry observations.
Finally, this chapter concludes by joining smooth manifolds and group theory in the shape
of global smooth functions, special subset of them being later consider as the physical field
content. Hereby, differential geometry is the mathematical setting of this coalescence and its
application to modern QFT has brought fruitful insight into the possible foundations of Nature.
This final series of sections contains an introduction to fiber bundles, field space, operators, and
functionals.
1.1 Spacetime
Spacetime is the fundamental ingredient our theoretical description is (literally) based on. Start-
ing in the category of sets we continue the journey to the mathematical branch of topology and
finally the theory of differential geometry. With each additional structure we attach to the notion
of spacetime we are confronted with an important choice about its very nature, and it is the aim
of this section to make these choices transparent.
While strengthening the ties between spacetime points, we will see how the rich structure
of mathematical description yields redundancies that are related to isomorphisms in the corre-
sponding category. The equivalence relations defined by those isomorphisms will be used to
2 Chapter 1. Physical Content
group indistinguishable structures together and encode the essential information in its associ-
ated equivalence classes.
The final notion of spacetime and the summary of this section can be found in subsection
1.1.5. For more details and references to the present discussion see [1–4].
1.1.1 Point sets
The most elementary object in the construction of spacetime is based on the theory of sets. This
axiomatic system is situated at the very root of mathematics and is related to its severest limi-
tations in the shape of Gödel’s incompleteness theorems [5].1 Even though sets are intuitively
comprehensible, the axiomatics of (standard) modern set theory were established in its full
beauty not before 1921s. This systematic approach – pioneered by Zermelo [6] and sharpened
by Fraenkel [7] – consists of eight axioms, forming the Zermelo–Fraenkel (ZF) set theory, to-
gether with the famous axiom of choice (AC)2. The advent of the Zermelo–Fraenkel set theory
with the axiom of choice (ZFC) led to an axiomatic description of sets that is so far found to be
free of paradoxes, in contrast to naive set theory.3
The precise definition of a set is in fact difficult and we will make use of the following
indirect version:
Definition 1.1.1 — Set. Let SET be a class of collections containing distinct objects and
which fulfill the axiomatics of ZFC. A set with respect to SET is a member of the class
SET.
It is now natural to use the class of functions – interconnecting two sets – as morphisms to
establish the category of sets.
Definition 1.1.2 — Category of sets. The category of sets CSET ≡ (SET,mSET,◦) is given
by the class of sets SET, the class of functions mSET, and the operation ◦, being the associa-
tive composition of functions in the usual sense.
There is an infinitely large variety of sets one can build within the ZFC. The most often encoun-
tered ones are given by sets whose members share a common feature or requirement.
In the construction of spacetime starting with a point set, we assume that the members, i.e.
the underlying building blocks of spacetime, do not carry individual information but are essen-
tially the same. In other words, labeling the elements of a set is superfluous in our description
of spacetime. In mathematical terms this translates to invariance of spacetime under bijection
(the isomorphisms of CSET):
Definition 1.1.3 — Bijection. Let A and B be sets. A function f : A→ B is a bijection :⇔
∀b ∈ B ∃ a ∈ A : b= f (a) (surjective) (1.1a)
∀a,c ∈ A : f (a) = f (c)⇔ a= c (injective) (1.1b)
For our purpose two sets give rise to the same spacetime whenever there is a bijection connecting
both. Grouping together bijective sets actually defines an equivalence relation ∼SET within the
ZFC. The category of sets can thus be partitions into equivalence classes of ∼SET which can
be classified by the so called cardinality.
1In short the theorems state that a set of axiomatics cannot produce all true statements about itself without
contradiction.
2In some axiomatic systems of set theory the axiom of choice (AC) is replaced by an equivalent axiom.
3For instance, Russell’s paradox exploits the inconsistent formalization of older set theory. Russell constructed a
set S ≡ {X |X ∈/ X} containing all sets that are not elements of themselves (which was allowed by older set theory).
Then, however, S ∈ S⇔ S ∈/ S and S has to be the empty set for which obviously counterexamples exist.
1.1 Spacetime 3
Definition 1.1.4 — Cardinality. Given the category of sets CSET, we can partition the sets
with the relation ∼SET. Such equivalence classes are classified by the cardinality # of the
contained sets. (In the finite case this corresponds to the number of elements.)
Thus, on the level of point sets distinct spacetime constructions can only differ by their ‘size’
(cardinality). Nevertheless, this already distinguishes in particular the sets of countable and
uncountable infinite sets.
R As weak as the selection of a specific point set and thereby a certain cardinality may
seem, it is already at this level where the distinction between finite and infinitely extended
lattice-based, as well as continuous spacetime construction arises.
In the process of attaching further mathematical structure to spacetime, we have to make sure
that the entire prescription retains the invariance under bijections. Lifting this symmetry to
other categories corresponds to the invariance under their respective isomorphisms. Hence,
topological, or smooth manifolds which are homeomorphic or diffeomorphic, respectively, will
give rise to the same spacetime construction.
S In the category of sets, spacetime is classified by its cardinality; it is an element of
SET∼ ..= {#(S) |S is in SET} ≡ {[S]∼ |S is in SET}SET
and the morphisms are functions with composition.
1.1.2 Topological spaces
Endowing spacetime with a topology opens up a rich structure of new classification schemes
and introduces a first idea of locality. It seems natural to equip spacetime with a formalism
that relates its elements in a suitable way. The first step is grouping together distinct points in
so called open sets that form a topology on a point set. Abstractly speaking, a topology is a
collection of subsets satisfying the following conditions:
Definition 1.1.5 — Topological space. Let X be a set. τ ⊆ 2XX ≡ {S ⊆ X} where 2X is
the power set of X . Then τX ⊆ 2X defines a topology on X :⇔ 0/ ∈ τX and X ∈ τX and
∀S⊆ τX : (∪U∈SU) ∈ τX (closed under union)
∀S⊆ τX , #(S)< ∞ : (∩U∈SU) ∈ τX (closed under finite intersection)
The ordered pair (X ,τX) is called a topological space, and elements of τX are open sets.
This construction gives rise to the notion of neighborhoods and naturally provides concepts of
compactness and connectedness, as well as a variety of derived properties. Once we have open
sets at our disposal a property exists locally if for each point x ∈ X there is an open set,U ∈ τX
with x ∈U , for which this property is satisfied.
Connected spaces
Connectedness is one such topological characteristic with which we can decide whether our
concept of spacetime consists of separated (disconnected) parts or is entirely ‘glued together’.
Definition 1.1.6 — (Dis-)Connected space. A topological space (X ,τX) is disconnected
if there exist two disjoint (non-empty) open sets that cover X , i.e.
∃U,V ∈ τX/{X ,0/} withU ∩V = 0/ and X =U ∪V
4 Chapter 1. Physical Content
A connected space is a topological space that is not disconnected.
This reflects the intuitive picture of connectedness translated into the language of topology. If
we can separate a part of spacetime without affecting its remaining portion we would have at
least two distinct non-communicating ‘worlds’. The sets V and U in definition 1.1.6 are open
and closed at the same time, a consequence of the very general idea of closed sets in topology:
C ⊆ X is a closed set if there exists U ∈ τX such that C = X/U . Only when considering a
connected topological space (X ,τX) we retain the separation of both concepts, by assuring that
only X and 0/ can be both open and closed sets in τX . In what follows we will usually assume
spacetime to be a connected topological space.
Compact spaces
Another very important concept in the category of topology is compactness, a combination
of generalized aspects of finiteness and closeness. It is therefore not astonishing that many
theorems rely on this property in one or the other way. Explicitly, we have:
Definition 1.1.7 — Compact space. A topological space (X ,τX) is a compact space :⇔
∀U ⊆ τX , ∪A∈UA= X :
=⇒ ∃U ′ ⊆U, ∪A′ ′A′∈U = X with #(U ′)< ∞ (finite open subcover of (X ,τX))
That is, every open cover of (X ,τX) has a finite open subcover.
Usually, whenever infiniteness or a lack of completeness enters the formalism one has to be
very careful to transfer well known features of finite sets or R to the space at hand. In topology
the transition from local to global aspects usually invokes patching together an infinite number
of open neighborhoods which may spoil local valid properties in the global picture. However,
for compact spaces this transition is in a certain sense ‘well-behaved’ for the prescription can
always be reduced to a finite number of open sets that still cover the entire space. This finiteness
may allow to lift local properties to global statements and this is what many theorems are based
on.
R Once the underlying point set X has finite cardinality even the discrete topology τ
X
X = 2
will give rise to a compact space. For uncountable sets compactness introduces a notion
of finiteness into the infiniteness.
In functional analysis compactness is the crucial ingredient for generalizing theorems on topo-
logical spaces, for instance the existence and uniqueness of functions solving differential equa-
tions or convergence issues. It is for this reason that in the consideration of, for example, field
content the formalism will be sensitive to the status of compactness of the underlying spacetime.
Continuous functions and homeomorphisms
Functions between different sets establish relations and classification schemes in the category
of sets. We can use special maps, bijections, to reveal an ordering of sets by their cardinal-
ity. In topological spaces, where we have an additional structure consisting of a collection of
subsets, the open sets, we have to look for functions that also take care of the topology. To
preserve the topological information as well we have to restrict to continuous functions in the
first place:
Definition 1.1.8 — Continuous function. Let A≡ (XA,τA) and B≡ (XB,τB) be topological
1.1 Spacetime 5
spaces. A function f : XA → XB is continuous with respect to A and B :⇔
∀U ∈ τB : f−1(U)≡ {x ∈ XA |∃y ∈U : f (x) = y} ∈ τA
That is, the pre-image of each open set in B is open in A.
This inverted looking definition corresponds to the one in metric spaces where the notion of
locality is supplemented with a distance. The true power of continuous functions unfolds when
considering the net they spread over the set of topological spaces. It turns out that the composi-
tion of continuous functions is again a continuous function, and so is the identity- and constant
map, id(x) = x and c(x) = c for all x ∈ XA, respectively. This allows to build long chains of
continuous functions over the entire class of topological spaces, thereby relating their notions
of locality. Some of the main theorems of topologies are concerned with the preservation of
topological properties, as e.g. compactness, under continuous maps. It is quite astonishing
that a local concept as continuity actually embeds global topological information into its target
space and thereby retains compactness and connectedness properties, for instance.
If two topological spaces can be mutually related by continuous functions they locally agree.
Extending this concept to the global picture allows us to find topologically identical spaces. As
we have taken care of topological information by continuity we have to assure that the set’s
specific information is as well retained, as is done by bijections. The combination of locality-
preserving and bijective maps leads to the isomorphisms of the category of topological spaces:
homeomorphisms.
Definition 1.1.9 — Homeomorphism. Let A ≡ (XA,τA) and B ≡ (XB,τB) be topological
spaces. A function f : XA → XB is a homeomorphism :⇔
( f is a bijection) ∧ ( f and its inverse function f−1 are continuous)
Then, A and B are called homeomorphic.
Since our entire formalism is based on the category of sets, all isomorphisms we encounter are
subclasses of bijections, preserving additional structures. Each set in a specific cardinality give
rise to a variety of topological spaces. For the class of finite sets this is merely a combinatorial
task. For the equivalence classes of countable and uncountable sets this will result in an infinite
number of topologies we generate from those sets. Homeomorphisms provide the means to
clear up the matter and distinguish the different classes of topologies one can obtain for each
cardinality.
Two topological spaces of a particular cardinality are the same if they can be transformed
homeomorphic into each other and thereby keep their local properties intact everywhere. They
share, by definition, the same topological invariants, i.e. quantities that are preserved under
homeomorphisms. The set invariants were similarly defined to be the quantities preserved un-
der the isomorphisms of the category of sets, i.e. bijections. The simplicity of sets gives rise to a
single invariant: the cardinality. It becomes apparent from the vast number of topological invari-
ants what kind of rich structure is encoded in the category of topology. Besides compactness-
and connectedness-properties, several separation-, countability-, and metrizability-conditions
were found to be invariant under homeomorphisms. They are usually the starting point to fig-
ure out if two topological spaces are not homeomorphic. Notice that even though we may
find topological spaces that agree in all topological invariants we know they do not have to be
homeomorphic. Only in rare cases we can characterize special topological spaces using certain
invariants.
S In the category of topological spacesC
.
TOP .= {(X ,τX) |X is in SET∧ τX is topology on X}
with morphisms mTOP = { f ∈mSET | f is continuous }, spacetime is classified by its car-
6 Chapter 1. Physical Content
dinality and a class of homeomorphic topologies; it is an element of
TOP∼ ..= {[(X ,τX)]∼ |(X ,τX )∼TOP (Y,τY ) :⇔ ∃ f : X → Y homeomorphism }TOP
1.1.3 Topological manifolds (with boundary)
A topology attaches a diversity of different structures to the simple equivalence classes of sets
– the cardinalities. Thereby, a wide range of new possibilities to classify spacetime arises from
the very definition of open sets. In fact, the number of those new constructions is related to the
cardinality of the power set and will be infinite for infinite underlying point sets. It is us who
have to decide among those topological spaces by gathering an amount of topological properties
we want to have satisfied by our model of the Universe. Usually one is keen to stay close to
the very familiar Euclidean space and only depart from this guiding principle when it seems
unavoidable. Following this path, we focus on topological spaces that reflect our (local) intuitive
understanding of the world we are living in, i.e. resemble an Euclidean type structure. Framed
in the language of topology, this entails a consideration of so called topological manifolds. Even
though the majority of topologies is dropped by those restrictions that keep certain Euclidean
features (at least locally) intact, we are still left with a wealth of globally and/or locally distinct
spaces.
Before finally giving the definition of topological manifolds (with boundary) we will moti-
vate its additional topological attributes and why they seem to naturally reflect spacetime prop-
erties. Thereby, we keep the notation more general in order to include boundaries (in the sense
of manifolds) into our description, which will be relevant for the later investigations.
Locally Euclidean space (with boundary)
When perceiving the world with our senses we deduce a continuity of space and time, which
seems to be perfect on our scales of observation. Whether or not this description of spacetime
holds on a deeper level is one of several questions physicists try to answer by studying theories
of quantum gravity. In this work we assume our intuitive (local) picture of the world to hold
down to any scales, i.e. we adopt the perspective that spacetime is locally homeomorphic to
Euclidean space.
In the construction of those Euclidean spaces two fundamental branches of mathematics
are combined, i.e. topology meets linear algebra. Whereas in topology we extend point sets
by properties of its subsets, in algebra operations and relations take this role. At this stage
fields, vector spaces and in general algebraic structures appear that equip the simple point sets
with additional concepts, different to the ones we have encountered so far. It is the purpose of
manifolds to enrich topological spaces with those algebraic properties and unfold a variety of
possibilities mimicking the Euclidean space.
Let R = (R, ·,+,≥, ‖ · ‖R) denote the totally ordered, complete field of the real numbers
with a norm induced by its multiplication (as scalar product). This is a vector space on its own,
endowed with further operations that are familiar to us from every day life. This plenitude
of additional structures allows us to introduce a chronology necessary for the description of
any evolution process, in particular for the direction of time. It further provides a measure of
distance and thereby further orders the unrelated elements of the underlying point set R.
Once we framed this natural appealing space R into mathematical language, we can derive
further algebraic structures from it. The d-dimensional vector spaces Rd = (R,+Rd , R, ◦R×Rd )
are important examples of this procedure where a first notion of dimensionality appears on the
cost of some properties of R. It is most instructive to compare the field R with the vector spaces
Rd.
A. First of all notice that the underlying point set, more precisely its cardinality, is the same
for R and all Rd with d ≥ 1, namely the uncountable infinite sets. Thus, in the category
1.1 Spacetime 7
of sets R and Rd are isomorph and the discriminability is only due to their algebraic
(operational) properties.
B. For d = 1 the vector space coincides with the field, i.e. R1 ≡ R. With d = 0 one usually
denotes a singleton set A, i.e. a set consisting of a single element: #(A) = 1. Its cardinality
obviously does not coincide with the one of R.
C. In the general case d > 1 we have lost the total order property of the field. What still
remains is an ordering induced by the metric (distance function) which however is one-
dimensional only.4 We have the following definition:
Definition 1.1.10 — Metric space. Let X be a set and dX : X ×X → R a function.
The ordered pair (X ,d) is a metric space :⇔ d is a metric on X , i.e. ∀x, y, z ∈ Y :
dX (x,y) = dX (y,x) (symmetric)
dX (x,y) = 0 ⇐⇒ x= y (equal points)
dX (x,z) ≤ dX (x,y)+dX (y,z) (triangle inequaility)
The non-negative function dX thus ‘embeds’ the real numbers along with its rich structure
into a subset of X . Any measurement relies on this kind of embedding and is therefore
described within a metric space.
D. A vector space intrinsically carries an algebraic dimension, given by the maximal number
of linearly independent elements (vectors). Given a vector spaceV ≡ (XV , ◦V , eV ,K,◦K·V)
over the fieldK, a subset B= {b j | j ∈ J ⊆N} ⊆ XV is a linearly independent basis :⇔
◦V
ak ∈K, ∀K ⊆ J with #(K)< ∞ : ∑ ak ◦K·V vk = eV =⇒ ak = eK, ∀k ∈ K
k∈K
Each such (basis) vector generates a direction isomorphic to R and thereby gives rise to
the above mentioned one-dimensional ordering. This algebraic dimension defines also a
topological invariant restricted to the category of topological manifolds (with boundary).
Based on these special algebraic objects we continue our journey in the field of topology. In
order to define topological spaces that locally reflect the vector- and metric space properties
of Rd we have to equip Rd with a notion of open sets that respects its algebraic properties.
Fortunately, there is a standard way to proceed here, once we have a metric space at our disposal.
Though not every topological space can be promoted to a metric space the other way around is
straightforward once we have defined a topological basis:
Definition 1.1.11 — Topological basis. Let N ≡ (XN ,τN) be a topological space. A collec-
tion of sets B⊆ 2X is a topological basis :⇔
B⊆ τN (all its elements are open)
∀U ∈ τ : ∃B′N ⊆ B such that U = ∪V∈B′V (open sets are unions of V ∈ B)
We say τN is generated by B, i.e. τN = genB.
The two main benefits a topological basis provides are concerned with the construction of
topologies and its reduction of information.
Therefore, let us choose a collection of subsets B∈ 2X that completely covers X . If it is
chosen such that the condition
∀B1, B2 ∈ B, ∀x ∈ B1∩B2 : ∃B3 ∈ Bwith x ∈ B3
4This is a common practice we will encounter whenever we need an order principle in our formalism. To
chronologically order time we have to relate this concept to R. For general orders within a set we have to find an
isomorphism to R in the corresponding category.
8 Chapter 1. Physical Content
holds true, then the collection of all unions of elements of B uniquely defines a topology on
X . The second advantage refers to its generating nature. A topological basis contains the
condensed information of all open sets, similar to a linearly independent maximal set of basis
vectors allows to reconstruct the full vector space. It therefore usually suffices to test certain
properties for the basis only and deduce global statements of the full space. One has to be
careful though when the cardinality of the basis is infinite.
R In vector spaces there is a canonical upper limit of basis elements given by the maximal
number of linearly independent vectors, which defines the dimension of vector spaces.
In contrast, for topological spaces it is the lower bound on the number of basis elements
that is of interest. The requirement that a basis should cover the full space X leads to a
least cardinality of a basis: the weight of a topological space – invariant under homeomor-
phisms.
A subset of real space R is open if it can be written as a union of open intervals, (a,b) with
a ≤ b ∈ R. A metric relates the structure of R to a subset of its elements and thereby allows to
inherit its topology. For the real vector spaces Rd with the standard Euclidean metric dEd this
leads to the following metric induced topological space:
Definition 1.1.12 — Euclidean space. The ordered pair dE ≡ (Rd , τEd , dEd ) is the Eu-
clidean space of dimension d :⇔
◮ dEd is the standard metric on R
d, i.e. given a vector space basis {v j | j ∈ [1,d]}, for all
q, p ∈ Rd we have with q= ∑k∈[1,d] qk ◦R·Rd vk:
√√√
dEd (q, p) .
.=√ ∑ (qk− 2pk)
k∈[1,d]
◮ τEd is the standard topology induced by dEd , i.e. open balls of varying radii
τ .d .= gen{Br(p) ..E = {q ∈ Rd |dEd (p,q)< r}|r ∈ R, p ∈ Rd}
So far we have modeled the world of our (local) perception with the mathematical concept of a
Euclidean space that combines algebraic structures, as the vector- and metric space properties,
with a correlative topology. Our objective is the local embedding of the Euclidean richness to
more general topological spaces.
To generalize these ideas to include boundaries for future investigation, we have to first
introduce a mechanism to construct new topological spaces from existing ones. For our purpose,
the subspace topology suffices, for more detailed account see [1].
Definition 1.1.13 — Subspace topology. Let N ≡ (XN ,τN) be a topological space. A
collection τ | ⊆ 2XNN A is the subspace topology for a subset A⊆ XN :⇔
τ .N | .A = {U ∩A |U ∈ τN}
The ordered pair (A, τN |A) is the induced topological subspace of N on A.
If A ∈ τN is an open set, the very definition of a topology guarantees that also the subspace
topology is a subset of the deduced one, i.e. τN |A ⊆ τN . In general however, their intersection
may consist of only the empty set and thus τN |A * τN . In order to introduce a boundary into
the description we have to confine one direction of the vector space Rd to the positive (or
equivalently negative) real numbers only. Breaking down the topology of the corresponding
Euclidean space to the remaining Rd−1×R+0 using the subspace topology yields
1.1 Spacetime 9
Definition 1.1.14 — Euclidean space with boundary. We say the ordered pair dE∂ ≡
(Rd−1×R+0 , τ d , d d ) is the d-dimensional Euclidean space with boundary :⇔E E∂ ∂
◮ R+0 is the Euclidean half-space, i.e. its elements are given by {p ∈R | p≥ 0},
◮ dE is the standard metric on Rd restricted to the subset Rd−1×R+0 ,
◮ τ d is the subspace topology induced by the standard topology τ dEd on E , i.e.E
∂
τ ..= {U ∩ (Rd−1d ×R+0 ) |U ∈ τ d}E E∂ ∂
The boundary of this space is given by ∂ dE ..∂ =
d
E∂ ∩ (Rd−1×{0}).
For a long time it seemed to be appropriate extending our local intuitive picture of a Eu-
clidean space to a global construction and assuming that spacetime is simply given by 4E . At
least since the advent of General Relativity this idea began to unravel and the distinction be-
tween locality and the global patch turned out to be of uttermost importance in understanding
the fundamentals of space and time. This presumable insight brought physicists to a more
tedious but general program, a route we are currently following. Instead of making global
statements about spacetime, we are modest and only make assumptions about its local struc-
ture:
Definition 1.1.15 — Locally Euclidean space (with boundary). Let dH be the Euclidean
space dE (with boundary dE∂ ). A topological space N ≡ (XN ,τN) is locally Euclidean (with
boundary) :⇔
∃d ∈ N, ∀x ∈ XN : ∃Ux ∈ {U ∈ τN |x ∈U}
∃V ⊂ dH , ∃ fU ∈ {g :Ux →V |g is homeomorphism }x
Then d is called the topological manifold dimension of N and (Ux, fU ) is a chart of N.x
We can thus find open neighborhoods for every point of XN which are topological equivalent to
the same d-dimensional Euclidean space (with boundary). Locally, we regain the nice Euclidean
structure along with its vector- and metric space properties, however sewing together the local
patches usually results in global networks that topologically differ severely from Euclidean
space.
The manifold boundary of topological spaces N ≡ (XN ,τN) that are locally homeomorphic
to dE∂ consists of points in the inverse image of Euclidean boundary, i.e.
∂N ..= {x ∈ X |∃(Ux, fU ) chart of x, ∃q ∈ ∂ dE∂ : fU (x) = q} (1.2)x x
Conversely, the union of all pre-images of dE defines the manifold interior Int(N). It seems natu-
ral to conclude that the manifold boundary and manifold interior do not overlap, i.e. are disjoint
sets whose union compose the entire space. The subtlety in proving this statement lies in the
local nature of charts and possible non-empty intersections of them: For a single chart (U, fU )
we have a homeomorphism that excludes manifold boundary points being simultaneously man-
ifold interior points w.r.t. (U, fU ). However, one has to make sure that for another overlapping
chart this separation maintains (the actual proof involves some deep aspects of topology). This
confirms the intuitive picture that manifold boundary and -interior are disjoint sets and in fact
are topological manifolds without boundary of dimension (d−1) and d, respectively [2].
R Notice that even though every locally Euclidean space is also a locally Euclidean space
with boundary, the reverse only holds if the manifold boundary is empty.
10 Chapter 1. Physical Content
At this point we have made our topological space locally Euclidean. Still the global result
may reveal certain unexpected properties that does not fit our concept of spacetime. We will
therefore add two more restrictions that excludes pathological topological spaces.
Hausdorff spaces
The infiniteness of the underlying point set and the ignorance of topology about measures or
distances yield to a peculiar feature of Euclidean spaces for any dimension d ≥ 1. In Euclidean
spaces locality is equal to global picture, meaning that any open set of dE is homeomorphic
to dE itself. For example the unit ball around the origin B (0) ..= {q ∈ Rd1 |dEd (0,q) < 1} is
homeomorphic to dE , and so they are topological equivalent. Open sets in the Euclidean space
thus exhibit a self-similarity property:
∀U ∈V ∈ τ : U =∼V ∼ dV Ed V = E
One direct consequence is that there is no smallest open set, in particular single points are not
open, {p} ∈/ τEd .
In order to (locally) obtain a duplicate of Euclidean space we have to implement this self-
similarity property into the global picture of spacetime. Hence, we require our topological space
to fulfill the Hausdorff property:
Definition 1.1.16 — Hausdorff space. Let N ≡ (XN ,τN) be a topological space. N is a
Hausdorff space (has the Hausdorff property) :⇔
∀x 6= y ∈ XN ∃Ux,Uy ∈ τN with x ∈Ux, y ∈Uy : Ux∩Uy = 0/
The Hausdorff property is a topological property that is preserved under homeomorphisms. Be-
sides the closeness of single points this guarantees the uniqueness of limiting points and selects
only those topological spaces that have ‘enough’ open sets.
Second-countable spaces
With the Hausdorff property we have a lower bound on the number of open sets that define
a topology. However, an upper bound is similarly important to avoid infinity issues. This be-
comes particularly problematic when considering topological bases. Such a basis generates the
topology and contains the essential information of its global features. If the basis is uncount-
able infinite one has to be very careful in the transition towards the full topology. In Euclidean
spaces a countable infinite number of basis elements suffices to represent the collection of all
open sets. This property comes under the name of second-countability and is defined as fol-
lows:
Definition 1.1.17 — Second-countable space. Let N ≡ (XN ,τN) be a topological space.
N is a second-countable space :⇔
∃B⊆ τN , #(B)≤ #(N), ∀U ∈ τN : ∃B′ ⊆ BwithU = ∪V∈B′V
With the restriction to second-countable spaces we ensure the generating basis to be ‘well-
behaved’ by the existence of an upper bound (though countable infinite in general).
Topological manifolds (with boundary)
The notion of a topological manifold combines the local concept of being Euclidean with global
restrictions. All local neighborhoods share the same dimensionality d, thus the local require-
ment actually contains already some global information. We further mimic real space by the
global properties of a Hausdorff- and second-countable space. Together they combine to a topo-
logical manifold which forms a generalization of Euclidean spaces to account for the modern
1.1 Spacetime 11
understanding of spacetime.5
Definition 1.1.18 — Topological manifold (with boundary). Let N = (XN ,τN) be a topo-
logical space. Let dH be the Euclidean space dE (with boundary dE∂ ).
N is a topological manifold (with boundary) of dimension d :⇔
∀x ∈ X ◦ ◦N ,y ∈ XN\{x} ∃Ux,Uy ⊆ XN , x ∈Ux , y ∈Uy : Ux∩Uy = 0/ (1.3a)
∃d ∈ N∀x ∈ XN : ∃Ux ∈ {U ∈ τN |x ∈U}
∃V ⊂ dH ∃ fU ∈ {g :Ux →V |g is homeomorphism } (1.3b)x
∃B⊆ τN , #(B) = #(N), ∀U ∈ τN : ∃B′ ⊆ BwithU = ∪V∈B′V (1.3c)
That is, a topological space with Hausdorff property (1.3a), which is locally Euclidean (with
boundary) (1.3b) and second-countable (1.3c).
It is apparent that the dimension of the topological manifold (with boundary) is inherited from
the local vector space. Notice that the cardinality of the underlying point set is a severe restric-
tion to the possible dimensions that can occur. Whenever the cardinality of the set is finite or
countable infinite, the dimension of the vector space and thus the one of the deduced topological
manifolds are zero. On the other hand, all real vector spaces of any non-vanishing dimension
are obtained by the same equivalence class of uncountable infinite sets. The main concerns in
studying Euclidean spaces are therefore sets of uncountably infinite cardinality, its zero dimen-
sional limit includes all the remaining finite and countable infinite cases.
The dimensionality of a manifold, as well as the Hausdorff- and second countable property
are topological invariants and are thus preserved under homeomorphisms.6 Hence, the class of
topological manifolds do not stretch over different topological equivalence classes but rather
define a subset in each element of TOP∼ separately.
We have further introduced the concept of a manifolds boundary and its interior and thereby
laid the grounds for later cosmological spacetime models.
S In the category of topological manifolds (with boundary), spacetime is classified by its
cardinality and a class of homeomorphic topologies that are Hausdorff, locally Euclidean
(with boundary) of dimension d and second-countable; it is an element of
d
TOPM .∼ .= {[(XN ,τN)]∼ |τN topological manifold of dim. d }TOP
Md
TOP∼∂ ..= {[(XN ,τN)]∼ |τN topological manifold with boundary of dim. d }TOP
1.1.4 Smooth manifolds (with boundary)
Our mathematical formulation of our world, N ≡ (XN ,τN), is made of local pieces with Eu-
clidean structure. Every such local patch is described by a chart (U, fU), an open set U ∈ τN
with a homeomorphism to dE∂ that locally promotes the multifacetedness of real vector- and
metric-spaces to a topological space. The diversity of possible topologies that emerges from
this generalization of Euclidean spaces can be attributed to their transition behavior between
different patches, i.e. the global variety of topological manifolds arises due to differently glu-
ing together local pieces. While the Hausdorff- and second-countable properties retain certain
local features even on the global level, especially non-topological aspects of Euclidean spaces
are usually lost. This has in particular consequences for studying the theory of functions in the
5Here we denote the topological interior of a set by A◦ ..= ∪(U∈τN)⊆AU .
6The theorem stating that different dimensional topological manifolds cannot be homeomorphic is known as
invariance of domain. In particular nE is not homeomorphic to any mE whenever m=6 n.
12 Chapter 1. Physical Content
important mathematical branch denoted analysis. Most of the functional calculus that was intro-
duced to describe (smooth) functions on Euclidean space, needs modifications on topological
manifolds, at least on a global scale.
Homeomorphisms are insensitive to differential aspects and we have to restrict the possible
transitions to spread a consistent calculus over an entire topological space. Therefore, let us
choose a collection of charts that cover a topological manifold and form what is known in
mathematics as an atlas:7
Definition 1.1.19 — Atlas. Let N ≡ (XN ,τN) be a d-dimensional topological manifold.
A collection of ordered pairs A≡ {(U, fU )} is a P-atlas on X :⇔
◮ Every element of A is a chart, i.e.
∀cU = (U, fU ) ∈ A : (U ∈ τX)∧ (∃V ∈ τ d : f :U →V is homeomorphism )E
∂
◮ The collection of open sets in A is an open cover of X , i.e. ∪cU=(U, f )∈AU = X .U
◮ For any two charts cu = (U, fU ), cW = (W, fW ) with intersecting sets,U ∩W 6= 0/ :
∀q ∈U ∩V with x= fW (q) : ( fU ◦ f−1W has P-property at x)
The P-property is a placeholder that specifies the kind of changes which are admissible in the
transition from one patch to another. For instance, if we consider a smooth (analytic) atlas, this
requires the transition functions to be smooth (analytic), etc. This defines what is known as a
P-structure on our notion of spacetime. Two P-atlases A1 and A2 on a topological space are
compatible, i.e. they give rise to the same P-structure, if any chart in A2 fulfills the P-property
with any intersecting chart8 ofA1. Since this similarity of atlases defines an equivalence relation
we can partition the totality of possible P-atlases in equivalence classes, known as maximal P-
atlases. The underlying topological manifold determines what kind of different P-structures we
can attach to our notion of spacetime, in particular whether there are any p-atlases at all. While
for the topological manifolds of dimension less than or equal to 3 there is a unique differential
structure [8, 9], higher dimensional classifications are still unresolved. Nevertheless, the non-
existence of smooth structures on certain topological manifolds has been shown, in particular
for some higher dimensional compact manifolds (for further details consult [2]).
For our purpose it is of uttermost importance to preserve some differential aspects of Eu-
clidean spaces. This requirement translates in a smooth structure we want to endow topological
manifolds with, i.e. we require all transition functions f −1 d dU ◦ fW : R → R to be smooth in the
C∞ sense. With a selection of a maximal C∞-atlas we designate which functions on and to a
topological manifold are infinitely differentiable. Distinct C∞-structures will alter the notion of
differentiability on the manifold.
Spacetimes equipped with a C∞-structure are described by smooth9 manifolds, i.e. [2]
Definition 1.1.20 — Smoothmanifold (with boundary). LetN=(XN ,τN) be a topological
manifold (with boundary). The ordered pair M = [(N, A)]∼ is a smooth manifold :⇔MC∞
A is aC∞-atlas on N.
We have ultimately reached the final mathematical formulation of our spacetime model
by the class of smooth manifolds. Thereby we invoke a global differential calculus on the
underlying topological space. In order to treat only distinguishable descriptions of spacetime we
have absorbed equivalent smooth structures in the concept of a maximalC∞-atlas. The confining
criterion for this partition is the existence of diffeomorphisms, i.e. smooth homeomorphisms
that retain theC∞-structure by requiring its inverse to be smooth. This further refines the notion
7For each connected component an atlas contains either only one chart or if it has more than one, these charts
overlap. Otherwise we would have a contradiction to the connectedness criterion.
8With intersecting charts we mean charts whose sets intersect.
9Smooth in what follows always refers toC∞.
1.1 Spacetime 13
of bijections for smooth manifolds, resulting in the concept of isomorphisms in the category of
C∞-structures: the diffeomorphisms which intuitively speaking reflects a deformation without
cracks.
The standard notion of differentiability we want to promote on the level of topologies refers
to Euclidean space without boundary. A function is called differentiable if a certain limit exists
in an open neighborhood. However, when a topological boundary enters the formalism there
appear additional open sets for which this differentiability condition cannot be applied, those
containing boundary points. The reason lies in the self-similarity of open sets in dE and the
prescription of deducing subspace topologies. In order to lift the standard differentiable struc-
ture to dE∂ and thereby to topological manifolds with boundary, we simply denote a function f
to be differentiable whenever there is an extended C∞-function on dE that agrees with f on the
restricted set. One implication that results from this definition is the invariance of the manifolds
boundary under diffeomorphisms.[2]
Attaching an additional differential structure to a topological manifold results in our final
definition of spacetime:
S In the category of smooth manifolds (with boundary), spacetime is classified by a topo-
logical manifold with a maximalC∞-atlas; it is an element of
d ∞ . { | ∞ ∈ M
d
M C∼ .∂ = [(X ,τX ,A)]∼ ∞ A is C -atlas on (X ,τ ) TOP
∂ }
MC X ∼
The isomorphisms of this category are the diffeomorphism defining indistinguishable
spacetimes. Here (X ,τX ,AX )∼MC∞ (Y,τY ,AY ) :⇔
(X ,τX )∼TOP (Y,τY ) ∧ AX compatible with AY .
1.1.5 Spacetime
In this section we have undertaken a journey to explore our model of spacetime in more detail.
On the way we have encountered three main branches of mathematics that all add their parts to
constitute our notion of the arena our entire formalism is based on.
Initially, we started with a point set whose entire information was contained in its cardinal-
ity, since we assumed the labeling of spacetime’s building blocks to be artificial. The invariance
under bijection was one of the main results that propagates through all categories we passed. In
topology we enriched the simple sets by exploiting its power set and introduced a first notion of
locality. Topological manifolds then intertwine linear algebra and topology by locally translat-
ing the ‘manifold’ properties of Euclidean space (with boundary) to more general topological
constructions. In a last step we attached a global calculus by generalizing the standard differ-
entiable structure to those topological manifolds. While homeomorphisms are used to select
indistinguishable spacetimes in the category of topological manifolds, the additional require-
ment of preserving the smooth structure results in the equivalence under diffeomorphisms. The
final model of spacetime – translated in the language of mathematics – is then:
Definition 1.1.21 — Spacetime. A spacetime is an equivalence class of smooth manifolds
of dimension d defined on a topological space (X ,τX), i.e.
[ ]
M≡ (X , τ ⊆ 2XX , A= {(U ∈ τX , fU)} ∼ (1.4)MC∞
with the following ingredients:
◮ A set X with cardinality #(X)
◮ A topology τX over X with the properties:
A. (X ,τX) is locally a d-dimensional Euclidean space (with boundary),
14 Chapter 1. Physical Content
B. (X ,τX) is a Hausdorff space,
C. (X ,τX) is second countable.
◮ A maximal C∞-atlas [A] on (X ,τX).
The diffeomorphism invariance that appears in our later formalization of quantum gravity pre-
cisely reflects this definition where spacetime is actually an equivalence class of indistinguish-
able smooth manifolds, rather than a single representative. It is important to be aware that the
arising symmetry of the formalism is part of the arena itself, rather than being a feature of field
space.
Notice that very often spacetime is defined as a Riemannian- or pseudo Riemannian ge-
ometry, which is always possible as long as the underlying smooth manifold is paracompact.
However, from the perspective presented here, introducing a Riemannian metric defines a con-
nection on the tangent space ofM and as such is a constituent of field space.
Furthermore, one can extend the present definition based on real Euclidean models to com-
plex spaces which then results in similar structures that nevertheless reveal some uncommon
features not present in the real case. The combination of quantum field theory and Lorentzian
metric field content is best studied using complex time coordinates. While the real part still
reflects the usual concept of time, its imaginary part can be related to temperature and thus a
connection between quantum field theory and statistical physics can be established. Isotropic
transformations in this time-temperature subspace are described by Wick-rotations. Whether or
not a complexification of spacetime may provide some insights into questions about the attach-
ment of electric charge to massive fields, or the prominent role of the spin group rather than
the special orthogonal group which is more directly connected to spacetime, remains to be seen.
For more details on complex spacetimes we refer to [10] and references therein.
1.2 Symmetries
One of the guiding principles in the construction of theoretical formulations of Nature are sym-
metries. Either experimentally observed or theoretically predicted they introduce severe con-
straints on the class of possible theories. In mathematics symmetries are described within the
theory of groups and its affiliation with other branches usually results in strong simplifications
and fruitful insights. The strength and wealth of symmetries are founded in its group properties
that either act as invariance conditions to reduce complexity or as powerful tools to extract gen-
eral forms of objects purely by their transformation behavior. Group theory is that important in
today’s physics, because it allows to assume less and simultaneously deduce more: The diver-
sity of mathematical formalizations of Nature is broken down by invoking additional (physical)
constraints, out of which symmetry constraints are usually the most restrictive ones. The glory
of the Standard Model of particle physics is strongly correlated with its formulation in terms of
symmetry groups that have led to many astonishing predictions verified by experiments.
In this section we touch only those aspects of group theory that are of particular importance
in studying QFT. Besides a brief introduction into the basics of group theory, we consider its
application to other algebraic structures, as for example vector spaces, and relate some physical
notations to the emerging mathematical concepts. Finally, we exemplify this abstract construc-
tions for several isomorphism groups equipped with different algebraic structures.
This section is mainly based on the following references [11–15], to which we refer the
reader for more details.
1.2.1 Groups
Nowadays, group theory is incorporated in many fields of mathematics, physics, and chemistry
to mention only a few. Compared to the wide-ranging consequences one can deduce from it, its
1.2 Symmetries 15
basic concepts rely on a very simple algebraic structure:
Definition 1.2.1 — Group. Let X be a set. The ordered pair G ≡ (X ,◦G) is a group :⇔
◦G : X → X is a relation on X with properties:
◮ ∀x,y,z ∈ X : (x◦G y)◦G z= x◦G (y◦G z) (associativity)
◮ ∃eG ∈ X , ∀x ∈ X : x◦G eG = eG ◦G x= x (identity element)
◮ ∃( ·)−1G : X → X such ∀x ∈ X : (x)−1G ◦G x= x◦G (x)−1G = eG (existence of inverse)
The beauty that appears once group structures are uncovered within a problem, is related to
the invariance and reduction that comes along. Groups are such powerful aspects because they
are based on a binary, well-formed operation in the sense of being invertible, stable, and suc-
cessively applicable. It therefore provides the natural grounds for studying transformations,
inevitable in quantum field theory.
The classification of point sets is entirely due to their cardinalities #(X). When augmented
with a binary relation these simple constructions develop a plenitude of (#(X))3 new structures.
Restricting to those relations that fulfill the group properties reduces this number considerably.
Two groupsG≡ (X ,◦G) and F = (X ,◦F) of the same underlying finite point set X are equivalent
if there is an isomorphisms in the category of groups that connects both :⇔
∃ f : X → X bijection, ∀x,y ∈ X : f (x◦G y) = f (x)◦F f (y) (homomorphism)
This condition ensures that group structures remain intact under homomorphisms while the
bijective property takes care of the point set correspondence. In the case of infinite cardinal-
ity homomorphisms are still the valuable maps that categorize equivalent groups, however the
number of distinct structures exceeds any limit.
Classification of groups
The classification of groups depends on the properties of their binary relation. There are two
options to extract the intrinsic features of a group G= (X ,◦G) either by directly studying ◦G or
by constructing new group specific (sub-)sets. An example for the first approach is given by the
Abelian property with which commutative group structures are labeled, i.e. groups for which
x◦G y= y◦G x holds for all x,y ∈ X .
Concerning the latter classification we may consider the conjugacy classes of G= (X ,◦G) –
subsets of X defined by the equivalence relation of conjugation:
Definition 1.2.2 — Conjugacy class. Let G ≡ (X ,◦G) be a group. A subset CG(x) ⊆ X is
a conjugacy class of G at x :⇔ :
CG(x) = {y ∈ X |∃g ∈ G : y= (g◦ x)◦ (g)−1G G G }
The totality of all conjugacy classes {CG(x) |x ∈ X} provides a deep insight into the underlying
group structure and will be relevant, in particular, in representation theory. Notice that for
Abelian groups each conjugacy class consists of a single element, CG(x) = {x}, a feature that
holds forCG(eG) = {eG} even in the general case.
Groups are something very special in the variety of all possible binary relations and they
provide the mechanism to reduce the complexity of problems significantly. Once we hold such
a valuable object in our hand it is worth to extract further related groups from it.
Subgroups
The classical way to construct groups from G ≡ (X ,◦G) is based on searching for closed re-
strictions of ◦G for some subset of X , i.e. investigating the symmetries of the binary relation
◦G. The only singleton that forms a group consists of the identity element of G: {eG} ⊆ X , the
trivial subgroup ({eG},◦G). While ({eG},◦G) satisfies the group properties, the extension to
16 Chapter 1. Physical Content
larger subsets severely depends on ◦G. Adding another element x ∈ X to the trivial subgroup in
general results in introducing a chain of new elements in order to close the algebra and to satisfy
Definition 1.2.1. For instance the inverse (x)−1G usually differs from x itself so does x◦G x.
Depending on the interaction of a subgroup H = (S ⊆ X ,◦G) with its parent group G =
(X ,◦G) there are some very special types among them:
Definition 1.2.3 — Normal subgroups. Let G ≡ (X ,◦G) be a group. The ordered pair
N = (0/ 6= S⊆ X ,◦G) is a normal subgroup of G :⇔
( )
∀x, y ∈ S : (x◦G y ∈ S) ∧ (x)−1G ∈ S (subgroup of G)
∀x ∈ S, g ∈ G ∃y ∈ S : (g◦G x)◦G (g)−1G = y ∈ S (normal)
One usually emphasizes N being a normal subgroup by using the notation N E G.
A normal subgroup is the union of conjugacy classes of its elements and thus closed under
conjugation. Notice that in general neither N EG is Abelian nor is an Abelian subgroup normal.
In fact combine the requirement of a conjugate closed and Abelian subgroup yields the center
of a group:
Definition 1.2.4 — Center of a group. Let G ≡ (X ,◦G) be a group. The ordered pair
Z(G)≡ (S,◦G) is the center of G :⇔ :
S= {x ∈ X |∀g ∈ G : x◦G g= g◦G x} ⊆ X
Besides its obvious Abelian nature, the center also commutes with the entire group and is there-
fore trivially normal. Moreover, the center of a group is non-empty since it always contains the
identity element and whenever G is Abelian the group coincides with its center.
Finally, let us have a glimpse on a special class of orbits (see below), the cosets of sub-
groups.
Definition 1.2.5 — Coset. Let G ≡ (X ,◦G) be a group, g ∈ G, and H ≡ (S ⊆ X ,◦G) a
subgroup.
A subset gH ⊆ X (Hg⊆ X ) is a left (right) coset of H in G w.r.t. g :⇔
gH = {g◦G h |h ∈H} (left coset) Hg= {h◦G g |h ∈ H} (right coset)
In general, a coset endowed with the group operation ◦G does not lead to a subgroup. Nev-
ertheless, consider a subgroup H for which left cosets and right cosets agree, i.e. gH = Hg
∀g ∈ G. This reflects the property of H to be a normal subgroup and the cosets give rise to a
group denoted as factor (or quotient) group.
Product groups
Another way to generate new groups is to extend the underlying point set by direct or semidirect
products of existing groups:
Definition 1.2.6 — Semidirect product. Let N ≡ (A,◦N) and H ≡ (B,◦H) be groups. The
ordered pair G≡ (N×H,◦G) is a semidirect product of N and H w.r.t. ϕ :⇔
ϕ : H →{ f : N → N | f and (∀h ∈ H : ϕ(h)≡ ϕh) are group isomorphism}
∀g1 = (n1,h1), g2 = (n2,h2) ∈G : g1 ◦G g2 = (n1 ◦N ϕh1(n2),h1 ◦H h2)
In abbreviated form one writes G≡ N⋊ϕ H .
It is straightforward to verify that the group isomorphism property of ϕ ensures thatG=N⋊ϕ H
1.2 Symmetries 17
factually defines a group.10 The structure of the original groups N and H is encoded in the
subsets {(n,eH) |n ∈ N} and {(eN ,h) |h ∈ H} that form a normal and a generic subgroup of G,
respectively.
There is a shorthand notation to illustrate the interrelation of groups using what is called
exact sequences. Within group theory an exact sequence is an ordered set of groups {G j} and
homomorphisms { f j} written as
−→f1G0 G1 −→
f2 −→f3G2 · · · −→
fn
Gn with Im( f j) = ker( f j+1) ∀ j
In more general cases group homomorphisms are replaced by the morphisms of the correspond-
ing category. Using this abbreviate form the semidirect product G = N⋊ϕ H is expressed in
terms of
−1
1→− N −→f1 G−→f2 fH →− 1 and H −−2→ G,
( )
where the group homomorphism is given by ϕ (n) = f−1 f−1h 1 2 (h)◦ f −1 −1G 1(n)◦G f2 ((h)H ) . An-
other example is the central extension given by the following short exact sequence:
1→− A⊆ Z(E)−→ E −→ G−→ 1
It has some relevance in physics associated with central charges, the generators of the center of
the central extension of a (local) symmetry group.
A more common procedure to extend groups is a special case of Definition 1.2.6: the direct
(group) product G = N×H defined by the trivial homomorphism ϕh(n) = n ∀h ∈ H, n ∈ N or
in terms of the exact sequence as follows:
1→− N×H −→f1 G−→ 1
It is this construction of directs products that describes the symmetry group of the Standard
Model of particle physics, the prosperous candidate for explaining all fundamental interac-
tions excluding gravity. But once symmetries of spacetime are considered, especially those
of Minkowskian spacetime, we are confronted with (a generalization of) the Poincaré group – a
semidirect product of rotation- and translation symmetries.
1.2.2 Group actions
Though powerful on their own, groups unfold there full beauty when revealed within other
structures. The general recipe to extract those hidden features is based on group actions:11
Definition 1.2.7 — Group actions. Let G ≡ (X ,◦G) be a group and S a set. A function
ρ : X×S→ S is a (left) group action of G on S :⇔
∀g,h ∈ X , ∀s ∈ S : ρ(g◦G h,s) = ρ(g,ρ(h,s))
∀s ∈ S : ρ(eG,s) = s
The set S is denoted as a G-set. For a right group action the first condition has to be replaced
10The iden(tity element of the s)emidirect product is given by eG = (eN ,eH) and its inverse operation reads
((n,h))−1G = ϕ
−1 −1
(h)−1((n)N ),(h)H .H
11In what follows we focus mainly on left group actions, though the same results apply for the right group actions
with some minor adaptions. In particular, every left group action ρL gives rise to a right action ρR and vice versa:
ρ (g,a) .R .= ρL((g)
−1
G ,a).
18 Chapter 1. Physical Content
by ∀g,h ∈ X , ∀s ∈ S : ρ(g◦G h,s) = ρ(h,ρ(g,s)) (reversing the order).
There are intrinsic actions of a group onto itself given by the inner-, left-, and right-action fixing
a specific member of the group. Now, using group actions we also have a tool to extrinsically
attach symmetry properties to other structures with the aid of homomorphisms. This is a very
strong mechanism to implement the deep insights we gain from group theory to other objects,
in particular those we encounter in physics. Nevertheless, the mere presence of a group action
on some set is insufficient to draw conclusions about their intertwined relation.12 Rather, we
have to consider group actions equipped with additional properties that bring those correlations
between the underlying group and set to light. In the remainder of this subsection we will
emphasize some of the most important features of group actions and finally focus on sets with
vector space character that lead to the study of representation theory.
Types of group actions
Group actions as special types of (binary) functions inherit the corresponding characterization
scheme of functions, however to stress the group structure in the first argument one uses a
different notation.
Definition 1.2.8 — Types of group actions. Let ρ : X×S→ S be a left (right) group action
of G= (X ,◦G) on the non-empty set S. The map ρ is . . .
. . . faithful :⇔ ∀g ∈ X/{eG} ∃a ∈ S : ρ(g,a) =6 a
. . . free :⇔ ρ(g,a) = a =⇒ g= eG
. . . transitive :⇔ ∀a, b ∈ S ∃g ∈ X : ρ(g,a) = b
. . . regular :⇔ ∀a,b ∈ S, ∃̇g ∈ X : ρ(g,a) = b
Notice that a faithful group action ensures that each group element gives rise to a different map
on S, a feature naturally fulfilled by the stronger condition of being free. With this property it
is guaranteed that no group element is superfluous, but provides a different transformation rule
on S. A free or a transitive group action are the correspondence of injectivity or surjectivity in
the language of functions, respectively, while the regular condition combines both concepts and
translates bijections into the present context.
In general group actions lack those features and their appearance is very welcome. Once we
have establish a free group action on a set we can proceed along the lines of injective function
to establish regular (bijective) group actions on subsets of S using orbits. Thereby, the transitive
property is gained in the process of generating those subsets by fixing an element of S and let
the group act on it, in detail we have:
Definition 1.2.9 — Orbits. Let G ≡ (X ,◦G) be a group, S a set, and ρ : X ×S→ S be a left
group action. A subset Ox ⊆ S is the orbit of a ∈ S w.r.t. ρ :⇔
Ox ≡ {ρ(g,a) |g ∈ X} ≡ Im(ρ(•,a))⊆ S
This procedure divides the entire set in subsets on which the group acts transitive (surjective)
and in fact defines an equivalence relation on S. For a free group action those equivalence
classes, i.e. orbits, are the aforementioned partitions on which the group action is regular. In the
general case however, the group action will have some fixed points that will spoil its injectivity,
being not free. This is similar to functions which have a non-trivial kernel and in the context of
group actions we refer to it as having a non-trivial stabilizer subgroup:
12For example the trivial group action exists for every group and non-empty set, where each element of the former
acts as the identity on the set.
1.2 Symmetries 19
Definition 1.2.10 — Stabilizer subgroups. Let G ≡ (X ,◦G) be a group, S a set, and ρ :
X ×S→ S be a left group action. The ordered pair Ga ≡ (Aa,◦G) is the stabilizer subgroup
(isotropy group or little group) of a ∈ S w.r.t. ρ :⇔
Aa ≡ {g ∈ X |ρ(g,a) = a} ≡ ker(ρ(•,a)) ⊆ X
At this point the group-structure-preserving nature of homomorphisms, as here given by ρ ,
becomes apparent: As morphisms of group theory they lift the category of sets to the category
of groups and in shape of group actions we have seen that orbits and stabilizer subgroups take
over the concepts of images and kernels of a map. The additional requirement of a function to
be homomorphic actually yielded a partition in equivalence classes of the ‘images’ and ensures
that its ‘kernels’ Ga are always subgroups of G.
Combining the ideas of orbits and stabilizer subgroups we can form a regular group action
at each element of a∈ S by dividing out its stabilizer subgroup Ga and finally restrict the domain
of ρ(•,a) to the orbit Oa. Especially in gauge theory a suitable knowledge of the underlying
orbits and isotropy groups is of particular importance.
Linear representations
Besides specifying group actions by its function properties we can endow the G-set with addi-
tional structure and let group homomorphisms preserve it. Since in physics vector spaces play
an essential role in understanding Nature, the way groups interact with them is a well studied
subject in mathematical physics: the theory of (linear) representations. In fact, from a mathe-
matical perspective representations provide a tool to understand groups in the way they (re-)act
with the very familiar branch of linear algebra. Vector spaces come along with fields and a set
of algebraic operations including an Abelian one, usually denoted by addition. In order to keep
this tied construction intact under group homomorphisms we have to restrict the action of group
elements to those functions that are linear, in other words automorphisms in the category of
vector spaces:
Definition 1.2.11 — General linear group. LetV ≡ (S,◦V ,K,◦K·V ) be a vector space over
the field K. A function f : S→ S is a linear function in V :⇔
∀a,b ∈ S : f (a◦V b) = f (a)◦V f (b) (additivity)
∀a ∈ S∀λ ∈K : f (λ ◦K·V a) = λ ◦K·V f (a) (homogeneity of degree 1)
This subset of all linear functions of a vector space V forms the general linear group Gl(V ), the
automorphism group in the category of linear spaces. While the group operation, given by func-
tion composition ◦, is closed for all (linear) functions, the existence of an inverse requires the
functions to be bijections. Gl(V ) is a significant example of a (non-Abelian) group that is at the
very basis of representation theory, as can be seen from the following definition:
Definition 1.2.12 — Linear representation. LetG≡ (X ,◦G) be a group,V ≡ (S,◦V ,K,◦K·V )
a vector space over the field K. A left group action ρ : X ×S→ S is a linear representation
of G on V :⇔
∀g ∈ X : ρ(g,•)≡ ρg ∈Aut(V )≡ Gl(V ) ..= { f : S→ S | f is a linear bijection on V}
Thus, (linear) representation can be understood as group actions that are specially designed to
suit the needs of linear algebra.
The structure of a general linear group derives from its associated vector space that is com-
pletely characterized by the cardinality of its point set and its vector space dimension, at least in
the finite-dimensional case. The variety of seemingly different linear spaces of the same dimen-
20 Chapter 1. Physical Content
sion are only different facets that all correspond to the same equivalence class. When changing
the basis one transitions into another representative of this class using an element of the general
linear group. For finite-dimensional vector spaces, say dimV = d < ∞, the distinct possibilities
for such a basis transformation – hence the size of Gl(V )≡ Gl(d;K) – is also finite and can be
represented by invertible d×d-matrices.13
At this point it is instructive to illustrate the coalescence of a free and a faithful group action
within the language of representation theory. Since in this context group actions are rather
considered as maps from a group into a function space Gl(V ), a faithful representation (group
action), which addresses the group argument, actually translates into injectivity, i.e. each group
element is mapped to a different function. For a free group action, usually a stronger condition,
every group member is associated to a distinct and injective function. Within representation
theory this does not amount to an additional restriction since all functions are bijections, thus a
faithful representation is a free one and vice versa.14
Representations play an important role in todays physics, as symmetry considerations lie
at the very heart of its mathematical foundations. Having establish such a connection between
a symmetry group G and a vector space V , one is keen to derive new representations from
the original one by modifying G and/or V . Hereby, the term ‘new’ refers to distinguishable
constructions in the category of representations, i.e. inequivalent linear, bijective group ac-
tions:
Definition 1.2.13 — Equivalent representations. Let G≡ (X ,◦G) be a group. Further, let
V = (SV ,◦V ,K,◦K·V ) andW = (SW ,◦W ,K,◦K·W ) be vector spaces.
Two (linear) representations ρV : G→ Gl(V ) and ρW : G→ Gl(W ) are equivalent :⇔
∃ f : SV → SW linear ∀g ∈ X : f ◦ρV = ρWg g ◦ f (equivariant map from V toW )
f is an isomorphism, i.e. ∃ f−1 : SW → SV linear
Notice that equivariance is the natural way to preserve the structure within representation theory
and the bijective property lifts this correspondence to isomorphisms. The natural object to
distinguish inequivalent representations are their characters:
Definition 1.2.14 — Character of a representation. Let G ≡ (X ,◦G) be a group, V =
(SV ,◦V ,K,◦K·V ) a vector space, and ρ : X ×Gl(V ) a (linear) representation of G on V . The
function χρ : G→K is the character of ρ :⇔
∀g ∈ X : χρ(g) = Tr(ρ(g)),
whereby Tr denotes the trace defined on endomorphisms of V (see Subsection 1.4.2).
Since the trace is cyclic two equivalent representations, ρV = f ◦ρW ◦ f−1, clearly have the same
character. The treasure provided by group theory is that also the reverse holds true, namely if
the character of two representations is the same both are equivalent.15 Our objective in the
following is to extend and simultaneously reduce the representation as much as possible to
deduce its inequivalent descendants.
In a first step we focus on a modification of the group and retain the full vector space
V for a moment. If we take only a subset of group members to act on V we have to make
13At this step Gl(V ) needs vector space properties, which linear functions inherit pointwise from its underlying
vector space.
14Transitivity is also part of the definition of a linear representation and thus the notion of regular group actions
also coincides with faithful or free representations in this case.
15In what follows we always assume that the underlying field has characteristic zero, that is its multiplicative
identity does not add up to give the additive identity.
1.2 Symmetries 21
sure that the group operations are closed, i.e. consider subgroups and their restricted group
action:
Definition 1.2.15 — Restricted representation. Let G ≡ (X ,◦G) be a group, V a vector
space, and ρ : X ×Gl(V ) a (linear) representation of G on V . Given a subgroup H ≡ (A ⊆
X ,◦G) of G, the map ρH : A×Gl(V ) is the restricted representation of ρ on H :⇔
∀g ∈ A : ρH(g) = ρ(g)
This is a very natural restriction of the group action to only a reduced number of symmetry
transformations. For the opposite direction there are in principle several ways to extend the
linear group actions to a larger group. The induced representation reflects the reverse of a
restricted group action, loosely speaking given a subgroup H of G with a representation the
representation ρH one searches for an extension on G for which ρH is a restriction; formally,
the induced representation is defined to be the left adjoint functor of ρH . Both concepts play
an important role in the study of phases and symmetry breaking, as well as in the study of the
diffeomorphism group.
Besides generating new representations by altering the group, an even more interesting
approach consists in a variation of the underlying vector space. We can gain deep insights of
abstract groups by simply investigating their imprint on structures of linear algebra, in particular
their actions on linear spaces. The difficulty resides in the search of a suitable vector spaces
that is sufficiently condensed while simultaneously being large enough to carry all essential
information of the group.
Let us assume we are given two representations ρV and ρW of a group G≡ (X ,◦G) acting on
vector spaces V andW , respectively. The direct sumV ⊕W yields a new, extended vector space
and we can also endow it with a representation ρV⊕W = (ρV ,ρW ) of G induced by the existing
ones that may result in new insights of the group G. However, the crucial point to notice here
is the mutual ignorance of both subspaces V andW under group action – their non-interaction
– implying that once we have perfectly understood ρV and ρW individually, there is no more
we can learn from studying ρV⊕W . This suggests to find the smallest possible building blocks
that in its totality still contain the essential information to built up the full group structure.
The objective is to spot subspaces that are invariant under group actions and decompose the
full vector space using the group action. This procedure will terminate at some irreducible
subspaces which – taken together – reveal the abstract group and its properties:
Definition 1.2.16 — (Ir-)reducible representation. Let V ≡ (S,◦V ,K,◦K·V ) be a vector
space, G≡ (X ,◦G) a group. A representation ρ : X ×Gl(V ) of G on V is reducible :⇔
∃W = (A⊂ S,◦W ,K,◦K·W ) non-trivial subspace of V , i.e. A 6= {eV}, such that
∀g ∈ G∀a ∈ A : ρg(a) ∈ A
If ρ has no non-trivial sub-representation (i.e. not reducible) it is called irreducible.
These fundamental building blocks contain all essential information of a group in a very con-
densed form, i.e. a full knowledge of irreducible representations is equivalent to the full knowl-
edge of all representations which in turn implies understanding the group.
The characterization of inequivalent irreducible representations thus lies at very heart of
representation theory and is based on Schur’s lemma [16]. It states that for finite dimensional
irreducible representations16 equivariance implies equivalence, i.e. if two irreducible represen-
tations are related by an equivariant map f this relation is either an isomorphism or f is the
16A representation is called (in-)finite dimensional when the associated vector space is (in-)finite dimensional.
22 Chapter 1. Physical Content
zero function. The theory of irreducibility is centered around this lemma and we will come to
appreciate it while studying its implications.
We have seen that the content of representation theory is fully described by fundamental
building blocks, the irreducible representations of a group, and it is our aim to find and classify
all of them. The necessary information needed is the number of those basic ingredients, the
associated vector spaces they naturally live in, and their interrelation. We focus on each of
these issues in the following list by taking an arbitrary group G≡ (X ,◦G):
◮ Number of irreducible representations nirrep: On the search for irreducible representa-
tions we first of all have to know how many we have to look for. Fortunately, the number
of these group actions is equivalent to a group property: the number of conjugacy classes.
As mentioned above, inequivalent representations give rise to different trace functions
(characters) and we can thus characterize each of the nirrep inequivalent irreducible repre-
j
sentations by exactly those characters {χ ( j) ≡ χρ |ρ j irreducible representation of G}.
◮ Dimension of irreducible representations: As soon as we have pinned down the count
of different characters and thus the number of irreducible representations, it is important
to know their associated vector spaces. In the finite dimensional case, a vector space is
fully characterizes by its dimensionality. Let us assume a group has nirrep inequivalent
irreducible representations ρ ( j) that map into vector spaces of dimension d( j), which
is denoted the dimension of the representation ρ ( j). Now, the dimensionality theorem,
which is a direct consequence of the orthogonality theorem we will study in a second,
relates the number of elements in G to the sum of all d( j)’s:
( )2
#(X) = ∑ d( j) (1.5)
j∈[1,nirrep]
Clearly, if the group is infinite, i.e. it is based on an infinite set, we have little to gain
here. Nevertheless, for any finite group we have strong restrictions for the dimension d( j)
of each irreducible representation ρ ( j).
◮ Relation between irreducible representations: Finally, a great deal of information is
obtained by studying the correlation of distinct building blocks. We have already account
for the fact that inequivalent representations, in particular irreducible ones, give rise to dif-
ferent characters and are factually encoded in those 1-dimensional representations. From
Schur’s lemma one can deduce the following important orthogonality theorem:
∫
δ jk = 〈χ ( j)χ (k)∗〉 ..= dµ(G)(g) χ ( j)(g)χ (k)∗G (g)
∫ g∈X
≡ dµ(G)(g) χ ( j)(g)χ (k)(g−1) (1.6)
g∈X
∫
Here, the normalized integral g∈X dµ
(G)(g) = 1 generalizes the summation over group
elements, ‘ 1#( ) ∑g∈X ’, in order to cover also compact, but infinite groups. (We will haveX
a closer look at measures in section 2.1). The statement of eq. (1.6) is a vital guiding
principle to deduce the full structure of irreducible representations using orthogonality of
their characters. In addition there is a similar relation between the conjugacy classes of the
group. Given a conjugacy class of a group member g∈X , i.e. CG(g) = { f ◦ g◦ f−1G G | f ∈
X}, and another group element h ∈ X we have:
∫ {
( j) 0, h ∈/ CG(g)∑ χ (h−1) dµ ( j)G( f ) χ ( f ) = (1.7)
j∈[1,n ] f∈CG(g) 1, h ∈CG(g)irrep
It is quite astonishing that most of the prescriptions for classifying the fundamental ob-
jects of representation theory relies on scalar functions, the characters.
1.2 Symmetries 23
In principle, representation theory reduces to answering the above questions for each group to
obtain the full set of inequivalent irreducible representations. Once at hand, any finite dimen-
sional representation ρ : G → Gl(V ) of a (finite) group can be completely decomposed into
those fundamental blocks (Maschke’s theorem [17, 18]):
⊕ ( j) ( j)
ρg = aρ ρg
j∈[1,nirrep]
( j)
The coefficients aρ are non-negative integers that stand for the number of times the respective
representation ρ ( j) is present in the decomposition of ρ . Their actual values can be deduced
again with the help of characters and the orthogonality theorem:
∫
( j)
aρ = dµ
G(g) χρ(g)χ ( j)(g−1)
g∈X
Taking all together, characters that symbolize irreducible representations play the most impor-
tant role in studying finite group actions on vector spaces, which in fact reveals a vast amount of
the group properties itself. Before coming to an end of this brief introduction of representation
theory let us consider some very important application of group actions in physics: symmetry
phases.
Phases and phase-transitions
A very important aspect of group actions in physics is associated to symmetry phases and the
corresponding phase-transitions [19]. Its has many applications in different branches of physics
– high- or low energy physic, quantum or classical in nature – and thus underlines the influence
of group theory in today’s sciences. To fit all those approaches into one general description we
have to introduce an abstract notation of constraints defined on a G-set S and some parameter
space Λ :
( f ,W Truej j ⊂Wj) with f j : S×Λ →Wj
Whenever f j(x,u) ∈WTruej we say that (x,u) fulfills the constraints. In practical cases f j will
be related to observable quantities and its explicit form will be quite intuitive, see the examples
below.
Depending on the value of the parameters u the constraint functions will allow different
elements of S, the order parameter space, to be admissible. The collection of all those order
parameters x ∈ S that satisfy the constraints defines the space of solutions:
Sol(u) ..= {x ∈ S | f j(x,u) ∈W Truej ∀ j} ⊆ S (constraints-fulfilling solutions)
Assume we have a theory where U(x,T ) represents the potential that depends on an external
parameter T ∈ R+, standing for the temperature of the system, say. Furthermore, let f : S×
R →W ≡ S be defined by f (x,T ) = ∂ U(x,T ) andW True+ 1 1 x 1 = {0} then all elements of Sol(T )
are extrema of the potential (for minima we need an additional constraint f2 that restricts to
positive second derivatives of U(x,T )). Now, the space of solutions sensitively depends on the
order parameter space. For instance, letU(x,T ) = (x3+ x) =U(x) be of polynomial type, then
Sol(T ) consists of {+ı,−ı} in case of S= C while it is empty for the real numbers R.
In the general case, Sol(u) will also depend on the parameters u and thus it will exhibit
different symmetries for different values of u:
XSol(u) = {g ∈ XG |ρ(g,x) ∈ Sol(u) ∀x ∈ Sol(u)} ⊆ XG (symmetry of solutions)
24 Chapter 1. Physical Content
Basically, this contains all group elements that map a solution of f j to another one, and thus
it is the intersection of all stabilizing subgroups of elements in Sol(u). Using this information
the symmetry phases partition S by means of an equivalence class of connected spaces on Λ
all sharing the same symmetry of solutions. In full generality, the division of the parameter
space is encoded in the symmetry phases with respect to the considered constraints, defined as
follows:
Definition 1.2.17 — Symmetry phases. Let G ≡ (XG,◦G) be a group, S = (YS,τS) and
Λ = (YΛ ,τΛ ) be topological spaces. Furthermore, let ρ : XG×S→ S be a (left) group action
and ( f : Y ×Λ →W ,W Truej S j j ) a family of constraints. Define an equivalence relation on the
parameter space Λ as follows:
v∼Λ w :⇔ ∃U ∈ τΛ , v,u ∈U , connected with ∀w ∈U ∪{v} : XSol(u)≡ XSol(w)
The ordered pair (ΛA,GA) is a symmetry phase of (S,Λ ,G,{( f j ,WTruej )},ρ) :⇔
∃u ∈ ΛA : ΛA ≡ {v ∈ Λ |v∼Λ u} (equivalence class)
∀u ∈ ΛA : GA = (XSol(u),◦G)⊆ G (symmetry of solutions)
The set S is the order parameter-, Λ the parameter-, and Sol(•) the solution space.
In this general case the constraint functions can assume values in any set and the corresponding
constraint is implemented by requiring the image to be in W Truej ⊂Wj. Once we are dealing
with a normed vector space most often the above form reduces to some constraints over the real
space.
The set of solutions Sol(u) respecting the constraints is in general very sensitive to the
parameter choice u and so is the set of its invariant transformations XSol(u). In physics the
transitions from one symmetry phase to another provides some very interesting insight into the
underlying theory and its properties. As we have topological spaces at our disposal these phase
transitions occur at the overlap of topological boundaries of two or more phases, say (ΛA,GA)
and (ΛB,GB):
Λ .A↔B .= ΛA∩ΛB
If this set is empty both phases do not share a direct phase transition. In other cases one distin-
guishes the change that happens at such a ΛA↔B by classifying the continuity of the transition.
Basically modern schemes invoke the notation that a discontinuous change of Sol(u) through
the boundary between phases corresponds to a first order phase transition while a continuous
change is denoted to be a higher order phase transition. In the present context, for ΛA↔B 6= 0/ ,
this translates into the following statement for a higher order phase transition:
∃U ∈ τΛ with ΛA↔B ⊆U : {x ∈ Sol(u) |u ∈U} ∈ τS
Whenever this condition is not fulfilled we encounter a first order phase transition.
We have seen that in the general setting we detect phases of a group action ρ : S×G →
S using order parameters and the classification of transitions is due to their continuity along
the boundary of phases. In quantum field theory or statistical physics, the field expectation
values are usually the constrained solutions that, depending on the explicit form of the measure
µ(•;{u}), exhibits different symmetry invariances.
It is instructive to give some very simple examples of symmetry phases that however exem-
plify all the abstract concepts we have introduced previously. Let us define the following system
and later specify the missing ingredient, a differentiable function η : S×Λ →R, to generate all
1.2 Symmetries 25
η(x,u) = u x22 +u1x η(x,u) = u x
3
3 +u1x η(x,u) = u4x
4+u 22x
u2 u3 u2
− √u
Sol 1A = { 2 } SolC = {
−u1 } −u1
u2 2u
Sol
2 A
={ 3 }u3 SolA = {0}
XSolA = {1} XSolC = {1} XSolA = {1} XSolA = XG
u1 u1 u4
Sol = 0/ Sol = 0/ Sol = 0/ √ √D E B −u1 SolB = 0/
SolC={− }
−u2
SolC={± }
XSol Sol Sol 3u3 Sol 2u4D = 0/ XE = 0/ XB = 0/ XB = 0/
XSol = {1} XSolC C = {±1}
SolB = {0}
XSolB = XG
Figure 1.1: The phase diagrams for different choice of η(x,u). Symmetry phases are indicated
by different shadings. Dashed (solid) lines correspond to first (higher) order phase transitions.
the different examples:
Order parameter space: S ≡R Parameter space: Λ ≡R×R
Symmetry group: G≡ (R/{0}, ·) Group action: ρ(g,x) = g · x ∀g ∈ G, x ∈ S
2
Constraints: {( f1(x,u) ..= d η(x,u),W Trued 1 ≡ {0}),( f2(x,u) = d2 η(x,u),W True2 ≡ R+x d x 0 )}
The constraints are written in the general notation used above, which at this point hides its
simple meaning. Actually, the requirements we impose is minimizing the function η , as can be
seen by translating them into standard form:
{0} ∋ d 2d η(x,u) = 0 ∧ R
+ ∋ d0 2 η(x,u) ≥ 0 ∀x ∈ Sol(u)x dx
The phase diagrams for three different types of η , thus different types of constraints, are de-
picted in Figure 1.1.
R Changing the order parameter space from R to C will result in very different equiva-
lence classes. In the present case the minima condition is quite restrictive for it involves
solutions of polynomials that for some parameters are imaginary and thus are excluded.
1.2.3 Lie groups & Lie algebras
The reason for studying groups in physics is their relation to symmetries which helps revealing
the properties of a mathematical structure. Again the cardinality of the point set determines
the variety of symmetries that may appear, which is finite for any finite set. In quantum field
theory infinities are part of the game and the symmetry groups we encounter are usually based
on uncountably infinite sets. Fortunately, the ‘physical’ groups are endowed with additional
properties that simplifies matter a lot. The infiltration of topology and smooth structures into
group theory gives rise to the concept of Lie groups introducing the notion of continuity and
differentiability:
Definition 1.2.18 — Lie groups. Let (X ,◦G) be a group. The ordered pairG≡ (X ,◦G,τX ,A)
26 Chapter 1. Physical Content
is a Lie group :⇔
τX ⊆ 2X with τX a topology
A= [{(U ⊆ X , fU) |(U, fU ) isC∞-chart }]∼ is maximal C∞-atlasMC∞
∀x,y ∈ X : (x,y)→7 (x)−1G ◦G y isC∞-function
G is thus a group and a smooth manifold for which the group operations are C∞-functions.
Lie groups are such powerful objects because their algebraic construction is compatible with its
topological and differential structure. Their inherited features are manifold. We have a closed
binary relation that allows to propagate through the set in a continuous and smooth way and
there is a local connection to Euclidean space with its properties, in particular a metric- and
vector space construct attached to every point in the group. It is the latter, the (local) vector
space structure, that ties another very important concept to each local piece of a Lie group, a
special vector space named Lie algebra:
Definition 1.2.19 — Lie algebras. Let V = (X ,◦V ,K,◦K·V) be a vector space over the field
K. The ordered pair g≡ (V, [• , •]g : X ×X → X) is a Lie algebra :⇔
∀x ∈ X : [x , x]g = eV (alternating)
∀x,y,z ∈ X , a,b ∈K : [(a◦K·V x)◦V (b◦K·V y) , z]g = (a◦K·V [x , z]g)◦V (b◦K·V [y , z]g)
(bilinearity) [z , (a◦K·V x)◦V (b◦K·V y)]g = (a◦K·V [z , x]g)◦V (b◦K·V [z , y]g)
∀x,y,z ∈ X : [x , [y , z]g]g ◦V [z , [x , y]g]g ◦V [y , [z , x]g]g = eV
The last property is the Jacobi-identity that is respected by the Lie bracket [• , •]g.
The connection between a Lie group and its Lie algebra can be best understood within the theory
of differential geometry using fiber bundle constructions. At this point we only scratch the
surface of this deep relation and postpone a more consistent treatment to section 1.3. The basic
idea is to consider the group action w.r.t. a specific element as an isomorphic transformation of
the group to itself i.e. maps of the form ρ :G→{ f :G→G | group isomorphism}, for instance:
. . . the left action: g→7 L(g)≡ Lg with Lg(x) = g◦G x ∀x ∈ G,
. . . the right action: g→7 R(g)≡ Rg with Rg(x) = x◦G g ∀x ∈ G,
. . . the inner action: g 7→ ad(g)≡ adg with adg(x) = (g)−1G ◦ x◦g ∀x ∈ G.
Now, the differential nature of Lie groups gives a meaning to an infinitesimal transformation,
that is a group isomorphism whereby the elements are shifted to its closest neighbors. The ho-
momorphism properties of the above actions imply that the local vicinity of the identity element
eG is responsible for these small shifts with the limit of eG inducing no change at all. Since we
have vector space structures locally attached everywhere, we can generate infinitesimal transfor-
mation using the vector space at the identity eG that actually fulfills the Lie algebra properties.
Thus, the local vicinity at each point on the group manifold is fully determined by its associ-
ated Lie algebra. However the Lie algebra may loose global information of the underlying Lie
group, for example its connectedness status. Nevertheless, from a group theoretical point of
view there are certain relations that are preserved within the Lie algebra. For instance, in the
case of a connected Lie group G the Lie algebra of the (Abelian) subgroup Z(G), the center of
G, coincides with the center of the full Lie algebra g of G.17 Hence, whether or not a Lie group
is Abelian can be read off by the Abelian property of its Lie algebra, whenever G is connected.
17It is common to emphasize the allocation of a Lie algebra to a Lie group by choosing small Gothic letters, i.e.
the Lie algebra of a Lie group G is denoted by g.
1.2 Symmetries 27
In conclusion, by fusing group theory with the concept of smooth manifolds we enter the
realm of Lie groups, a rich and powerful structure that seems to be at the core of todays under-
standing of Nature. The theory of Lie algebras is closely attached to these constructions and
will be explored later on. For now, we conclude with a classification of those structures.
Classification
While topological features as, for instance, compactness and connectedness determines the
global nature of a Lie group, the algebraic information are almost entirely encoded in its associ-
ated Lie algebra. Classification of Lie groups thus is closely related to a suitable classification
of Lie algebras, at least in the finite dimensional case.18
For finite groups we have seen that representation theory reveals the group properties by
studying its irreducible group actions in a very condensed form using scalar functions, the char-
acters. However, the interesting part of Lie groups concerns infinite groups, where the local
vector space dimension is usually finite but non-zero, thus its underlying point set has uncount-
able infinite cardinality. Therefore, we have to proceed on a different – though closely related
– path that is concentrated more on roots and fundamental weights than solely on character
functions.
Though based on an infinite group, Lie groups come along with a naturally related vector
space, their Lie algebra, which provides a direct route to extract the infinite number of irre-
ducible representations using the set of infinitesimal transformations. It is the classification of
Lie algebras that will answer the question of irreducible representations for those infinite, but
nicely equipped Lie groups.
To characterize Lie algebras, let us start by its vector space properties. The rich algebraic
properties of vector spaces allows to reconstruct the full space using a sufficiently small number
of elements that form a linearly independent basis, which in the context of Lie algebras is
usually referred to as a set of generators. Each basis can be transformed into another one using
an element of the associated general linear group and is thus not a unique construct the vector
space gives rise to. Nevertheless, since the general linear group consists of the automorphisms,
the vector space results are in general preserved under a change of basis.
Though for vector spaces the basis is sufficient for its full reconstruction, a Lie algebra is
additionally equipped with a binary operation, the Lie bracket that is not fixing by choosing
a basis. Fortunately, a Lie bracket is bilinear and hence for a given set of generators we can
encode its complete information into a set of structure functions:
Definition 1.2.20 — Structure functions. Let V ≡ (X ,◦V ,K,◦K·V ) be a vector space of
dimension d, g≡ (V, [• , •]g : X×X → X) a Lie algebra, and B≡ {Tj ∈ X} j=[1,d] a linearly
independent basis that spans V .
Functions f mjk : [1,d]× [1,d]× [1,d]→K are the structure function of g w.r.t to B :⇔
∀Tj, Tk, Tm ∈B : [Tj , Tk]g = ∑ f mjk ◦K·V Tm (1.8)
m
The special properties Lie algebras are endowed with allow to reduce its entire information, up
to isomorphisms, into a set of d3 scalar coefficients along with the associated vector space basis,
i.e. g=̂({T } , f mj j∈[1,d] jk ).
At this step it is instructive to consider the defining equation of the structure function,
eq. (1.8) in the light of representation theory. The two necessary ingredients a representa-
tion is based on, a group and a vector space, can be conflated into the same object when con-
sidering Lie algebras. One intrinsic group action of g is given by the adjoint representation
18There exist infinite dimensional Lie algebras that are not associated to a Lie group, see [20].
28 Chapter 1. Physical Content
where the Abelian group of the underlying vector space induces a natural action using the Lie
bracket:
Definition 1.2.21 — Adjoint representation of a Lie algebra. V ≡ (X ,◦V ,K,◦K·V ) be a
vector space of dimension d, g≡ (V, [• , •]g : X ×X → X) a Lie algebra. The smooth group
action ad : (X ,◦V )→ Gl(V ) is the adjoint representation of g :⇔
∀x ∈ X : ad : x 7→ adx ≡ [x , •]g
The bilinearity of the Lie bracket manifests an interpretation of structure functions as generators
for the adjoint representation by evaluating ‘ad’ on the basis elements:
∀Tj,Tk ∈B : ad mT (Tj k) = [Tj , Tk]g = f jk Tm
This intrinsically present Lie algebra endomorphisms adx is the key to understand the irreducible
representations of the associated Lie group. As a particular form of endomorphisms adx can be
chained and traced over, and when both operations are combined it gives rise to the Killing
form:
Definition 1.2.22 — Killing form. Let g≡ (V, [• , •]g : X×X → X) be a Lie algebra over the
fieldK. A binary function K : X ×X →K is the Killing form of g :⇔
∀x,y ∈ X : K(x,y) = tr(adx ◦ ady)≡ tr ([x , [y , •]g]g)
The function inherits the bilinearity and symmetry property of the Lie bracket and the trace, re-
spectively. Using the Killing form we obtain a first characterization of Lie algebras:
Definition 1.2.23 — Semisimple Lie algebras. Let g ≡ (V, [• , •]g : X ×X → X) be a Lie
algebra over the field K and K its Killing form. The Lie algebra g is semisimple :⇔
∀x ∈ X : K(x,•) is isomorphism
A more common definition involves the non-existence of non-zero Abelian ideals of g which –
due to Cartan’s criterion – is equivalent to the more practical form used above. If the restriction
to Abelian ideals is dropped than a Lie algebra is simple, meaning it has no non-trivial ideal at
all.
Now, Maschke’s theorem concerning the full reducibility of finite groups can be extended
to all finite dimensional Lie groups that have a semisimple Lie algebra: A semisimple Lie alge-
bra can be fully decomposed into simple ones that corresponds to a decomposition into a direct
sum of irreducible representations. In the language of Lie groups reducible and irreducible rep-
resentations are replaced by semisimple and simple Lie algebras, respectively. Finally, instead
of studying only characters we have to introduce another concept the highest weights which
take over the role to classify irreducible subspaces. We now summarize the algorithm to obtain
a full characterization of finite-dimensional, irreducible representations of any semisimple Lie
algebra g≡ (V, [• , •]g : X×X → X) over the complex fieldK≡ C:
A. Semisimple: Check whether or not the Lie algebra is semisimple using the Killing form
which has to be non-degenerate:
∀x ∈ X : K(x,•) is isomorphism
B. Cartan subalgebra: Amaximal Abelian subalgebra h≡ (Vh, [• , •]g : Xh×Xh→Xh)⊂ g:
∀H,H ′ ∈ X ′h : [H , H ]g = eV
∀a ∈ X : [H , a]g = eV ∀H ∈ Xh =⇒ a ∈ Xh
1.2 Symmetries 29
is called a Cartan subalgebra of g and the dimension of the linear subspaceVh ofV defines
its rank rkg. The dual space of Vh denoted V
∗
h , in short h
∗, contains all linear functions
that map Xh to the complex numbers:
V ∗ ..h = {α : Xh → C | linear function}
C. Root system: Decompose the full Lie algebra g into the Cartan subalgebra and associated
root spaces:
( )
⊕
g= h⊕ gα
α∈R
For a d-dimensional (finite) Lie algebra this Cartan decomposition involves only a finite
number of gα ’s implying that the cardinality of R ⊂ V ∗h is d− rkg. Each additional sub-
space we attach to the Cartan subalgebra has the following property:
gα = {Eα ∈ X |∃α ∈V ∗h , ∀H ∈ Xh : adH(Eα)≡ [H , Eα ]g = α(H)◦C·V Eα}
The roots of the Lie algebra α ∈ V ∗h can be interpreted as non-zero eigenvalues of the
adjoint representation and the corresponding eigenspaces gα are referred to as root spaces.
One of the fundamental statements for these root spaces is their one-dimensionality and
therefore their associated eigenvalues α are all non-degenerate (Cartan’s theorem, see ref.
[12]). Thus, each gα is spanned by a single basis element Eα , the root vector, which in
physics is usually denoted as step (or ladder) operator.
Furthermore, the roots exhibit a Z2-symmetry, meaning that for every α ∈ R also its
inverse root (−1)◦C·V∗ α ≡−α is in R but at the same time any other multiple c◦C·V∗ α 6=
±α is not a root. While the adjoint representation of any Eβ corresponds to a translation
in h∗ from gα to gα+β , its inverse root space E−β is mapped to the Cartan subalgebra. It
therefore appears that the direct sum
sβ ≡ gβ ⊕g−β ⊕ [gβ , g−β ]g with [gβ , g−β ]g ≡ {c◦C·V [Eβ , E−β ]g |c ∈ C} ⊆ h
forms a subalgebra of g isomorphic to the simple Lie algebra su(2) (see below). This has
some important and very nice features on which we briefly explore in what follows. First
of all, there is a unique element19
. 2Hα .= 2 ∑ α(H j)◦C·V H j ∈ [gβ , g−β ]g ⊆ hα
j∈[1,rkg]
that fulfills the requirement α(Hα) = 2 and leads to the standard description of su(2):
1
[H 2α , E±α ]g =±2E±α , [Eα , E−α ]g = α Hα
2
While this shows the closure of the subalgebra sα , for general root vectors we have
[Hα , Eβ ]
2
g = 2 ∑ j α(H j)β (H j)Eβ . The revelation of those simple Lie algebras is a greatα
fortune in understanding the irreducible representations of g. One direct consequence is
that for any vector space the representation of Hα will have integer eigenvalues.
19Here and in what follows we use the short hand notation α2 = ∑ j∈[1,rk ] α(H j) ·α(H j).g
30 Chapter 1. Physical Content
D. Simple roots & root lattice: Cartan’s decomposition involves a sum over d− rkg roots
α ∈V ∗h contained in the selection R, whereas its vector space V ∗h is only rkg-dimensional.
To construct a suitable basis on the root lattice ΛR, the span of R,
{ ∗ }Vg
Λ ..R = ∑ cα ◦C·V∗ α |cα ∈C ⊆V ∗g ,
α∈R
we have to implement an ordering principle on ΛR to define positive and negative direc-
tions within the root lattice. This requires an embedding of the real numbers R into ΛR,
in fact a linear one to preserve the vector space properties. In detail we have to select a
linear function fR : ΛR → R that gives rise to a splitting:
R= R+∪R− , with R+ = {α ∈ R | fR(α)> 0} and R− = {α ∈ R | fR(α)< 0}
A direct way to define such a function fR involves an ordering of the basis elements in h,
that is a labeling H1, · · · ,Hr and then define fR(α) positive (negative) whenever the first
non-vanishing value of α(Hk) has positive (negative) real part. Though fR introduces an
auxiliary ingredient into the prescription, it turns out that the following result does not
depend on its particular choice. Having divided the roots in positive ∈ R+ and negative
∈ R− roots what remains is the reduction of R+ to its basis elements. In a final step we
search for simple roots in R+ which are elements that can not be expressed in terms of
two or more other positive roots, i.e.
R+ ..= {αS ∈ R+S |∄β ,γ ∈ R+ : αS = β ◦V ∗ γ}
It turns out that R+S forms a linearly independent basis for ΛR and all positive (negative)
roots can be written as
α = ∑ n j ◦ SC·V∗ α j
j∈[1,rkG]
with non-vanishing positive (negative) integers n j.
There is a nice schematic description to characterize Lie algebras using Dynkin diagrams
[14]. We start with the Cartan matrix of simple roots given by
2
M ..= ∑ αSjk 2 j (Hr) ·α
S
k (Hr)αk r∈[1,rkg]
which is a non-singular square matrix of rank rkg. Furthermore, it can be shown – using
the Weyl group – that it fulfills the following constraints [12]:
∀ j 6= k ∈ [1, rkg] : M jk ∈ [0,−1,−2,−3], M jkMk j ∈ [0,1,2,3], M j j = 2
The important information encoded in the non-diagonal elements can be cast into the
form of a graph where each circle corresponds to a simple root and the number of con-
necting lines illustrate their relations, thus represents the off-diagonal values. Hereby the
convention is such that whenever M jk > Mk j holds true, an arrow is superimposing the
lines pointing towards the simple root αSj . If and only if the Dynkin diagram is connected
we are dealing with a simple Lie algebra which can fall into nine different classes.
As an example consider the Lie algebra so(5) with simple roots αS1 and α
S
2 (using the
notation that H∗j (Hk) = δ jk), Cartan matrix M
so(5), and corresponding Dynkin diagram:
( )
αS
2 −2
2 = H
∗
1 ◦V ∗ (−H∗) ∧ αS = H∗ : ⇐⇒ Mso(5)2 1 2 ≡ −1 2
⇐⇒
1.2 Symmetries 31
Every form can be recovered by any other up to isomorphisms and so they are all equiva-
lent descriptions of the same underlying Lie algebra.
E. Weights, highest weights & physical charges: It now appears that once we have suc-
ceeded in characterizing the adjoint representation of a Lie algebra (as done in the previ-
ous steps) it is straightforward to deduce irreducible representations on any finite-dimen-
sional vector spaceW . Therefore consider a faithful but otherwise arbitrary representation
ρ : g → Gl(W ). We can adapt the results of the adjoint representation to the more gen-
eral case while only slightly changing the notation to retain the special status of the root
structure.
Let us start by looking for some special vector in W called a highest weight vector:
Definition 1.2.24 — Highest weights. Let ρ : g → Gl(W ) be a finite-dimensional
representation of the Lie algebra g onW ≡ (A,◦W ,K,◦K·W ). Let h⊂ g be the Cartan
subalgebra and g= h⊕ (⊕α(gα∈R) its decomposition into root spaces gα with R+ its
positive roots. Furthermore, let µ• : k→ C.
An element w ∈W is a highest weight vector of g onW :⇔
∀H ∈ h : ρH(w) = µw(H)◦C·W w (eigenvector of h)
∀α ∈ R+, : ρEα (w) = eW (w is in the kernel of gα )
Any finite-dimensional representation of a semisimple complex Lie algebra gives rise to
at least one highest weight vector and if it is in addition irreducible there is a unique
such element. Instead of abusing the notation of roots, which are reserved for the adjoint
representation, one denotes eigenvalues of ρH as weights µ , and in the special case of the
highest weight vector it obtains the supplementary adjective ‘highest’. Notice that when
choosing the linear combination Hα the weights µ(Hα) have to be of integer type.
A highest (or dominant) weight vector symbolizes the anchor where our formalism is
attached to. While elements of h simply scale this vector w, all positive root vectors
Eα map w to the identity eW . The only possibility for a non-trivial transformation of a
highest weight is an (successive) application of negative root vectors E−α where α ∈ R+
(so −α ∈ R−). Remarkably, any irreducible representation of the Lie algebra g can be
obtained by this latter procedure while restricting to simple negative roots only. Thus
given any representation ρ : g→ Gl(W ) and a highest weight vector w w.r.t. µ ∈ h∗. The
subspace related to the associated irreducible representation ρ irrep : g→ Gl(Wirrep) is then
given by:
µw
Wirrep ≡{w, ρE (w), ρ (w), · · · , ρ (w)} , αS ∈ R+, n ∈N−αS E E j 01 −αS2 −n αSr r −···−n1αS j S1
The string of simple root compositions −n Srαr −·· ·−n S1α1 will ultimately terminate for
some values of n j which is bounded by the fact thatWirrep is of finite dimension. Due to
the uniqueness condition mentioned above each irreducible representation is fully char-
acterized by its highest weight. Any such highest weight can be uniquely rewritten in
terms of a linear combination of non-negative integer coefficients factoring the so called
fundamental weights given by ω .r(HαS) .= δr j. Notice that in physics, highest weightsj
correspond to the respective charges of the underlying gauge theory.
F. Multiplicity of weights: Characters provide a condensed form of the information con-
tained in a representation and their knowledge allows us to decompose an arbitrary repre-
sentation into irreducible ones. In case of a compact Lie algebra, the character is given by
Weyl’s character formula, which we will not present here, because far more terminology
on e.g. the Weyl group is needed. We focus here only on the multiplicity of a certain
32 Chapter 1. Physical Content
weight within a representation and give the dimensionality associated to them. First of
all, the dimension of a finite representation with highest weight µW is given by the Weyl
dimension formula:
∏ W Ww
dim µ( )≡ α∈R+ K(µ +ρ ,α)Wirrep ∏α∈R+K(ρW ,α)
with ρW ≡ 12 ∑α∈R+ α . This provides the initial point from which to successively derive
w
the multiplicity of the weight spaces associated to a weight µµ withinWirrep, the Freuden-
thal’s multiplicity formula:
( )
‖µW +ρW‖2−‖µ +ρW‖2 ·dim(W µ) = 2 ∑ ∑ K(µ + j ·α ,α)dim(W µ+ j·α)
α∈R+ j≥1
The full decomposition of any representation into its irreducible counterparts can be car-
ried out by the Steinberg formula [12].
1.2.4 Examples of Groups
In this final remark on symmetries and groups, we will exemplify the above mentioned concepts
for some relevant groups essential in the study of physics. We start with finite groups, then move
on to infinite groups that however can be associated to a finite dimensional vector space, and
finally conclude with the isomorphism group of functions, being infinite with any respect.
Symmetric groups and Young tableaux
The symmetric groups Sn are the prototypes for any finite group due to their relation to the
isomorphisms of sets, the bijections:
Sn = ({ f : X → X | f is bijection },◦) with #(X) = n< ∞
Since we have a finite number of elements the set of Sn has also finite cardinality n!. The
cyclic notation is a very instructive description of bijections f : X → X with #(X) = n< ∞ that
allows to classify the group elements of Sn. The underlying observation is that any bijection
can be represented by a successive application of transpositions or in general permutations in
the following way (1)f ≃ (a1 · · ·
(1) (2) (2)
a j )(a1 · · ·a j ) · · · with
(r) (r) (r) (r)
f (an ) = an+1 and a j +1 = a1 . Since1 2 1
interchanging the order of cycles does not affect the function we choose the convention to order
them in a natural way by requiring jn ≥ jm whenever n < m.20 The group operation of two
bijections g and f on X is the usual composition of functions ◦S ≡ ◦, thus in cyclic notationn
(1) · · · · · · (ra) · · · ◦ (1)we have (a1 ) (a1 ) (b1 · · · ) · · ·
(r
(b a
)
1 · · · ).
For finite groups it is usually the first step to determine the conjugacy classes, since they
correspond to the number of inequivalent irreducible representations. It turns out that bijections
are categorized under conjugation into several classes each sharing the same partition of n,
meaning the possible ways to split n be a sum of real numbers. For a systematic approach
we go back to the cyclic notation and order their internal structure such that largest blocks of
permutations always stand on the left as mentioned above. In the case of S6, for instance, instead
of writing f ≃ (1)(352)(46) we reorder this cycle such that the largest permutation comes to
the left, the second largest follows, etc. giving f ≃ (352)(46)(1) – an element of the conjugacy
class CS(3+ 2+ 1). Hence, bijections that split into different cyclic shapes are elements of
different conjugacy classes, e.g.
g≃ (234)(156) ∈CS(3+3) while h≃ (156)(24)(3) ∈CS(3+2+1)
20Notice that each number from 1 to n can only appear once in this cyclic notation, since f is a bijection.
1.2 Symmetries 33
The labeling of the elements is irrelevant for this classification and usually each conjugacy
class is represented in a diagrammatic way by means of Young diagrams. This method is best
illustrate listing some examples, say this time for S4:
λ1 ≡CS4(4)≃ , λ2 ≡CS4(3+1)≃ , λ3 ≡CS4(2+2)≃ ,
λ4 ≡CS4(2+1+1)≃ , λ5 ≡CS4(1+1+1+1)≃ (1.9)
A particular representative is given by a filling of the boxes with the elements of X .
The symmetric group is actually an exceptional case for which we can move on and system-
atically construct its irreducible representations. Therefore, we have to define a standard Young
tableaux, some special representative of a conjugacy class whereby the filling is in a particular
order, precisely it decreases along the row from left to the right and along a column from top to
bottom. For instance λ ≡CS4(3+1) contains three standard tableaux, denoted (λ )ST,
T1 ≡ 1 2 3 ≡ 1 3 4 1 2 4T2 T4 2 3 ≡ 3
Together they generate the span of the irreducible representation corresponding to λ ≡CS4(3+
1), referred to as Specht module. The number of such standard tableaux can in general be
computed directly from the Young diagram itself, using the product of hook length of each box.
For a given conjugacy class λ ≡CS (nn 1+ · · ·+nr) we obtain:
S(n)
d ..λ = #{ |
n! (λ)
Tj Tj ∈ (λ )ST} ≡ , hook ..j,ℓ = (n(λ) j− ℓ)+ ∑ Θ(nk+1− ℓ)
∏ j,ℓ hook j,ℓ k≥ j
S(n)
Thus the Specht module represents a dλ -dimensional C-vector space explicitly given by
 
 
Specht(λ) ..= ∑ cr ∑ sgn(π) [π ◦T
−1
r ◦π ]∼row |Tr ∈ (λ )ST ∧ cr ∈ C
S(n)
r=[1,d ] π∈col(λ)
λ
whereby col(λ ) and row(λ ) are the stabilizer group of the Young diagram that interchange
elements between columns or rows, respectively, without deforming the shape of λ . Corre-
spondingly ∼row is the equivalence relation of Young tables under the action of row(λ ).
Finally, the full character table can be derived using the Murnaghan–Nakayama rule [21]. It
is again based on the Young diagrammatic approach constructing some (non-standard) fillings
from which to compute the characters. Basically, for an irreducible representation represented
by a Young diagram λ , one obtains the character for a conjugacy class µ ≡ CS (n1 + · · ·+n
nr) in the following way. Find the set of possible Young tableaux of λ with n j boxes filled
with j. While equally filled boxes have to be connected one has to avoid 2× 2 squares of
equal entries, yielding what is known as border-strip tableaux. Then one determines their total
(µbs)
heights H(µbs) ≡ − (µ )( 1)r−∑ j=[1,r]Hr , whereby H bsj agrees with the number of rows the entries
j populate. The characters for elements of µ in the representation λ are thus determined by
χλ (g ∈ µ) ≡ ∑ H(µbs)µ . This completes the full investigation of the symmetric group for anybs
finite n. We will see that these considerations have also impact on infinite groups, especially
those which play an important role in today’s physics.
34 Chapter 1. Physical Content
Let us conclude by listing the full character table of S4 obtained by the described methods.
We will use (the order of conjugacy classe)s given in eq. (1.9) so that each χ
λ j defines a tuple
with entries χλ j(g ∈ λ1), · · · ,χλ j (g ∈ λ5)
χλ5 = (1,1,1,1,1) (trivial)
χλ4 = (−1,0,−1,1,3) (standard)
χλ3 = (0,−1,2,0,2)
χλ2 = (1,0,−1,−1,3) (standard + signed)
χλ1 = (1,−1,1,−1,1) (signed)
Similarly, for S2 we obtain:
χ (g ∈ ) = 1 χ (g ∈ ) = 1 (trivial)
χ (g ∈ ) =−1 χ (g ∈ ) = 1 (signed)
Notice that in both cases the characters are orthogonal, as required.
Young diagrams are a great tool specially tailored to the need of determine irreducible rep-
resentation, decomposition, etc. of the symmetric group. We will encounter them at various
other occasions, where an intrinsic relation to the symmetric group is present. For a more detail
account on Young diagrams, Young tableaux, and Young operators we refer to [11–15]
The general linear group Gl(d,K)
Leaving the case of finite cardinality, in the next step we consider isomorphisms on finite-
dimensional vector spaces: the group of linear bijections. Finite vector spaces are uniquely
characterized by their dimensions and thus the general linear group can be defined on any of its
representatives, say V with dim(V ) = d:
Gl(d;K) ..= ({ f :V →V | f linear bijection },◦) with dim(V ) = d < ∞
For d = 0 we recover the symmetric group of order #(V ) = n.
The (reducible) representations of the linear group Gl(d;K) can be constructed on the basis
of tensor products V ⊗·· ·⊗V →K of the associated vector space V of dimension d. Tensors
are multilinear maps of vectors defined by some equivalence class over the Cartesian product
of V ’s. For a rank r tensor T (r), the group action of g ∈ Gl(d;K) is described by
ρ (r)g(T )(v j1 , · · · ,v ) ..= T (r)j (g(v j1), · · · ,g(v j )) ∀v j ∈Vr r
While the general linear group is unconstrained and there is thus no seed of irreducibility present,
one can show that permutations of the arguments actually commute with ρg due to the multi-
linearity of the representations. Explicitly for any tensor T (r) of rank r the (linear) symmetric
group action is given by
ρ̃ (T (r)σ )(v j1 , · · · ,v j ) ..= T (r)(vσ( j ), · · · ,vσ( j )) ∀v j ∈Vr 1 r
(r)
and we denote Tσ ≡ ρ̃σ (T (r)) for simplicity. Thus, the irreducible representations of S(r) lead
to a reduction of any r-tensor on the basis of some (anti-)symmetrization procedure. In fact, it
turns out that this decomposition is actually also irreducible in Gl(d;K) [11]. With this knowl-
edge, we can use all the methods we have at our disposal to characterize the symmetric group
1.2 Symmetries 35
and apply them to Gl(d;K), in particular the diagrammatic description via Young tableaux. Any
tensor of rank r can thus be split into a set of irreducible tensors classified by some associated
Specht module. For instance, rank 2 or rank 3 tensors are represented as follows
T (3) = T +T +T T (2) = T +T
The subscript indicates the invoked symmetrization, meaning the combination of permutations
that act on the respective arguments. Thus T stands for the symmetric rank 2 tensor while
T represents the anti-symmetric part of T (2). If the underlying field is the space of complex
numbers, K≡C, transposed Young diagrams correspond to complex conjugate representations.
Notice that the row count of any tensor can not exceed the dimension of the vector space, for
then we have at least one anti-symmetrization of equal indices.
What differs from the prescription of S(n) in the present case is the computational rule for
the dimensionality of an irreducible representation. The reason is simply that the symmetriza-
tion contains only half the truth, while the remaining freedom is hidden in the tensor structure
(r)
itself, that has in general dr free parameters. Given an irreducible tensor Tλ for some Specht
module corresponding to the Young diagram λ ≡CS (n1+ · · ·+nr) we obtain the dimensional-n
ity as follows:
Gl(d,K) ≡ ∏ j ((d− j+n j)!/(d− j)!)d with hook(λ) ..λ = (n − ℓ)+ Θ(n +1− ℓ)(λ) j,ℓ j ∑ k
∏ j,ℓ hook j,ℓ k≥ j
Next, we discuss some other, still uncovered aspect of Young diagrams that is intrinsically
encoded in this diagrammatic representation. The so called Clebsch-Gordan series provide a
way to decompose inner products of irreducible representations such that the result is again
a sum of such fundamental building blocks. By means of Young diagrams this split is easily
performed by simply following some very elementary multiplication rules of boxes. One of
the diagrams that participate in the inner product is labeled row-wise with distinct numbers or
letters. Then one starts to row-wise successively attach boxes of this enumerated diagram to
the other while avoiding equal letters or numbers in the same column and constructions that
are not Young diagrams at all. In the end one dismisses all diagrams where, counting from
right to left and top to bottom, at any point we have more labels of a certain kind than from
any preceding ones, i.e. more b’s or c’s then a’s, for instance. Furthermore equally labeled
diagrams are condensed to a single tableaux and finally one lifts the enumeration and yields
the full Clebsch-Gordan series. If there are more then two irreducible representations involved
in the inner product, one proceeds pairwise. As an example consider the following product of
S(2):
( )
× ≡ × a = a ⊕ × b
b a
= a ⊕ ⊕ a ≡ ⊕ ⊕
b a b
b
Upon only slightly changing the notation we can thus generate irreducible representations of
Gl(d;K) from existing irreducible tensors of lower rank.
Due to the simplicity of this diagrammatic technique, the geometric side falls a bit behind.
In certain situations, however, the topological or differentiable aspect of the Lie structure of
Gl(d;K) is far more appropriate especially in contact with differential geometry. Considered as
a Lie group Gl(d;K) is a d2 dimensional smooth non-compact manifold. When K is complex
this Lie group is connected, in the real case however there are two disconnected components,
36 Chapter 1. Physical Content
one Gl+(d;K) associated to matrices with positive the other Gl−(d;K) to matrices with negative
determinant. The Lie algebra consists of all d×d K-matrices, that do not have to be invertible,
and the canonical commutator as Lie bracket:
gl(d;K) ..= ({g :V →V |g is linear ∧ dim(V ) = d}, [g , f ]gl = g◦ f − f ◦g)
The intrinsically relation of Gl(d;K) to vector spaces makes it such an important Lie group
and some of its subgroups play a central role in many branches of physics. In particular the
maximal compact subgroup O(d) and U(d) of Gl(d;R) and Gl(d;C), respectively, and their
further descendants appear in almost every field of physics. As an example, let us consider the
most interesting candidate that appears in the formulation of the Standard Model of particle
physics, the special unitary group.
The special unitary group SU(d) derives from the unit(ary g)roup U(d) for which we have
to consider a vector space equipped with an inner product • , • . On the basis of this scalar
V
product we can judge whether or not an element of Gl(d;C) is unitary. Its group structure is
given by
( ) ( )
U(d) = ({g ∈ Gl(d;C) | g(v) , g(w) = v , w ∀v,w ∈V},◦)
V V
As such, it is the maximal compact subgroup of Gl(d,C) as mentioned above and in the finite-
dimensional case it can be identified with the matrix algebra of d× d-unitary matrices. The
simplest example, namely U(1), is an Abelian group famous for its association to Quantum
Electrodynamics (QED). Concerning SU(d), the adjective ‘special’ indicates that it is a sub-
group of U(d) containing only elements with determinant equal to one, i.e.
SU(d) = ({g ∈ U(d) | det(g) = +1},◦)
Notice that this condition is the only one that fixes the determinant completely while retaining
the group structure of U(d). Concerning its topological aspects, SU(d) inherits compactness
from U(d) and is furthermore simply connected as a d2 − 1 real smooth manifold. From the
algebraic side its center is isomorphic to the cyclic group Zd consisting of the d-square roots of
unity times 1. Let us now turn our focus on its Lie algebra and classify SU(d) by means of the
properties of su(d) given by
SU(d) = ({g ∈ gl(d;C) |Tr(g) = 0 ∧ (gT)∗ = A}, [g , f ]gl)
Pursuing the strategy of deducing the root space for SU(d) requires at first a semisimple Lie
algebra, a criterion that is perfectly satisfied for su(d) which in fact is also simple. Identifying
the subalgebras SU(2) within SU(d) is a straightforward calculation that for d > 1 yields the
following Dynkin diagram
α1 α2 α3 αd−3 αd−2 αd−1
Hereby, αr denotes the roots, hence the rank of the Lie algebra, i.e. the dimension of its Cartan
subalgebra, is equal to d− 1. In physics, the Standard Model of particle physics is built on
SU(3)×SU(2)×U(1), explaining the prominent status of SU(d) in physics literature.
While U(d) and SU(d) have important applications in modern QFT their real counter-
parts O(d) and SO(d) are closely tied to the geometry of spacetime. They describe certain
linear transformation of real vector spaces and as such are very important especially in classical
physics or the theory of quantum gravity. Besides these interrelations of (special) orthogonal
groups and natural science, the concept of spin in quantum mechanics is constructed by means
1.2 Symmetries 37
of the double universal covering of SO(d), denoted spin group Spin(d). Their implementation
on smooth manifolds M as spin structures, requires the first and second Stiefel-Whitney class
of M to vanish. It can be shown that the Lie algebra SO(d) of SO(d) is semisimple and gives
rise to the following Dynkin diagram for :
αr
α1 α2 α3 αr−3 αr−2
αr−1
for SO(2r)
α1 α2 α3 αr−2 αr−1 αr
for SO(2r+1)
The Cartan subalgebra has thus rank d/2 or (d−1)/2 for d even or odd, respectively. Notice that
for the orthogonal group O(d) and its descendants there is an additional operator that commutes
with the group action, namely the contraction. This is a typical example where irreducible
representations of a group (here Gl(d)) can be further reduced when restricted to a subgroup
(here O(d)). For details on the irreducible representations of O(d) we refer to [11]
Diffeomorphism group Diff
We have already mentioned that isomorphisms of a category form a group, that is they can be
successively joined to obtain new isomorphisms in the same category and by virtue of their bi-
jectivity there exists an inverse for each of them. In the previous two cases we have encountered
two special types of isomorphism groups, the class of finite group over finite sized sets and the
class of finite dimensional Lie groups over finite vector spaces with infinite cardinality. In the
final step of this series of examples we consider the next level of extension for which the group
is even not locally compact. This may happen when the dimension of the underlying vector
space turns infinite, as for example for function spaces.
We will give a brief overview for the diffeomorphism group, the isomorphisms on a smooth
manifold, that has naturally a distinguished status in the present work, however the construction
of irreducible representation is beyond the scope of this thesis. So, let us assume a smooth
manifold M. The diffeomorphism group acts on the respective equivalence class of M and is
defined by
Diff(M) ..= ({ f :M →M | f diffeomorphism },◦)
In order to describe the group structure of Diff(M)we have to give one more definition for some
special topological spaces that is also relevant in the construction of field space:
Definition 1.2.25 — Fréchet-space. Let V ≡ (X ,◦V ,K,◦V ·K) be a vector space. A topo-
logical vector space V ≡ (V,τV ) is a Fréchet-space :⇔
∃dV :V ×V → R metric, with dV (v◦V a,w◦V a) = dV (v,w) :
(V,dV ) is metric complete ∧
∀U ∈ τV ∀u ∈U ∃ε > 0 : {v ∈V |dV (u,v) < ε} ⊆U
∀U ∈ τV with eV ∈U : (∀u,v ∈U ∀ t ∈ [0,1] : (t ◦V ·K u)◦V ((1− t)◦V ·Kw) ∈U) ∧
(∀u ∈U ∀ t < 0 : (t ◦V ·K u) ∈U) ∧
(∀v ∈V ∃ t > 0 : (t ◦V ·K v) ∈U)
38 Chapter 1. Physical Content
The last three requirements for neighborhoods of the identity imply that V is a locally con-
vex.
In the same way we introduced a topological manifold along with a smooth differential struc-
ture, we can proceed and define a Fréchet manifold that simply replaces the locally Euclidean
property with the requirement of being locally a Fréchet space.
It turns out that the diffeomorphism group equipped with its weak topology can be described
by a Fréchet manifold if M is compact, thus it carries additional geometrical information that
allows to apply the machinery of differential topology. In fact, Diff(M) for M compact is a
regular (Fréchet) Lie group with a Lie algebra equivalent to the space of smooth vector fields on
M and the commutator given by the negative Lie bracket. Hence, infinitesimal transformations
close to the identity of Diff(M) can be described by the Lie derivative.
The classification of its irreducible representations is based on the general linear group act-
ing on each tangent space of elements in M and the transitive property of the natural group
action. Usually, one considers only compact manifolds M and generates a chain of represen-
tations for subgroups of Diff(M) that ultimately give rise to an induced representation for the
whole group. For some d-dimensional connected smooth manifold the orientation preserving
diffeomorphisms Diff0(M)21 give rise to an equivalence of the underlying spaceM and the quo-
tient group Diff0(M)/Diff1p(M), whereby p ∈M and Diff1p ≡ {g ∈ Diff0(M) |g(p) = p} is the
stabilizer subgroup of Diff0(M) at p. Decomposing smooth functions by means of jet equiva-
lence generates a chain of normal subgroups for which specific representation might be deduced.
On the other hand, there are investigations of non-compact manifolds invoking measure theo-
retical concepts in studying unitary irreducible representations of Diff(M), usually based on a
sequence of compactly supported diffeomorphisms. For more details we refer to [22, 23].
1.3 Field space
We have set the grounds to introduce fields into our formalism. Most of our today’s understand-
ing of fundamental physics is closely tied to the revelation of symmetries attached to objects
living on spacetime. This combines the algebraic component of group theory with the topolog-
ical and differential complex of smooth manifolds culminating in the concept of fiber bundles.
Lie groups studied previously already build a bridge to this end and it is the aim of this section
to extend this connection to smoothly affiliate with the differential structure of spacetime.
In this context a global notion of smoothness for functions can only be answered using
fiber bundles and we will start this section by introducing the necessary notation. As a special
and important example we consider the tangent space of smooth manifolds and thereby fill
the gap between the interrelation of Lie groups and their Lie algebras. Having established the
mathematical framework we will see how these concepts fit nicely into the physicist’s language
of quantum field theory. From the presented perspective it is quite clear what constitutes the
differences of interaction mediating- and simple matter fields. In the final part of this section
we take a closer look at the resulting field space and the role of diffeomorphism invariance.
Once equipped with a thorough understanding of physical fields, the remaining sections of this
chapter investigate endomorphisms and functionals of field space, paving the ground to the
foundations of quantum field theory.
This section is based on a list of excellent books on field theory or differential geometry,
[2–4, 24–28].
21Orientation preserving transformations are in the same homotopy class as the identity.
1.3 Field space 39
1.3.1 Fiber bundles
Our objective is to define smooth functions on an arbitrary curved spacetime M. The natural
candidates to populate spacetime with content are fields, which allocates to each position in
space and to each instance of time a particular value in a certain target space. The motivation
of fiber bundles is found in the subtleness to consistently combine both differentiable structures
to a global notion that will ultimately lead us to the formalism of gauge theory.
In the construction of spacetime the distinction between local and global requirements
opened up a variety of new structures described with the concept of smooth manifolds. While
the familiar picture of Euclidean space is retained in the vicinity of every point, the global struc-
ture in general reveals entirely different properties. All this information is encoded in the set of
functions that are called smooth within this manifold, i.e. C∞(M) and redefining this class will
affect the notion of differentiability on spacetime.
Fields ‘communicate’ between two smooth manifolds, spacetime and a target space which
actually defines the physical character of a field (e.g. a Higgs-, scalar-, . . . field). It turns out,
that in general sewing together spacetime and the target space of fields to a kind of product
manifold is not unique at all, but in fact gives rise to a number of distinct topological or smooth
manifolds that all differ in their notion of differentiability. The reason for this variety can be
found in regions of overlaps on the underlying manifolds, where the global notion of these
concepts deviate from the Euclidean versions due to additional twists in the combined total
space. In mathematics the notion of product topologies or product manifolds is reserved for
the so-called trivial bundle. Given two topological spaces M ≡ (XM,τM) and F ≡ (XF ,τF), the
product space is given by
M×F ..= (XM×XF,τM×F ≡ {(UM,UF) |UM ∈ τM ∧UF ∈ τF})
For products of smooth manifolds an additional Cartesian product of charts has to be imple-
mented.
However, gluing together the local structures in general results in a global object with a
topology and differentiability different to the trivial case. The corresponding information is
condensed in the concept of fiber bundles:
Definition 1.3.1 — Fiber bundles. Let E ≡ (XE ,τE), M ≡ (XM,τM) and F ≡ (XF ,τF) be
topological spaces, and πE : XE → XM be a surjective continuous map. The ordered pair
(E,M,F,πE) is a fiber bundle :⇔
∀u ∈ XE ∃UM ∈ τM, πE(u) ∈UM, ∃ϑ j : (U j ≡ π−1E (UM), τE |U )→ (Uj M, τM|U )×F :M
ϑ j is isomorphism ∧ proj1 ◦ϑ j = πE
Here proj j : A1×·· ·An → A j with u = (a1, · · · ,an) 7→ proj j(u) = a j is the projection on the
jth component of an ordered multiple. A fiber bundle is abbreviated with E(M,F,πE) or
F → πEE −→M and E ,M, and F are denoted total space, base space and fiber, respectively.
A fiber bundle promotes the concept of a function’s graph from a set theoretic perspective to the
category of topological or smooth manifolds. Intuitively speaking, one attaches a fiber (target
space) to every spacetime point in such a way that locally it reflects the standard Euclidean
picture of continuity and smoothness. Only in the global description fibers can be twisted and
deformed with respect to their neighbors.
The collection {(ϑ j,U j)} is called a local trivialization of E(M,F,πE) and can in principal
be deduced from the comparison of the topologies and smooth structures of E ,M, and the fiber
F . They imitate the coordinate charts of manifolds and reveal the local embedding of trivial
product spaces (UM = πE(U j), τM|U )×F within the global description of τM E .
40 Chapter 1. Physical Content
To extract the set of different fiber bundles compatible with base space M and fiber F , one
has to specify when two fiber bundles give rise to the same global differentiability and therefore
coincide. This definition reflects the concept of a maximal atlas of smooth manifolds, another
indication that fiber bundles are a generalization to products of these objects.
Definition 1.3.2 — Equivalence of fiber bundles. Two fiber bundles E(M,F,πE) and
E ′(N,F ′,π ′E) are said to be equivalent, i.e. E ∼FB E ′ :⇔
∃ fE : E → E ′ isomorphism ∧ π ′N ◦ f ◦π−1E E isomorphism
Within one equivalence class, all fiber bundles will define the same collection of functions to be
smooth and its representatives only differ by a ‘choice of coordinates’. The information of the
underlying differential structure, in particular the content of the transition functions, is encoded
in a set Γ (E) which denotes the collection of (local) sections:
Definition 1.3.3 — Sections. Let E(M,F,πE) be a (smooth) fiber bundle. A morphism
(continuous and/or smooth,. . . map) s :U ⊆ XM → XE is a section of E(M,F,πE) :⇔
∣
πE ◦ s= id ∣M U
The set of smooth (local) sections on E is denoted Γ (E). IfU ≡ XM, then s defines a global
section.
Notice that sections are smooth maps on E that in general fail to be differentiable in the standard
Euclidean way, though it holds locally true.
Tensor bundles
Tensor bundles are an illustrative but also very important examples of fiber bundles for they are
closely related to the intrinsic properties of a manifold. Given any smooth manifoldM it locally
coincides with a Euclidean space Rd by virtue of a set of charts (U j, fU :U j → Rd) that coverj
M. Thus, locallyM is equipped with vector- and metric properties compatible with the standard
notion of differentiability which provides a natural definition of smooth real-valued functions
C∞(M,R)≡{ f :M→R | f ◦ f−1 ∈C∞U (Rd ,R)∀U j ∈ τM}. The structural richness of their targetj
space R gives rise to an algebra on C∞(M,R) that is inherited by pointwise evaluation on R:
( f ◦+ g)(p) = f (p)+g(p) , ( f ◦• g)(p) = f (p) ·g(p), (α ◦R·C∞ f )(p) = α ·g(p)
The generalized notion of a derivative on C∞(M,R) is then obtained by a special subset of the
dual vectors of (C∞(M,R),◦+,R,◦R·C∞): the derivation.
Definition 1.3.4 — Derivations. Let M be a smooth manifold.
A linear map pD : C∞[γ ] (M,R)→ R is a derivation at p ∈M :⇔
∀ f ,g ∈ C∞(M,R) : p pD[γ ]( f ◦• g) = D[γ ]( f ) ·g(p)+ f (p) ·
p
D[γ ](g) (Leibniz-rule)
The additional requirement we impose on the dual vectors C∞(M,R) → R is the well known
Leibniz-rule that reflects the standard derivative properties. The space of derivations also
inherits further structure from R which leads to the algebraic definition of tangent spaces:
Definition 1.3.5 — Tangent spaces. Let M be a smooth manifold of dimension d. The
1.3 Field space 41
ordered pair TpM ≡ (X ,◦T M,R,◦R·T M) is the tangent space of M at p ∈M :⇔p p
∀ pD[γ ] ∈ X :
p
D[γ ] is derivation at p
∀ p , pD[γ ] D ∈ X , ∀ f ∈ C
∞(M,R), ∀α ∈ R :
( [σ ] ) ( )
p
D[γ ] ◦
p p p p p
T MD[σ ] ( f ) = D[γ ]( f )+Dp [σ ]( f ) ∧ α ◦R·T M D[γ ] ( f ) = αDp [γ ]( f )
In other words TpM is a vector space over R of dimension d. On the other hand, TpM can be
interpreted as the equivalence class of smooth curves γ : R→M passing p:22
{[γ ]∼curv |γ ∈ C∞(R,M)} with{ ∣ ∣ }
[γ ]∼ ..curv = σ ∈ C∞(R,M) |σ(0) = p ∧ d ∣ dd γ(t) = d σ(t)∣t t=0 t t=0
Every equivalence class of curves can be naturally identified with a derivation and vice versa.
The connection is established by specifying a map
∣
d ∣
[γ ]∼curv 7→
p
D[γ ] : ∀ f ∈ C
∞(M,R) pD ( f ) ..= ( f ◦ γ)∣[γ ] dt ∣t=0
and then check that pD[γ ] indeed defines a derivation in the above sense. By virtue of the chain
rule this correspondence is w∣ell defined, since each∣element σ ∈ [γ ]∼curv g∣ ives rise to the same
element pD ( f ) = d ( f ◦σ)∣ = f ′[σ ] d (σ(0)) dd σ(t)∣ = f ′(γ(0)) d ∣
p
t t=0 t t=0 d
γ(t) =D[γ ]( f ). There-t t=0
fore, we adopt the short notation [γ̇ ] ≡ pp D[γ ] to describe tangent vectors associated to curves γ
at p. Every tangent space describes a vector space over R, whereby the dimension agrees with
the one of the underlying manifold; in the finite dimensional case every fiber is thus isomorphic
to Rd .
The set of linear functionals, i.e. linear maps from a vector space V to the field R, forms
again a vector space, the dual space V ∗. We will encounter infinite dimensional dual spaces
in section 2.1 when studying measures on field space. In this case the set of continuous linear
functionals is only a subspace of V ∗. In addition, for infinite vector spaces, subtleties occur in
the construction of a suitable basis on V ∗. However, for the construction of geometric objects
on spacetime a treatment of finite spaces is sufficient. Associated to each tangent space we can
thus define a dual description that introduces a differentiable structure on the space of linear
functionals on every TpM:
Definition 1.3.6 — Cotangent space. Let M be a smooth manifold of dimension d. Then
T∗pM ≡ (X ,+,R,◦R·T∗M) defines the cotangent space of M at p :⇔p
∗
T∗pM ≡ (TpM)
Any smooth map f : M → N can be naturally promoted to a smooth function relating the re-
spective tangent and cotangent spaces. The covariant functor f∗ is the so called differential map
given by
f∗ : TpM → T f (p)N, [γ̇ ]p ∈ TpM →7 ( f∗[γ̇ ]p) with ∣∣
( f∗[γ̇ ]p)(g)≡ [γ̇ ]p(g◦
d
f ) = (g◦ f ◦ γ)∣∣ ∀g ∈ C∞(N,R)dt t=0
22Here and in what follows, local charts are implicitly used, as for instance dγ(t)/dt=̂d( fU ◦ γ(t))/dt for some
chart (U, fU ) with γ(t) ∈U .
42 Chapter 1. Physical Content
The corresponding contravariant functor f ∗, also known as pullback, connects the respective
cotangent spaces and by definition reverse the order of composition:
f ∗ : T∗ N → T∗f (p) pM, w ∈ T∗f (p)N 7→ ( f ∗w) with
( f ∗w)([γ̇ ]p) = w( f∗[γ̇ ]p) ∀ [γ̇ ] ∈ T∗p pM
It is easily verified that for f1 : M → N and f2 : N → Q we obtain ( f1 ◦ f2)∗ = f1∗ ◦ f2∗ and
( f ◦ f )∗ = f ∗ ◦ f ∗1 2 2 1 , which indeed classifies f∗ and f ∗ as covariant or contravariant functors,
respectively.
Thus, we have a number of natural vector spaces associated to a smooth manifold M at
our disposal that can be further combined to a multilinear tensor algebra, as follows. At first,
consider the Cartesian or direct product of cotangent and tangent spaces at p ∈ M, i.e. is the
ordered pair
T∗pM×·· ·×T∗M×T M×·· ·×T M︸ ︷︷ p ︸ ︸ p ︷︷ p ︸
m-times n-times
The tensor product establishes a special equivalence class of elements for this direct product
space that gives rise to a multilinear algebra:
∀ [σ̇ ]p, [γ̇ j]p ∈ TpM, ∀w′,wk ∈ T∗pM, ∀α ∈ R :
(w1, · · · ,wℓ, · · · ,wm, [γ̇1]p, · · · , [γ̇n]p)+ ′⊗ (w1, · · · ,w , · · · ,wm, [γ̇1]p, · · · , [γ̇n]p)
∼ (w1, · · · ,(wℓ+w′), · · · ,wm, [γ̇1]p, · · · , [γ̇n]p)
(w1, · · · ,wm, [γ̇1]p, · · · , [γ̇ℓ]p, · · · , [γ̇n]p)+⊗ (w1, · · · ,wm, [γ̇1]p, · · · , [σ̇ ]p, · · · , [γ̇n]p)
∼ (w1, · · · ,wm, [γ̇1]p, · · · ,([γ̇ℓ]p ◦T [σ̇ ] ), · · · , [γ̇ ]pM p n p)
α ◦R·⊗ (w1, · · · ,wm, [γ̇1]p, · · · , [γ̇n]p)∼ (α ◦R·T∗M w1, · · · ,wm, [γ̇1]p, · · · , [γ̇p n]p)
∼ ·· · ∼ (w1, · · · ,wm, [γ̇1]p, · · · ,α ◦R·T M [γ̇n]p)p
(n,m)
The resulting tensor product algebra T M ≡ T∗M⊗·· ·⊗T∗p p pM⊗︸TpM⊗·︷·︷·⊗TpM is the︸ ︷︷ ︸ ︸
m-times n-times
canonical construction of objects that are linear with respect to every constituent while retain-
ing the intrinsic structures of its component spaces. It can be used to define vector bundles
over spacetime that are very closely connected to the intrinsic properties of the underlying man-
ifold.
Definition 1.3.7 — Tensor bundles. Let M be a smooth manifold of dimension d.
Then TM(M,Rd,π ∗ dTM) or T M(M,R ,πT∗M) defines the tangent or cotangent bundle on M,
respectively, :⇔
⊔ ⊔
TM ≡ ∈ {p}×T M T
∗
p M ≡ ∈ {p}×T
∗M
p M p M p
Finally, T(n,m)M is a tensor bundle of type (n,m) on M :⇔
⊗
T(n,m)M = T∗
⊗ ⊔ ⊗ ⊗
M⊗ TM ≡ ∈ {p}× T
∗M⊗ TpM
m n p M m p n
A (global) section on T(n,m)M, TM ≡ T(1,0)M, and T∗M ≡ T(0,1)M defines a (global) tensor,
vector field or differential 1-form, respectively.
Tensor fields take a very prominent role in the formalization of today’s physics, for they provide
a natural description of movement or change on spacetime that has to be only supplemented with
1.3 Field space 43
an interaction to produce any matter fields of Nature. Intrinsically, they carry a rich structure,
in particular a concept of linearity, that is inherited by the space of sections, Γ (T(n,m)M). For
instance, a vector field v ∈ Γ (TM) defines a smooth collection of derivations
v :M → TM , with vp ≡ [γ̇p]p ∀ p ∈M
Thus, a vector field associates to every base point an equivalence class of curves. Notice that
in general γp describe different curves for every p. In the special case that vγ(t) ≡ [γ̇ ]γ(t) holds
true, γ defines the flow of v.
The linearity of TpM can be extended to the space of smooth sections Γ (TM) in a pointwise
manner. In addition, there is a natural commutator for vector fields, the Lie bracket, given by
[ ]
• , • : Γ (TM)×Γ (TM)→ Γ (TM)
L [ ]
(v,w)→7 v , w ..= v(w(•))◦+ (−1)◦R·C∞ v(w(•))L
The space of k-forms constitutes another class of relevant tensor bundles that frequently
appears in physics. It defines the irreducible fully antisymmetric subspace of T(0,k)M and is
denoted Λk(M), respectively. The associated Young operator that projects T(0,k)M onto Λk(M)
is given by the epsilon-tensor.
While vector fields and also k-forms are individual algebraic constructions, they all can be
embedded into the algebra of tensor fields. Additional operations, as for instance the wedge
product, multiplication and addition of tensor fields, allow for transitions from k-forms to k+ r-
forms, for instance, and also interrelates the important concept of Lie derivatives for these
subspaces:
Definition 1.3.8 — Lie derivative. Let M be a smooth manifold and T(n,m)M the tensor
bundle of type (n,m). A map L :Γ (TM)×Γ (T(n,m)M)→ Γ (T(n,m)M) is the Lie derivative
onM :⇔ ∀ v∈ Γ (TM)
[ ]
∀ f ∈ C∞(M,R) : Lv f = v( f ) ∧ Lv , d = 0L
∀ t ∈ Γ (T(n,m)M), w ∈ Γ (TM), α ∈ Γ (T∗j ℓ M) :
Lv(t(w1, · · · ,αn)) = (Lv(t)) (w1, · · · ,αn)+⊗ t(Lvw1, · · · ,αn)
+⊗ t(w1, · · · ,wm,α1, · · · ,Lvαn)
∀ t , t ∈ Γ (T(n,m)1 2 M) : Lv(t1⊗ t2) = (Lv(t1)⊗ t2)+⊗ (t1⊗Lv(t2))
Notice th[at for]vector fields these properties give rise to an identification of Lv(w) and the Lie
bracket v , w .
L
A very special type of smooth manifolds are Lie groups that will describe the symmetry
principals our fundamental interactions are based on. In the discussion of group theory we
already mentioned the close connection between certain Lie algebras and Lie groups that turns
out to be a geometric relation. When considering a Lie group G as a smooth manifold the
full apparatus of tensor bundles applies. The Lie algebra associated to G defines differentiable
changes in the group and thus has to be related to the tangent space of G. In fact, the set of
left-invariant vector fields, as well as the tangent space at eG are all isomorphic to the group
theoretical construction of the Lie algebra g in section 1.3. We are going to complete the story
of the close affiliation of G and g when having a more elaborated formalism at hand to study
tangent spaces: connections and exponential maps.
Principal bundles
In the context of physics, fields are in general required to be smooth functions – usually equipped
with additional symmetry constraints – and thus are sections on the corresponding fiber bundle.
44 Chapter 1. Physical Content
Hence, if we were able to acquire the full knowledge of the set of physical fields, we could de-
duce the associated equivalence class of global topologies and the differential structures of their
target space and spacetime. However, it is usually the other way around in that we first obtain
the mathematical description for some very general constraints, derive all admissible theories,
and finally reduce this number by additional observable data. Thus, in the maze of all such
possibilities, symmetries will guide us to obtain a first (physical) reduction of mathematical for-
malism, hence, once again, group theory enters the very fundamentals of our description of the
Universe. From the very beginning of natural science symmetries have taken an important part
in deducing the laws of Nature and lighting the way to the very foundations of physics. With the
arising of quantum field theory this tied connection has reached a new level in that interactions
and symmetries become severally intertwined. As a result, it is of uttermost importance to keep
the notion of smooth functions invariant under the symmetry transformations that are preserved
in Nature.
R Let us assume G is a symmetry group realized in Nature and we have found a selection of
functions X ≡ { f :M → F} that represent physical fields and are therefore designated to
be smooth. Thus, there should be an equivalence class of fiber bundles [E(M,F,πE)]∼FB
that define a corresponding notion of differentiability, i.e. X ⊆Γ (E). Under the symmetry
action ρ of G on X we obtain a new set of physical fields gX ≡ {ρg( f ) | f ∈ X}. While
in general gX 6= X may hold, it is important that the transformed set satisfies gX ⊆ Γ (E)
to assure that gX still describes physically admissible fields. Hence, we have to make
the notion of differentiability invariant under the relevant symmetries, since otherwise
we may start with a set of physical (i.e. smooth) fields and end up with unphysical (i.e.
non-differentiable) ones.
In order to implement symmetry considerations into the formalism, we have to reduce the
differential structures to those that are compatible with a selection of group actions related to
symmetry constraints observed in Nature. A unified theory of the Universe would consist of
a number of fields interacting with each other. Each field is a function from spacetime to a
certain target space and thus a section in a specific equivalence class of fiber bundles that differ
for the various fields in general. In principal we could decide about the smoothness properties
of the distinct field spaces separately but to obtain a consistent picture that gives rise to the
same differentiable structure and assures that all fields are invariant under the same symmetry
groups we have to merge those smoothness concepts in a natural way. This is accomplished by
a principal bundle that, in a nutshell, contains all essential information of a class of associated
fiber bundles in a very condensed and universal way.
Definition 1.3.9 — Principal bundles. Let P ≡ (XP,τP) and M ≡ (XM,τM) be topological
manifolds. Let G ≡ (XG,◦G,τG) be a topological group and πP : XP → XM a surjective
morphism. The ordered set (P,M,G,πP,ρRP ), denoted P =M
πP×ρRG, is a principal bundleP
:⇔ :
G→ πPP−→M (is a fiber bundle)
ρRP : P×G→ P morphism ∧ πP ◦ρRP = πP (preserves fiber)
∀ p,q ∈ P, ∃̇ g ∈ XG : ρRP (p,g) = q (regular group action)
G is called the structure group of the principal bundle. In the category of smooth manifolds
we have to replace continuity condition with differentiability.
A section of a principal bundle corresponds to a choice of an identity element over each base
space point. It is like fixing the origin of an affine space in a consistent way, meaning that a
global differential structure arises. Remarkably, the very existence of a global section on P is
1.3 Field space 45
equivalent to P being a trivial principal bundle.
It turns out that a Lie groupG, a smooth manifold P, and a free, proper group action ρRP : P×
G→ P are sufficient to characterize a smooth principal bundle. The missing information can be
reconstructed by considering P/G= {{ρRP (p,g) |g ∈G}| p ∈ P} as base space and πP : P→ P/
G with π (p) = {ρRP P (p,g) |g ∈ G} – the orbit map – as the projection.
From this perspective it is straightforward to reduce the structure group G by or to a sub-
group H while retaining the smooth structure of the underlying principal bundle P. The first
restriction results in a new principal bundle P/H = (P/G)πP/H×ρR (G/H). Hereby, the fiber
P/H
of this new construction is reduced to the coset space F ≡ G/H ≡ {{g ◦G h |h ∈ H}|g ∈ G}
which coincides with the quotient group whenever H is normal w.r.t. G. Using the homomor-
phism property of ρRP we indeed find that the resulting base space agrees with P/G. On the
other hand, it might be possible that all transition functions are compatible with a subgroup H
of G, i.e. the structure group can be reduced to H without changing P at all: P =MπPH ×ρR H ≃PH
PG =M
πP×ρR G. This construction which is reasonably denoted the reduction of the structurePG
group to H is, if it exists, in general not unique. Prominent examples occur in theories of grav-
ity where one usually reduces the general linear group to its special orthogonal subgroup, for
instance, thus the tangent spaces carry a fiberwise inner product.
Principal bundles intertwine the differentiable structure of the basis with a symmetry group
in a specific way such that associated fiber bundles share a consistent notion of smoothness.
Given a particular principal bundle P=MπP×ρRG the set of bundle isomorphisms will generateP
an entire class of equivalent principal bundles that constitute an equivalence class [P]∼FB . The
natural and important question to ask at this point, is howmany non-equivalent principal bundles
emanates upon fixing a base space M and a Lie group G, that is we are concerned about the
cardinality and structure of the following set:
PBs(M,G)≡ {[P=MπP×ρRG]∼FB |P principal bundle over M with structure group G}P
In general there is more than one equivalence class, implying that the full set of distinct differ-
entiable structures has cardinality more than 1. The classification of principal bundles over a
given base space is in fact a hot topic in algebraic topology and involves the study of homotopy
theory, classifying spaces, and characteristic classes. The latter are elements of cohomology
groups giving a measure for the twist of fiber bundles in an invariant way. For instance, the
Stiefel-Whitney classes are a collection of topological invariants related to the classifying space
of real vector bundles, the respective Grassmannians (generalizations of the projective space).
Only if the first and second Stiefel-Whitney class vanish, the base space admits a global spin
structure and thus is suitable to construct spinors. A more detailed account on the classification
of principal bundles can be found in [4, 29], for instance.
Associated bundles
With the introduction of a structure group we have set the scene to all possible fiber bundles
that are compatible with the arising smoothness conditions. Choosing a specific equivalence
class out of all principal bundles we end up with a fixed notion of G-invariant differentiability
that also specifies the shape of admissible G-invariant transitions. The strategy of producing
consistent descendants of such a class of principal bundles is given by the concept of associated
bundles.
For a general fiber bundle with typical fiber F transition functions are constraint to be ele-
ments of the corresponding isomorphism group in order to maintain the structure of F . Thus,
one imposes the symmetry invariant description of a principal bundle to a general fiber bundle
by establishing a (faithful) action of the structure group G on the isomorphism group Iso(F), i.e.
ρ : G→ Iso(F). We are mainly interested in fields assuming values in a linear space and thus
46 Chapter 1. Physical Content
focus on fibers possessing a vector space structure in what follows. So, let us start by choosing
a topological vector space V ≡ (XV ,◦V ,F,◦F·V ) that represents the target space F of a certain
(physically relevant) field. The domain should be given by the same base space that underlies
the principal bundle P=MπP×ρRG. Hence, the graph of those fields lives in the Cartesian prod-P
uct XM×XV . In order to define smooth fields we have to endow XM×XV with a suitable notion
of differentiability that in the present case should be compatible with the symmetry constraints
encoded in P. Therefore, we have to attach the structure group G to the vector space V by a
(faithful) left action ρV : G×V →V . The remaining ingredients of an associated bundle are . . .
◮ . . . the total space, given by
( )
E = (P×V )/G≡ {{ ρR −1P (u,g),ρV (g ,v) |g ∈ G}|u ∈ P, v ∈V}
◮ . . . the projection, given by
( )
πE : E →M, ∀q≡ { ρRP (uq,g),ρ −1V (g ,vq) |g ∈ G} ∈ E : πE(q) = πP(uq)
Notice that principal bundles and their associated bundles share the same base spaceM. Further-
more, πE is well defined for πE(q)≡ πP(uq) = πP(ρRP (uq,g)) = πE(q) holds true. To emphasize
the close connection between an associated and its principal bundle we write P×̃ρ V for theV
former.
The most important aspect of associated bundles for gauge theories are the (global) sections,
which ultimately defines the matter sector. Sections in P×̃ρ V have an equivalent description inV
terms of equivariant maps, i.e. functions that preserve the group actions of domain and target
space:
f : P→V, with f (ρRP (u,g)) = ρ −1V (g , f (u)) ∀u ∈ P, ∀g ∈ G
The correspondence between a section σ :M→ E of an associated bundle P×̃ρ V and an equiv-V
ariant map f : P→V is established as follows:
σ f (π (u)) .P .= [u, f (u)] ∨ [u, f (u)] .σ .= σ(πP(u)) ∀u ∈ P
Especially for computational purposes this provides a more convenient and adaptable way to
define and work with physical fields fulfilling a certain symmetry constraint.
Connections and covariant derivatives
So far we have a static construct that will allow us to introduce the physical matter content into
our theoretical setting. What is missing constitutes the interactions between matter fields which
should result in a change of the initial configuration. In mathematics the notion of ‘change’
is closely tied to ‘differentiation’, thus, we will present this concept for fiber bundles in the
following.
For a general smooth manifold M the tangent space fills the role of the linear space needed
to define a notion of differentiation. The directional derivative of smooth functions f :M → R
at a point p ∈M along a certain vector field v ∈ Γ (TM) is given by vp( f ). The specification
of a function can be absorbed in a cotangent vector d f ∈ Λ1(M) implicitly defined by d fp(v) =
vp( f ) for all v ∈ Γ (TM) and p ∈ M. Tangent spaces contain the local information of M
that are essential to extract infinitesimal changes, thus they locally characterize the underlying
base space M. Calling to mind that vector fields are equivalence classes of curves, the relation
between local patches of a manifold and tangent spaces is even intuitively graspable.
The linear operation d is the key to derive all derivatives of higher rank tensor fields using
the Leibniz-rule in a successive manner. Hence, for any principal bundle P seen as a smooth
1.3 Field space 47
manifold interactions are related to the corresponding differential d. In this case however a de-
composition of the the directional derivatives is more appropriate since a principal bundle is a
combination of two individual smooth manifolds that both allow for separate changes. For the
overall picture we have a certain freedom in splitting up the distinct contributions to differenti-
ation that is encapsulated in the vertical and horizontal parts of the tangent space TP.
The generation of the vertical subspace VP⊂TP only affects the structure groupG. Moving
vertical above a fixed base space point p ∈ M means propagating along the fiber such that
πP(u) = p. This is naturally provided by the Lie algebra g of the structure group that describes
infinitesimal displacements vertical to the base space M, i.e.
VuP= {[γ̇ ]u ∈ TuP |πP∗[γ̇ ]u = eTπ (u)M} ≃ g≃ Te G ⇐⇒ VP= ker(πG P∗)P
This space defines a direction on P perpendicular to the base space. The final equivalence
of the vertical tangent space and the Lie algebra of G is generated by fundamental vector
fields:
Definition 1.3.10 — Fundamental vector fields. Let P=MπP×ρRG be a principal bundleP
with G a Lie group and [γ̇ ]u ∈ Te G ≃ g. An element v ≡ [γ̇ ] ♯u ∈ TP is the fundamentalG
vector field w.r.t. [γ̇ ]u :⇔
∀u ∈ P : v ≡ [γ̇ ] ♯ ≡ ρRu u P (u,•)∗[γ̇ ]u ∈ TuP with
( )∣
∀ df : P→ R : v ( f ) = f ◦ρRu P (u,γ(t)) ∣dt t=0
Again, introducing a new structure opens a variety of different possibilities, which in the
present case are reflected in the different choices for HP. Fixing a horizontal subspace amounts
to defining an orthogonal decomposition of the tangent space w.r.t. VP and thus specifies di-
rections on P which are parallel to the base space. While the vertical subspace is given by
the kernel of the projection πP∗, the definition of a horizontal counterpart requires an addi-
tional piece of information that is not already present in the data of a principal bundle. This
missing ingredient is the connection one-form that acts as a projection of TuP onto its vertical
subspace.
Definition 1.3.11 — Principal connection. Let P =MπP×ρRG be a principal bundle withP
G a Lie group. Let Rg : G→ G be the right group action: Rg(h) = h◦G g for all h,g ∈ G. A
Lie-algebra valued one-form ω : TP→ g is a principal connection of P :⇔
∀V ∈ g : ω(V ♯) =V( ) (vertical projection)
∀g ∈ G : ω = adg∗ R∗gω (equivariant)
Notice that the second statement of definition 1.3.11 guarantees that horizontal subspaces HuP
and HρR(u,g)P on the same fiber are interconnected by a linear right action Rg∗.P
The tangent map πP∗ and a principal connection define ‘orthogonal’ constructions in the
sense that they specify the vertical and horizontal component of TuP as the kernel of πP∗ and
ω , respectively. A horizontal vector field induces no change when moving along the base space,
i.e. explicitly we have
HuP≡ {vu ∈ TuP |ω(vu) = eg}
For a given principal connection ω a general tangent vector vu ∈ TuP splits uniquely into its
vertical and horizontal component:
vu ≡ vH Vu +TuP vu with ω(vHu ) = e Vg ∧ πP∗vu = eTuP
48 Chapter 1. Physical Content
A principal connection establishes a consistent classification of change on a principal bundle
that leads to a more detailed account on the properties of an arbitrary vector field on P. In
particular the projection onto its two constituents, the base space and the structure group, are
uniquely defined by means of ω that further introduces the notion of parallelism. For instance,
let us consider a curve on the base spaceM given by γ :R→M. For each u0 ∈ π−1P (γ(0)) there
is a unique curve γ̃ ≡ h(γ) : R→ P, denoted the horizontal lift of γ , that satisfies
πP ◦h(γ) = γ ∧ h(γ)(0) = u0 ∧ [γ˙̃]γ̃(t) ≡ [ dd γ̃(t)] ∈ Ht γ̃(t)P
Such a curve is parallel to the base space and thus selects a set of elements {h(γ)(t) | t ∈R}⊆ P
that lie on the same sheet as u0. This in fact defines an equivalence relation and the repre-
sentatives of [u0] are called the parallel transports of u0. Now consider a particular class of
smooth curves that closes on M, the so called loops defined by γ : [0,1] → M at some point
p = γ(0) = γ(1). Depending on the connection ω on a principal bundle the horizontal lift of
a loop in M will not necessarily give rise to a close loop on P but rather introduces a trans-
formation on the fiber over p. By virtue of the Lie group structure in the vertical direction
this transformation corresponds to an element of G that in general varies for different loops.
Furthermore, the class of all loops emanating at a given point p is denoted
C∞ (M; p) ..= {γ : [0,1]→M |γ smooth, with p= γ(0) = γ(1)}
Loops
The set of all possible transformations that arise by the elements of C∞Loops(M; p) results in the
holonomy group at u ∈ π−1P (p), a subgroup of the structure group G:
{ }
Hol .u .= g ∈ G |∃γ ∈C∞ (M;πP(u)) with γ̃(0) = u ∧ γ̃(1) = ρRP (u,g)Loops
As we will see in a moment, the holonomy group is closely tied to the principal connection by
means of the Ambrose-Singer theorem.
However, let us first collect all the necessary ingredients to couple fields to interactions via
some ω . The first step is the covariant exterior derivative. It provides a consistent notion of
change for equivariant functions and general vector-valued forms:
Definition 1.3.12 — Covariant exterior derivative. Let P=MπP×ρRG be a principal bun-P
dle with G a Lie group and ω a connection one-form on P. Furthermore, let V be a vector
space and β ∈ Λr(P)⊗V be a V valued one-form on P. Then (dω β ) ∈ Λr+1(P)⊗V is the
covariant exterior derivative of f :⇔
(dω β )(v1, · · · ,vr+1)≡ (dβ )(vH1 , · · · ,vHr+1)
The exterior derivative d for β ∈ Λr(P)⊗V is given by:
( )
(dβ ) (v1, · · · ,v j+1r+1)≡ ∑ (−1) vj β (v1, · · · , v̂j, · · · ,vr+1)
j∈[1,r+1] ([ ] )
+ ∑ (−1)ℓ+ jβ vℓ , vj ,v1, · · · ,vL r+1
ℓ∈[1, j]
whereby v̂j denotes the omission of a vector field.
Covariant derivatives embed the notion of differentiation on a principal bundle to any associated
vector bundle. As a very special but likewise very interesting case we may consider the self-
interaction of a principal connection, resulting in the curvature two-form:
1.3 Field space 49
Definition 1.3.13 — Curvature two-form. Let P=MπP×ρRG be a principal bundle with GP
a Lie group and ω a principal connection on P. Then, a bilinear, skew-symmetric g-valued
2-form Ωω ∈ Λ2(P)⊗g is the curvature two-form of ω :⇔
Ωω ωu (v,w) = (d ω)u (v,w) ∀u ∈ P, ∀ v, w∈ TuP
The Ambrose-Singer theorem states that this curvature two-form contains the entire holonomy
information of the connection. More explicitly, the Lie algebra h of the holonomy group
H ≡Holu0 ⊆G at u0 ∈ P is equivalent to the Lie subalgebra of g that is spanned by the values of
Ωωu (v,w) with curves γ ∈C∞Loops(M;πP(u0)) such that γ̃(t) = u for some t. In other words, the
curvature provides a mean to measure the deviation of the initial and final point of a horizontal
lift.
Since the covariant derivative intrinsically carries the information of the underlying prin-
cipal connection there is a more explicit way to write down the curvature, a form known as
Cartan’s structure equation:
Ωω ≡ dω +ω ∧ω ∨ Ωωu (v,w)≡ (dω)u(v,w)+ [ω(v) , ω(v)]g
The second term originates from the subtraction of vertical vector field contributions to the
ordinary differential d. Finally, the Bianchi identities follow from d2 = 0 and the properties of
ω , in particular the zero projection of horizontal fields, ωu(vH) = eg:
( )
(dω Ωω) (v,w,y) = (dΩω)(vH,wH,yH) = d2u u ω +2dω ∧ω (vH,wH,yH) = eu g
Since this holds true for all tangent vectors v ∈ T P we obtain the identity dωΩωu = eΛ3(P) ⊗ eg.
However, notice that contrary to d the covariant derivative dω is in general not nilpotent.
Associated connections
Now, let us assume we are given a smooth manifoldM on top of which we have build a principal
bundle P = MπP×ρRG with a principal connection ω : TP → g. This suffices to construct anP
entire class of new associated bundles, related to some vector spaces V , all sharing a consistent
notion of differentiability with the induced covariant derivative
dω : Λr(P)⊗V → Λr+1(P)⊗V
The tied bound between a principal bundle and its associated descendants also appears on the
level of connections. For any principal connection ω that defines the concept of horizontality
on TP there is a unique induced linear connection on E . Therefore, denote πP.E :V ×P→ E and
(v) (v)
πP.E : P→ E the canonical projections of P to E , explicitly given by u 7→ πP.E(u) ..= [(u,v)] ≡
{(ρRP (u,g),ρ −1V (g ,v))} ∈ E . Then, the induced notion of horizontality on E reads
{
. (v)
} (v)
HaE .= va ∈ TaE |∃wu ∈ ker(ωu)≡ HuP : πP.E∗wus ≡ va with πP.E(u) = a
Due to the G-invariance of the horizontal space this definition is independent on the choice of
v.
From a geometrical point of view, the most interesting associated bundles are tensor bundles
of the base space, in particular TM, for they encode essential information ofM itself and are the
objects we encounter in Riemannian geometry (smooth manifolds with a metric on its tangent
space). A particular well known differential construction is the covariant derivative of tensor
fields ∇ω : Γ (TM)×Γ (T(p,q)M)→ Γ (T(p,q)M) that we are going to relate to a principal con-
nection ω on P. Essentially, we have to project ω on the tangent space of M using the intrinsic
information of P and G.
50 Chapter 1. Physical Content
Let us consider an arbitrary associated vector bundle E ≡ P×̃ρV for a moment. By the
uniqueness of the horizontal lift, a section s : M → P provides a local isomorphism between
Tγ(0)M and Hs(γ(0))P, i.e. with u0 = s(γ(0)) we have
vγ(0) ∈ Tγ(0)M ←→ γ : R→M ←→ hs(γ) : R→ P ←→ hs∗vu0 ∈ Hs(γ(0))P
Due to the right invariance of horizontal spaces this definition is in fact independent on the
section s.
The covariant derivative on E acts on the exterior algebra of V -valued r-forms. Any section
σ ∈ Γ (E) can be identified by an equivariant function fσ : P→V , a V -valued 0-form.
dω fσ ∈ Λ1(P)⊗V , with [u, fσ (u)] = σ(πP(u))
(dω fσ )u (v) = (d f
H
σ )u (v ) ∈V ∀ v∈ Γ (TP)
The important observation at this point is the equivalence between the horizontal subspace HuP
and the tangent space Tπ (u)M ≃ Rd of the base space. It suggests the following definition ofP
∇ω in case of E ≡ T(p,q)M:
( ) ( )
(∇ωvβ )
..
π (u) = d
ω f (h v) = d f (h (p,q)β ∗ β ∗v) ∀β ∈ Γ (T M), v∈ Γ (TM)P u u
Notice for that definition to hold, we need a group action ρ : G → Iso(Rd × ·· · ×Rd) with
d = dim(M) that ensures T(p,q)M being an associated bundle of P. Due to its vector space
character the natural structure group related to tensor bundles is based on Gl(d;R).
As an example, let us consider the tangent space TM, its smooth sections being vector fields
that locally are described by vp ≡ Dγ ;p. We thus obtain:
(∇ωvw)p ≡ (d fw) ss(p) (h∗(v)) ∈ TpM with [u, fw]≡ w(πP(u))
This definition provides a means to study the effect of interactions on the tangent space of
spacetime or more general of base space. It further allows to establish a relation between similar
shaped vector fields in a sense that ∇ωvw= eΓ (TM) holds true, i.e. w is parallel transported along
v. Very special vector fields are those based on curves that define the ‘closest’ path between
different points on the manifold with respect to the interaction ω , the geodesics which are
parallel to themselves everywhere:
Definition 1.3.14 — Geodesics. Let P≡ P=MπP×ρRG be a principal bundle with G a LieP
group and ω a principal connection on P. A curve γ : S ⊆ R→M is a geodesic w.r.t. ω :⇔
∇ωvv= eΓ (TM) with vγ(t) = [γ̇ ]γ(t) ∀ t ∈ S
The moment we have a distance measure on M induced by a Riemannian metric that is related
to the principal connection such that it fulfills the Levi-Civita property, corresponding geodesics
are the shortest paths to interconnect points on base space. Whether or not there is always such
a unique shortest path between two points on M and to what extend we can prolong a curve so
that it remains geodesic, is up to the manifold and its properties.
Levi-Civita connection and Riemannian metrics
The covariant exterior derivative and the linear connections it induces on associated vector bun-
dles provide a new classification scheme for principal connections. Imposing mathematical
constraints on the action of ∇ω or in general of dω restricts the set of all possible connections ω
on P to some subset which fulfills these additional requirements. One of the usual implemented
assumptions is about the vanishing of the torsion tensor – a concept encountered when consid-
ering principal bundles P which are frame bundles of a certain base space M, i.e. its structure
1.3 Field space 51
group has to be a subgroup of Gl(d;R) with d ≡ dim(M). The special status of frame bundles
is due to its close relation to the geometry of M which results in the existence of a Solder- and
thus torsion-form on P. Hence, in order to uniquely specify a principal connection on P by
its pullback on M we have to impose conditions on the torsion form and its action on a set of
smooth global sections
The Levi-Civita constraint is a very prominent example of this identification process where
a Riemannian metric, g ∈ Riem(M)⊂ Γ (T(0,2)M), plays a central role. Thus, let us first define
the important concept of a Riemannian manifold:
Definition 1.3.15 — Riemannian manifold. LetM ≡ [(XM,τM ,A)]∼ be a (real) smoothMC∞
manifold. Further let g : XM × (TM×TM) → R be a smooth function. The ordered pair
(M,g) is a Riemannian manifold :⇔
∀ p ∈M : gp : TpM×TpM → R bilinear, with ∀ vp, wp ∈ TpM :
gp(vp,wp) = gp(wp,vp) ∧ gp(vp,vp)≥ 0 ∧
gp(vp,v) = 0 =⇒ vp ≡ eTpM
Then g is called a Riemannian metric on M and defines an inner product on each tangent
space TpM.
In fact, every paracompact smooth manifold admits a Riemannian structure, which is by defini-
tion guaranteed in finite dimensions. Riemannain metrics have a special status in physics due to
Einstein’s theory of General Relativity, which is formulated for (pseudo-)Riemannian metrics,
i.e. bilinear forms with Lorentzian signature.
Now, coming back to principal connections and define the Levi-Civita property. First of all,
assume that the torsion form vanishes identically. Furthermore, we require the existence of a
Riemannian metric g which lies in the kernel of the induced affine connection of ωLCg , i.e.
ωLC
∇ gv g= 0 ∀ v∈ Γ (TM)
This turns out to be sufficient to uniquely specify the underlying principal connection on P.
Conversely, given a torsion free connection ω , it is fully determined by a Riemannian metric
whenever its holonomy group is a subgroup of O(d) [30]:
(Holω ,◦Od)⊆ O(d) =⇒ ∃g ∈ Riem(M) : ∇ωvg= 0
and thus ω ≡ ωLCg . This correspondence is however not unique, for instance any constantly
rescaled metric gives rise to the same Levi-Civita connection:
ωLC LCg ≡ ωc·g ∀c ∈ R+/{0}
In theories of gravity, one severally exploits the relation between connections of frame bundles
and Riemannian metrics in that ωLCg is fully replaced by the associated metric g. However,
since this relation is not bijective it will produce different results when averaging over the full
field space in both cases, since (at least) the entire set of global-conformally related metrics
correspond to only one Levi-Civita connection on P.
Exponential map
Before we turn our attention to the very interesting physical aspects of differential geometry,
we are going to add yet another piece of the puzzle. It concerns the relation between a smooth
manifold and its tangent space and is a very illustrative example of the described fiber bundle
techniques. Furthermore, it will play a major role in the study of quantum gravity, where field
52 Chapter 1. Physical Content
space is in fact an infinite dimensional non-trivial manifold that fails to be a global vector space.
The local correspondence between tangent space and its manifold allows for a local recovery of
the vector space structure that reopens all the possibilities we usually have at our disposal.
Thus, let us consider the part of tangent space that can be used as starting vectors of suitably
extended geodesics:
EM ..= {(p,vp) |∃γ : [0,1]→M, p≡ γ(0) : ∀ t : v ωγ(t) = [γ̇ ]γ(t) ∧ ∇vv= eΓ (TM)}
One can show that EM forms an open set in the topology of TM and in the case EM ≡ TM
one denotesM to be geodesically complete. For finite dimensional manifolds without boundary
this feature is equivalent to metric completeness of M in terms of Cauchy sequences via the
Hopf-Rinow theorem. However, for the infinite dimensional case that we encounter for field
space or in the case of spacetimes with boundary this correspondence fails in general and the
construction of the set of geodesics cannot be circumvented by means of establishing metric
completeness.
With EM we have a unique allocation of curves to tangent vectors. Since curves are de-
fined on the manifold M and tangent vectors on the attached linear spaces, this correspon-
dence translates to an isomorphism between a subspace of M and EM given by the exponential
map:
Definition 1.3.16 — Exponential map. Let P ≡ P =MπP×ρRG be a principal bundle withP
G a Lie group and ω a connection one-form on P. Then, the map expω MM : E → M is the
exponential map on M w.r.t. ω :⇔
(p, [γ̇ ]p) 7→ expωp ([γ̇ ]p)≡ γ(1) with γ geodesic
The range where this map provides an isomorphism between the tangent space and a neighbor-
hood of a point p on the manifold is estimated by the injectivity radius:
injp(M) .
.= sup{r ∈ R | expω+ p : B ωr(eTpM)⊂ TpM → Im(expp ) isomorphism }
It measures the size of the largest possible ball around the origin in TpM where we obtain a full
geodesic covering. On global scales we can look for the minimal radius that can be spanned by
the exponential map, which for a geodesically complete manifold should be half the diameter
of the full metric space:
inj(M) ..= inf{injp(M) | p ∈M}
The larger this quantity is, the more we can cover with a single tangent space TpM and thus the
better M can be approximated as it is diffeomorphic to a linear space.
R In finite dimensions and for empty manifold boundary, the Hopf-Rinow theorem [31]
provides a correspondence between metric completeness (a global topological property
of metric spaces) and geodesic completeness. Besides this the Cartan-Hadamard theorem
can partially substitute this global to local correspondence even in infinite dimensional
spaces, as e.g. field space. It states conditions under which the universal cover of a
manifold is diffeomorphic to (flat) Euclidean space, explicitly given by the exponential
map [4, 32].
Lie groups and Maurer-Cartan form
As a direct application of these concepts consider a Lie groupG as a smooth manifold that can be
artificially extended to a trivial principal bundle over a single point (a zero-dimensional smooth
manifold). The necessary ingredients are given by P ≡ {p} ×G ≃ G with ρRP ((p,h),g) ≡
1.3 Field space 53
(p,Rg(h)) = (p,h ◦G g) and constant projection πp(u) = p. As a result of the simple point
structure of the base space the tangent space TP agrees with TG and curves defined on P can
only propagate in vertical direction, i.e. for any γP : R → P there exists a unique γG : R → G
with γP(t) = (p,γG(t)). In order to construct the set of fundamental vector fields, consider g
as the tangent space Te G. For any element V ≡ [γ̇e]e ∈ Te G with γe(0) = eG the associatedG G G
fundamental vector field is simply given by its left translation ♯Vg = Lg∗V :
( ) d ( ) ∣
V ♯g ( f ) = ρ
R
P ((p,g),•)∗V ( f ) = f ◦ρRP ((p,g),γe(t)) ∣dt t=0
d ∣
= ( f (p,g◦G γe(t))) ∣ = (Lt=0 g∗V )( f )dt
An admissible connection θ : TP→ g on P has to revert this operation and in addition has to
satisfy the equivariance condition, i.e. it should be a solution to the following equations:
( )
! !
ω ∗g(Lg∗V ) =V ∀V ∈ g ∧ ωg(v) = adg−1 ∗ Rg−1ωe (v) ∀ v∈ Γ (TG)G
It turns out that the simplest guess gives rise to a unique connection, the so called Maurer-Cartan
form or canonical one-form θg ≡ Lg−1 ∗ [33]. Evaluating the above condition of θ indeed reveals
its connection character. For any f : C∞(G;R) and any γe : R→ G with γe(0) = eG we obtain
( ) d ( ) ∣
(θg(Lg∗[γ̇e]e )) ( f ) = Lg−1 LG ∗ g∗[γ̇e]e ( f ) = f ◦L ◦L −1 ◦ γ (t) ∣G g g edt t=0
d ∣
= ( f ◦ γe(t)) ∣ = [γ̇t=0 e]e ( f )dt G
which proves the first statement. In order to check the second condition we consider any tangent
vector [γ̇ ]g ∈ TgG and any smooth function f ∈ C∞(G;R) and evaluate both sides:
( ) d ( ) ∣ d ( ) ∣
(θg([γ̇ ]g)) ( f ) = L
−1
g−1 ∗[γ̇ ]g ( f ) = f ◦Lg−1 ◦ γ(t) ∣ = f (g ◦ γ(t)) ∣dt t=0 Gdt t=0
d ( )∣ d ( ) ∣
= f (g−1 ◦G γ(t)◦G g◦G g−1) ∣ = f ◦Rg−1 ◦ ad ∣d g−1 ◦ γ(t)t ( ) t=(0 dt ) t=0
= ad R∗g−1 ∗ g−1 Le ∗([γ̇ ]g) = adg−1 ∗ R
∗
g−1θe ([γ̇ ] )G G g
In the second last step we artificially inserted Le which then finally leads to the required equal-G
ity.
Hence, every Lie group has its unique connection, the Maurer-Cartan form that we now
analyze further. With a principal connection at hand, we should now be able to decompose the
tangent space in its vertical and horizontal part. The vertical subspace of TP is defined by the
kernel of the pushforward projection πP∗ while the horizontal part is given by the kernel of θ .
VpP= { ![γ̇ ]p ∈ TpP |∀ f ∈ C∞(P,R) : dd ( f ◦πP ◦ γ(t)) = 0} ≡ TpPt
The identification of the vertical and full tangent space is due to the fact that any γP : R → P
can be uniquely written as γP(t) = (p,γG(t)). Under projection, πP(γP(t)) = πP((p,γG(t))) = p,
we obtain a constant function, hence the restricting condition is redundant yielding ‘0= 0’. On
the other hand, the triviality of the horizontal space is demonstrated using an auxiliary function
f ∈ C∞(G;R):
( ) !
HpP= kerθ = {[γ̇ ]p ∈ TpP |θ([γ̇ ]p)( f ) = dd f ◦ (p,Lg−1γ(t)) = 0} ≡ {et TpP}
54 Chapter 1. Physical Content
The necessary condition for tangent vectors to be horizontal can only be satisfied for constant
curves γP(t) = γP, since the left-action on G is a group isomorphism. A direct consequence is
the vanishing of the curvature two-form since dθ is bilinear in its arguments which by definition
are horizontally projected. The result are the well known Maurer-Cartan structure equations:
Ωθ = 0= dθ +θ ∧θ
The missing ingredient, the exponential map of a Lie group, is however not associated to the
Maurer-Cartan form, for in this case the base space is the trivial point {p}. Therefore, we have to
reverse the description and define a connection with the Lie group as base space. The important
fact here is that a Lie group is always trivializable and thus its tangent space TG can be written
as the direct product TG≃G×TeG≃G×g by means of Lg∗ : g→ TgG. This correspondence
extends further in that the Lie algebra g is isomorphic to the space of left-invariant vector fields:
g≃ {vL ∈ Γ (TG) |L L Lg∗vh = vg◦ h ∀g,h ∈ G}G
The canonical construction on Lie groups with a bi-invariant form is based on a left-invariant
Levi-Civita connection resulting in geodesics starting at γ(0) = eG that are group homomor-
phisms from (R,+) to G, i.e.
γ(t1)◦G γ(t2) = γ(t1+ t2) = γ(t2+ t1) = γ(t2)◦G γ(t1) ∧ γ(0) = eG
Hence, the exponential map is simply given by the extension of curves on G associated to some
left-invariant vector field:
expG : Te G≃ g→ G , expG(t · [γ̇ ]e ) = γ(t) with γ(t1)◦G γ(t2) = γ(tG G 1+ t2)
Furthermore, fundamental vector fields on a principal bundle P over some Lie group G are
generated by elements of g and the exponential map of G:
∣
V ♯u ≡ ρRP (u,•) V ≡ d∗ d (u◦G expG(t ·V )) ∣ ∀V ∈ Te G≃ gt t=0 G
Every vertical subspace VuP is thus isomorphic to the Lie algebra g. Notice that the exponential
map commutes with the adjoint action and that in addition the following identity holds [24, 25]:
( ) ( )
ρRP (•
♯
,g) ♯∗ V = adg−1(V )
As an example, consider the group Gl(d;R) consisting of all real d×d matrices, Mat(d;R),
that are invertible with ◦G the usual multiplication. The left-invariant vector fields are isomor-
phic to the Lie algebra consisting of
( )
gl(d;R)≡ {Aγ ∈Mat(d;R) |γ(0) = 1 ∧ γ(ε) ∈ Gl(d;R)},+, [• , •]gl ≡ [•,•]
= (Mat(d;R),+, [•,•])
In this case, the exponential map is simply given by the exponential series of a matrix:
t j
exp jGl(t ·A)≡ ∑ A ∀A ∈ gl(d;R)
∈N j!j 0
and the fundamental vector fields are determined as
( )
∣ j ∣
A♯u ≡ dd (u◦G expGl(t ·A)) ∣ = u◦ d
t
A j ∣ = u◦ A
t t=0 G dt ∑ j! t=0 G
j∈N0
1.3 Field space 55
Summary
Fiber bundles introduce the possibility to enlarge our understanding of continuity and differen-
tiability and free those concepts globally from the standard Euclidean picture while retaining
most of its valuable features due to its local correspondence. Removing the global restrictions
provides a multitude of new structures that locally agree with the standard notion but globally
produce new and interesting properties that turn out to be at the heart of today’s physics. We
have to be aware that in curved spacetimes smoothness is a more flexible notion than a simple
non-interrupted curvy function. Indeed, even fields that have jumps or obvious zigzag patterns
may show up to be perfectly admissible when considered in their natural environment, i.e. a
suitable fiber bundle. The confusion that may arise is only due to the projection into the famil-
iar Euclidean structure that we should abandon whenever we leave the realm of locality. The
several patches of spacetime (the base space) that have to be glued together already reflects this
formalism, and here we extended this prescription to the space of functions.
While there is a variety of possible differential structures a target F and a domain space M
can be combined to, we have to focus on inequivalent ones that respect a certain symmetry con-
straint, leading to the concept of principal bundles. The trivial case coincides with the product
topology τM×F where all transition functions are identities and the only source of discontinu-
ities in the standard sense, i.e. compared with Rd, is due to a possibly non-trivial structure ofM.
Taken together, we have to decide about the differentiable structure of M, F , and E(M,F,πE)
independently in order to fix the notion of smooth functions f : XM → XF in the bundle context.
For our purpose, the most important examples of fiber bundles are tensor bundles. Combining
different fiber bundles that share the same symmetry group G requires a consistent notion of
differentiability compatible with G. Principal bundles, i.e. P =MπP×ρRG, provide this univer-P
sal notion and give rise to a class of descendants, the associated bundles, which are naturally
connected in their concept of smoothness.
An additional ingredient that has to be defined on top of a principal bundle are connections.
They define a zero line on the tangent space of a principal bundle from which we measure devia-
tions, by decomposing vector fields into horizontal and vertical components. Such connections
are the natural candidates to describe interactions in physics, for they introduce a consistent
notion of change into the framework of fiber bundles. A derived concept that attaches these
interactions to ordinary fields is the covariant derivative that we will encounter at several steps
of our investigations. Following this short summary, we consider next the implementation of
differential geometry into the context of field theory.
1.3.2 Field space
We have seen so far three main ingredients of the mathematical theory that turn out to be at the
root of the functional renormalization group approach to quantum gravity. The first addresses
topology and differentiability in the shape of the base space our physical field content lives
in. It sets the arena in which all the physics takes places. The second ingredient is a remark-
ably simple yet very powerful branch of mathematics that reflects symmetries. They appear as
structure groups in the construction of principal bundles and are directly related to the funda-
mental laws of Nature under which the physical content interacts. In many ways group theory
is an excellent guiding principle to extract very strong constraints on mathematical objects and
nowadays it seems that the interplay on an elementary level – that gives rise to the diversity of
phenomena we observe around us – actually emanates from some very simple symmetry princi-
ple. Fiber bundles are the natural construction to consistently combine symmetry constraints of
interactions and differential geometry, thus it is not astonishing that the very successful Standard
Model of particle physics is presented in this language. The necessary connection to quantum
field theory is established by giving the mathematical objects in the theory of fiber bundles a
56 Chapter 1. Physical Content
physical meaning, which leads to what is known as gauge theories.
At this point, we are going to present the missing link to physics by introducing the basics
of gauge theory. It turns out that the set of physical fields can be usually embedded into a
smooth structure, a Fréchet manifold. This equips field space with a large variety of additional
properties, in particular it is locally described by a vector space. In addition lifting the collection
of fields from a mere set to a Fréchet space allows to consider a notion of change within field
space itself. Thus, functional variation can be defined at least on a local patch generated by the
so called background field method, which provides the transition from dynamical to fluctuation
fields. The necessity of this patchwork is demonstrated in the presence of boundary conditions.
Fields in gauge theories
The two basic ingredients for the construction of a principal bundle are the base space and the
structure group. In physics, spacetime M constitutes the ground on which physics takes place
and the structure group is given by some symmetry G≡ (XG,◦G) most often denoted the gauge
or symmetry group of the theory. The latter can be a combination of direct and semi-direct
products of groups but it is usually assumed to be a Lie group. This information is enough to
generate the set of inequivalent principal bundles:
PBs(M,G)≡ {[P=MπP×ρRG]∼P FB}
Each equivalence class of principal bundles, an element of PBs(M,G), gives rise to a different
set of smooth functions and hence to different theories. The classification of all principal bun-
dles for some base space M and structure group G is a hot topic in differential geometry and is
based on the homotopy group of some classifying space that exists whenever the base space is
paracompact. In fact, the homotopy properties of the classifying space are a direct consequence
of the homotopy groups of base space M and structure group G. For instance contractible base
manifolds give rise to only trivial principal bundles and depending on compactness and connect-
edness properties of M and G further classifications can be deduced (see ref. [4, 29] for further
details).
At this point we have to make a suitable choice for [P] in order to proceed. Having selected
[P]∼FB ∈ PBs(M,G) the set of (local) smooth sections over some region U ⊆ XM defines the
admissible (local) gauges
s ∈ Γ (P) defines a gauge
Each (local) section defines a (local) reference scale and corresponds to a gauge choice in
physics. A global section however is only possible if P is trivial. Nevertheless, it is usually
sufficient to transfer the global theory into local coordinates using a covering of charts on M.
The theory should be gauge invariant under a change of section (gauge) which is generated by
a gauge transformation, an element of
G≡ { f ∈ Iso(Γ (P))} ≡ { f ∈ Iso(P) |π RP ◦ f = πP ∧ ρP ( f (u),g) = f (ρRP (u,g))}
This second equivalence renders the group of gauge transformation as the vertical automor-
phisms of P in that they shuffle only the structure group while acting as the identity on space-
time. A key requirement in gauge theory is the invariance of observables , especially expectation
fields, under G.23
R Notice that the horizontal automorphisms of P are given by the diffeomorphisms of M.
Whenever a special coordination of the spacetime is invoked, i.e. a representative of
23A pure gauge transformation describes the action of Gon a zero-section, i.e. s(p) = (p,eG).
1.3 Field space 57
M ≡ [M]∼ is chosen, we have to assure that the formalism is invariant under theDiff(M)
full diffeomorphism group of P. While for the general description it is mandatory to
define equivalence classes, in practical applications one usually resolves the equivalence
character, but ensures the results be independent on a particular choice.
In most of practical calculations gauge invariance is described or tested using infinitesimal
gauge transformations which form a Lie algebra. The transition to G is provided by the expo-
nential map, though in general the Lie algebra spanned by its generators covers only part of the
full gauge group, in particular only the identity component. Hence, we have set the scene to
introduce interactions in a smooth and symmetry preserving way, before we finally let matter
fields enter the game.
Interactions In quite general terms, interactions change the status quo depending on the
underlying rules. The diversity of opportunities one has in constructing these fundamental laws
is huge, however those that preserve the differential structure and map physical fields into new,
still physical, ones is severally constrained. Given a certain principal bundle P the collection of
distinct principal connections are exactly the different possibilities in defining smooth symmetry
invariant notions of change:
XC(P) ≡ {ω : TP→ g |ω is principal connection }
X ∗[C](P) ≡ XC/G≡ {{ f ω | f ∈ G}|ω ∈ C(P)}
Again, the mathematical redundancy of physically equivalent constructions is removed by build-
ing equivalence classes of principal connections. At this step additional (mathematical) require-
ments can be imposed, for instance a vanishing torsion or the Levi-Civita constraint, which
may single out a unique representative of X[C](P). For (anti-)self-dual principal connections one
refers to ω as instanton.
Each connection defines a notion of interaction and goes under the name of gauge connec-
tion in the physics literature. In a local frame U ⊆ M with a trivializing cover {(U j, f j)} and
s j ∈ Γ (P) local sections, the (local) gauge potential related to ω ∈ XC(P) is given by
Aj : TU → g with A .j .= s∗jω
Notice that Aj is in general a local object that can be combined to a global smooth 1-form only
on the principal bundle. In general, we have to combine different smooth functions Aℓ using
the local sections sℓ, which however have to fulfill a consistency condition in order to give rise
to the same ω :
A ∗j = adκ−1Aℓ+g κℓ jL•−1 ∗ ∀ j, ℓ : U j∩Uℓ =6 0/ℓ j
Here κℓ j(p) : G → G are the transition functions, defined by fℓ ◦ f j(p,g) ≡ (p,κℓ j(p) ◦G g)
whenever p ∈U j ∩Uℓ. The above transformation is obviously not linear, since it contains the
Maurer-Cartan form of G as an additional translation term. For a matrix Lie group G the em-
bedding of the Lie algebra into G is trivial and the gauge transformation of the gauge potential
assumes the form:
A = κ−1j ℓ j ◦G Aℓ ◦κ −1ℓ j+g κℓ j ◦dκℓ j ∀ j, ℓ : U j ∩Uℓ 6= 0/
Furthermore, the curvature Ωω in a local gauge is the well known (local) field strength
Fω = s∗j jΩ
ω
58 Chapter 1. Physical Content
Imposing some restrictions on Ωω is another way to reduce the set of allowed connections.
However, usually these constraints only appear in combination with some even stronger condi-
tion, as for instance the requirement of ω being Levi-Civita, i.e. related to a Riemannian metric
on M, which is encountered in theories of quantum gravity for instance.
Matter fields With a principal connection ω we have introduced change and thus inter-
actions into the description. The missing ingredient are the matter fields, the physical content
that interacts with each other. Notice however, that matter fields on their own do not require
a principal connection but are independent concepts that can be defined whenever a principal
bundle is selected.
Matter fields usually live in a certain vector space over spacetime, and if they take part in
the interaction defined by ω it should be a (local) section of an associated bundle, i.e. a matter
field of a gauge theory is given by
Φ̂ ∈ X . ∞ −1F .= {Φ̂ ∈ C (P,V ) |ρV (g ,Φ̂(u)) = Φ̂(ρRP (u,g))} ≃ {σ ∈ Γ (E) |E ≡ P×̃ρV}M
Hereby, the internal space V stands for a vector space, that when specified further gives some
particular meaning to the related matter fields. For instance, if V ≡R we speak of a scalar field,
more general terms are related to n-component vector fields in case of V ≡ Cn, and for V ≡ g
and the adjoint representation ρ ≡ ad physicists denote Φ̂ as Higgs field.
Gauge transformation equivalent matter fields represent the same physical field, so in prin-
cipal we work with an equivalence class of sections, that constitute the actual field content of
the system. Let {Vj} j∈J be the collection of all target spaces in which the relevant matter fields
are defined by some associated group action ρ j. The full set of independent physical fields
constitute the matter part of field space, and with V ≡ V1×·· ·×VJ and ρ ≡ ρ1⊗·· ·⊗ρJ , we
define
X .[F ] .= XF /G≡ {[Φ̂] |Φ̂ ∈ XF } ≡ {{Φ̂ ◦ f | f ∈ G}|Φ̂ ∈ XM M M F }M
They interact with each other via the covariant derivative induced by elements of X[C](P).
Summary Gauge theory, based on symmetry principles, is one of the great achievements
of the last century, translating QFT into the language of differential geometry. By this, re-
markable insight into the geometrical structure of fundamental theories might be obtained. The
information we need to specify the field content of a gauge theory consists of
◮ Spacetime: M an equivalence class of smooth manifolds
◮ Symmetry group: G, acts as the structure group
◮ Principal bundle: [P =MπP×ρRG]∼FB ∈ PBs(M,G), defines a G-invariant notion of dif-P
ferentiability and thus smooth functions
◮ Gauge fields: [ω ] ∈ X[C](P), a principal connection on P that defines interaction
◮ Matter fields: V and ρV :G×V →V , a vector space (target space) and group action that
defines an associated bundle along with admissible matter fields X[FM]
The remaining ingredients can be reconstructed by the above structure, in particular the gauges
and the associated gauge group of transformations.
R For Levi-Civita connections, which are often considered in quantum gravity, the gauge
fields can be rewritten in terms of ‘matter fields’, namely as a Riemannian metric. In those
cases, listing the gauge field is redundant and is usually omitted, though it is intrinsically
present.
1.3 Field space 59
Field space
We have already seen that the set of smooth functions or vector fields can be endowed with
additional structure and at least locally they define vector spaces. It turns out that the collection
of matter and gauge fields inherits a plenitude of rich structure from the fiber bundle construc-
tion and the base space properties. In physical applications, one usually has supplementary
constraints on the true set of relevant fields, for example in the shape of boundary conditions,
that restricts X[F ] and X[C] further, which may either spoil or enrich certain properties of fieldM
spaces.
In order to implement a notion of differentiation on field space, which is nothing else than
variation w.r.t. fields, a suitable topological structure is necessary. The actual choice severally
depends on the underlying spacetime and most of the discussion presented here is only valid
for closed manifolds, i.e. for compact manifolds with vanishing boundary. Notice however that
there exists extensions to non-compact manifolds and cases of non-vanishing boundary along
similar lines, usually based on a different topology. In most applications, the sets XF andM
XC are endowed with an extension of the Whitney c∞-topology for smooth functions C∞(P,V )
where the tensor structure is trivially added. Basically, the infinite dimensional space C∞(P,V )
is reduced to a finite dimensional jet space using an equivalence relation similar to the Taylor
expansion. Assume we are given a function σ ∈ C∞(P,V ) that in local coordinates (UP, fU )P
and (UV , f dU ) can be expanded around a point fU (u)≡ x0 ∈ R with u ∈U ⊆ P:V P
σ̃(x)≡ ( fU ◦σ ◦ f−1U )(x) = (jkx σ̃)(x)+O(xk+1) withV P 0
(jk σ̃)(x) ..= ∑ 1x α!∂
α σ̃(x0)(x− x α0 0)
|α |≤k
Hereby, we have used the notation of multivariate Taylor expansion with α ∈Rd and ∂ α f (x) =
∂ α1 · · ·∂ αdx x f (x). The map jkx simply reduces a function to its kth-order Taylor expansion, which1 d 0
gives rise to an equivalence relation on C∞(P,V ), the kth-jet space:
Jk(P,V ) ..= {{σ ∈ C∞(P,V ) |(jk σ̃ )(•) = (jk σ̃ )(•)}|σ ∈ C∞u 2 f (u) 1 f (u) 2 1 (P,V )}UP UP
It turns out that Jku(P,V ) is a real, smooth manifold with dimension dim(P)+dim(V ) ·(dim(P)+
k)!/(dim(P)! · k!). The Whitney ck-topology is then obtained using the topology of V :24
{ ⋃ }
τ ..= gen{σ ∈ C∞ck (P,V ) |(jkx σ̃)(x)} and τc∞ = gen τ k0 c
k∈N0
Lifting this topology to XF and XC gives rise to infinite-dimensional topological manifoldsM
FM ≡ (XF ,τc∞) and C≡ (XC,τc∞), respectively. The sensitivity of this construction on theM
properties of spacetime is encoded in the jet space. For non-compact or otherwise generalized
spacetimes the properties of Jku(P,V ) may not be sufficient to deduce a well behaved topology
on C∞(P,V ).
Once we have introduced a suitable topology on field space with a local vector space struc-
ture and Hausdorff character, there are several derived concepts at our disposal. With an addi-
tional differentiable structure a notion of field variation, which is of particular importance in
QFT, is introduced, that further supplements the construction of field space. For this, let us first
consider the space of connections25 and follow the lines of ref. [34, 35]. In ref. [36] the author
demonstrates that the group of gauge transformation can be used to define a fibration on C
24Here gen{U} denotes the topology generated by {U}, see definition 1.1.11.
25As in [34] we confine [C] to the space of generic connections. These are connections for which dω ∗dω has
trivial kernel.
60 Chapter 1. Physical Content
with the canonical map πC : ω ∈ C→7 [ω ] ∈ [C].26 The space of equivalence classes [C] turns
out to be a smooth manifold with an additional Riemannian structure [36–38]. The associated
Riemannian metric – a bilinear, symmetric, non-degenerated form on the tangent space of M –
is a consequence of the underlying constituents of C, namely a metric ḡ on M and the Killing
form K of G. To obtain its explicit form, notice that [C] carries an affine structure modeled on
Γ (Λ1(M)⊗E), the space of 1-forms onMwith values in the adjoint bundle E ≡P×̃adg. A trans-
lation of any connection ω along α ∈ Λ1(M)⊗Γ (E) gives rise to a new principal connection ω ′
and thus the tangent space of C is identical to Λ1(M)⊗Γ (E), i.e. Tω C≃ Λ1(M)⊗Γ (E). The
Riemannian metric G on C is induced by the inner product on Ak ≡ Λk(M)⊗Γ (E) restricted
to k = 1:
( ) ∫
α . d k1 , α2 .Ak = d x K(α1∧⋆α2) ∀α1, α2 ∈ A
x∈M
This inner product and thus the Riemannian metric on C are gauge invariant by construction.
As we will see, an inner product introduces a classification scheme on the space of operators,
in particular it gives rise to the adjoint operator of dω denoted dω ∗. A natural definition of
horizontality on Tω C is then provided by a principal connection on field space:
( )
χω ..= dω ∗
−1
C d
ω dω ∗ and Hω [C] ..= ker(χ
ω
C)
Horizontal vector fields on Cgenerate physically non-equivalent gauge fields, while the vertical
translations do not change the equivalence class [ω ] at all. The projection onto the horizontal
component explicitly depends on the field space connection χωC and is given by
ΠHω : Tω C→ H H ω ωω C, Πω ≡ id−d χC
which is easily shown to be idempotent and in fact also self-adjoint.
From now on, we assume that the fibration of C(P) actually defines a principal bundle C=
[C]πC×ρR Gwith the base space consisting of physically equivalent gauge fields. All theoriesC
we are going to construct actually live on this base space, however for practical calculations
and general considerations it is usually convenient to rather work in the total space C(P) and
ensure the formalism being invariant under G. While it simplifies the evaluation of the path
integral, at the same time this transition may introduce regularization issues and ambiguities,
due to the non-existence of a global section in [C]. These artifacts of gauge invariance cause
several problems in QFT and are very prominent for theories of quantum gravity. The actual
transition of theories defined on [C] to the total space C is postponed until we have the full
device of measure theory at our disposal, see subsection 2.1.5.
Whilst there are in general no global sections on C a horizontal local section is a set of
principal connections which have no vertical displacement, thus satisfy
sω0 :U0 ⊆ [C]→ C with [ωA]→7 π−1C ([ωA])∩
∗
Sω0 ∧ Sω0 ..= {ω |dω0 (ω −ω0) = 0}
This defines the covariant background gauge condition and fixes the gauge in field space, at
least locally. Based on this local section, an induced Riemannian metric on [C] is thus obtained
by a horizontal lift of the scalar product on A1 at some ω ∈ π−1C ([ω ]). Given two vector fields
on base space v, w∈ Γ (T[C]) there are horizontal vector fields ṽ and w̃ on TCwhich locally
describe the respective lifts. Then, pointwise we have:
( ) ( ) ( )
v[ω ] , w[ω ] .
.= ṽω , w̃ω ≡ ṽ , ΠHT [C] A1 ω ω w̃ω 1[ω] A( )
≡ ṽ , ΠH ΠH Hω ω Π0 ω ω w̃0 ω A1
26Actually, Ghas to be reduced by its center to provide a consistent fibration. We will assume this in what follows.
1.3 Field space 61
Due to the gauge invariance and the properties of the horizontal lift, this definition is indepen-
dent on the choice of ω and thus well suited for our purpose. However, this coordinate choice
fails whenever the Faddeev-Popov operator M(ω ,ω ) ..= dω0∗0 dω has a non-trivial kernel, i.e.
ker(M) =6 {eTω C} which corresponds to the appearance of the Gribov ambiguity. Notice that
ΠHω Π
H H
ω Πω is the local coordinate form of the metric induced on [C]. With (ω −ω0)≡ v0 0 ω0 ∈
Tω0 C this local expression can be recast in the form
( ( ) )
ΠH ΠHΠH ≡ ΠH id− ω ∗K ω −1 ∗ H .ω ω ω ω v .0 0 0 ω d d K Π with K (ξ ) = [v , ξ ]0 vω v0 ω0 ω0 ω0
Here, Kvω : A
0 → A1 denotes the infinitesimal generators of the gauge transformation, and in
0
the case of metric gravity it corresponds to Lvω ξ . Hence, the second term in the bracket0
subtracts the vertical component as required.
If the horizontal space Sω0 intersects each orbit on C it actually defines a global normal
coordinate system [34]. In fact, it turns out that geodesics are ‘straight lines’, namely curves
of the form ω0+ t · vω0 and once it is perpendicular to a specific orbit it will remain so for any
other orbit intersecting with Sω0 . Hence, the exponential map at ω0 is simply given by
ω
χ 0
exp Cω0 (t · vω0)≡ πC(ω0+ t · vω0)
The exponential map becomes singular at ωS = ω0+ tS · vω0 exactly ifM(ω ,ω0)+ t ·dω0∗S Kvω0
has a non-trivial kernel, which in any case will be finite dimensional. The associated conjugate
points ωS mark the Gribov horizon of ω0 along vω0 and their occurrence spoils the diffeomor-
phic identification of C with its tangent space at ω0. Explicitly, ωS corresponds to a ξ ∈ A0
with M(ω ,ω0)ξ = 0 and πC∗(dω ξ ) = 0, hence to a vanishing Faddeev-Popov determinant. As
such the existence of Gribov ambiguities and horizons is an intrinsic artifact of the theory.
For matter fields similar arguments holds, at least if the full set of sections is considered.
Then, FM has a fibration πF : Φ̂→7 [Φ̂]∈ [FM M]which in fact provides a fiber bundle construction
under some very weak restrictions:
FM = [FM]
πFM×ρR G
FM
As long as there are no further constraints imposed on the matter fields, the full set of sections
inherits many, very nice properties of the function space, in particular vector space and Rie-
mannian structures. Hence, the usual L2-scalar product can be extended to the space of matter
fields by means of the Killing form K. Global sections provide a way to evaluate the generat-
ing functional in QFT for the set of physically inequivalent fields, i.e. [FM]. However, usually
additional restrictions are invoked, as for instance boundary conditions in the case of ∂M 6= 0/ ,
that may spoil the trivial nature of field space. Some of the severe challenges on the way to
a quantum theory of gravity can be traced back to the non-trivial fiber bundle structure of FM
compared to the Standard Model of particle physics for instance. To maintain the covariant
and global descriptions (on field space) the formalism turns out to be far more complicated and
actual evaluation of the generating functional seem to be impossible. As history has given many
examples, it might be exactly this difficulty that will provide further insight into the underlying
principles of Nature for all interactions.
Background field method
We have seen that the space of principal connections and sections of associated bundles has
a wealth of mathematical structure that allows to introduce almost the entire apparatus of dif-
ferential geometry. In particular its fiber bundle and Riemannian manifold description are of
uttermost importance to define a suitable notion of field variation. While C carries an affine
62 Chapter 1. Physical Content
structure, the matter sector is usually given by a global smooth vector space, hence a trivial
principal bundle with structure group G. Though in general, we have to keep the notion of con-
nections and matter fields separately, for simplicity we combine both field spaces into a single
one, denoted Y, and only remove this covering cloak when a distinction seems to be unavoid-
able. One can also think about Y as the direct product of Cwith FM giving rise to a product
principal bundle
Y = [Y]πY×ρR G with [Y]≡ {{ρRY(Φ̂, f ) | f ∈ G}|Φ̂ ∈ Y}Y
It inherits all topological, differentiable and bundle properties from its constituents, in particular
the projection πY and the group action ρ
R
F are naturally induced by the respective objects on C
and FM.
For the gauge fields, we have already seen that there is no global vector space structure
present, however due to its affine nature it gives rise to a trivial principal bundle and hence –
without additional constraints – Y is also trivial. This provides a global section and [Y] can
be nicely embedded in Y, which is particularly relevant for the construction of a generating
functional in QFT. Unfortunately, it turns out that Nature seems to favor only a subset of
all mathematically possible principal connections and sections of associated bundles. Usually,
this restriction is best implemented into the formalism using an operator on Y with non-trivial
kernel:
B : Y→ Y with ker(B) 6= {eY}
One then defines the true set of physical fields by the kernel of these constraining operators, i.e.
⋂
X .F .= {Φ̂ ∈ Y |Φ̂ ∈ ker(Bj)}
j
In general, the reduction of the fiber bundle Y to the subset XF will spoil topological and some
derived properties, in particular the concept of smoothness and fibration. In order to maintain
the calculus of variation on XF one has to start from scratch and rebuild, if possible, a new
differentiable structure that hopefully promotes XF to a new equally equipped principal bundle.
While for a generic B this procedure might fail, it seems that most of the constraints we en-
counter in Nature lead to a field space that is open in Y, i.e. XF ∈ τY. The induced subspace
topology, resulting in F, keeps certain differentiable and fiber bundle properties intact, however
algebraic structures might differ for Y and F, as for instance vector space properties. Depending
on the explicit form of Bj this can have further (global) consequences for F, in particular the
triviality of principal bundle may be lost, as for example in the case of Levi-Civita connections.
Some very well known examples of constraining operators are boundary conditions. In
case of spacetimes with non-vanishing boundary ∂M 6= 0/ differential operators require that
the fields of interest fulfill some conditions on ∂M that render certain relevant operators self-
adjoint or elliptic. The most famous choices are Dirichlet or Neumann boundary conditions
that correspond to the following class of constraint operators:27
B∂M(D);Ψ(Φ̂)
..= (Φ̂−Ψ)(•)δ∂M(•) with ker(B∂M(D);Ψ)≡ {Φ̂ ∈ Y |Φ̂|∂M = Ψ|∂M}
B∂M ..(N);Ψ(Φ̂) = (d
ω0Φ̂−Ψ)(•)δ∂M(•) with ker(B∂M(N);Ψ)≡ {Φ̂ ∈ Y |dω0Φ̂|∂M = Ψ|∂M}
There are also e.g. mixed or non-local boundary conditions which seem relevant in physical
applications. However, it is already for these two simple and nicely behaved operators that
27Here δ∂M(•) denotes the restriction to the boundary ofM, hence a zero (identity) map for all interior (boundary)
points of M.
1.3 Field space 63
F
GF Gauge fixing condition
πF
GF [F]
Field space
×
Gauge sector Matter sector
× ×
P (1) E
ρ 1G
E
(2) 2
G ρGauge choice G E3
(3)
ρG
M
πM
Diff(M) M
Figure 1.2: The fiber bundle construction of field space based on the gauge theory over space-
time M and symmetry group G. The principal field space bundle F consists of base space [F],
inequivalent fields, and structure group GF.
certain structures of field space are lost. Therefore, notice that for any choice of Ψ 6= eY the
Dirichlet and Neumann boundary conditions are not linear but only affine operators. Reducing
field space by these supplementary constraints results at most in an affine space, even for the
matter sector. While this is no major drawback one can imagine that non-local or non-affine
operators will severally reduce the structure of F. Most important, due to the diffeomorphism
invariance of a manifold’s boundary the smoothness structure is kept intact, though one has
to adapt the group of gauge transformation appropriately, i.e. such that ρRY(XF,GF) ⊆ XF. An
illustration of field space and its constituents is given in fig. 1.2.
Besides the apparent negative effects that come along with B∂M( ); and B
∂M
( ); , there areD Ψ N Ψ
some very welcome implications on the level of operators, in particular concerning their spectra,
see section 1.4. The Riemannian manifold property of Y is nevertheless maintained in the
transition from Y to F and in fact imposing only Dirichlet or Neumann boundary conditions to
obtain field space does not affect the trivial nature of the principal bundle, even though it lacks
a global notion of linearity. Hence, in this case we obtain
F= [F]πF×ρRGF ≡ [F]× GFF
with a flat field space connection and a trivial exponential map that provides a diffeomorphism
for any TΦ̄F to F.
We have seen that the explicit form of the boundary operator, which in the above case
defines an affine transformation, decides about the inherited properties of F. For nicely formed
constraint operators F is a trivial principal bundle with an affine structure that in particular
allows to cover F by one of its tangent spaces, say at Φ̄. For more complex restrictions that
still fulfill ker(B) ∈ τY non-trivial fiber bundles emerge and also the affine property of F is
lost. Hence, in general we only retain a Riemannian manifold F with some fibration and the
injectivity region of the (non-trivial) exponential map might not cover full field space. In these
64 Chapter 1. Physical Content
cases we need several base points Φ̄ j and a partition of unity to span F by some U j ⊆ TΦ̄F.
Whenever field space turns out to be non-trivial, i.e. it carries a non-flat principal connection
χF, we will encounter a lot of difficulties, especially in the construction of a covariant definition
of field variations. Fortunately, for the Standard Model of particle physics on flat spacetime on
which most of our today’s understanding of Nature is based on, field space seems to be perfectly
trivial and no such problems occur.
However, attempts to construct a quantum version of General Relativity revealed the non-
trivial field space character and thus a non-vanishing curvature tensor. In what follows we
consider metric gravity, where we have a non-linear constraint on the vector space of symmet-
ric 2-tensor fields. Besides boundary conditions in the case of non-vanishing ∂M, the class of
(pseudo-)Riemannian metrics is not only symmetric but has a fixed signature along with cer-
tain non-degeneracy properties. The latter constraints turn the subset of (pseudo-)Riemannian
metrics into a non-trivial fiber bundle, which is still a Fréchet manifold and can be equipped
with a ‘Riemannian’ metric [39–41]. Conveniently, the field space connection is chosen to be
of Levi-Civita type w.r.t. a field space metric G. In addition the physical assumptions con-
cerning the nature of free theories impose further requirements on the connection – and thus
on G – in particular to be ultra-local (not-containing any field derivatives), trivial for free the-
ories, and fully determined by the classical action functional. The field space metric exhibits
all properties of a Riemannian metric. However, later we will enlarge field space to contain
also anti-commuting Faddeev-Popov ghost fields and thus its symmetry properties have to be
slightly modified. This also takes place when fermionic field content is added, for then addi-
tional factors of (−1) appear when two anti-commuting fields change position. Otherwise, we
have a symmetric, bilinear function which is in addition non-degenerate:
( ) ( ) ( )
GΦ̄ v , w =GΦ̄ w , v ∀v, w ∈ TΦ̄F , GΦ̄ v , v = 0 =⇒ v= eTΦ̄F
To establish a covariant description of field space based on a Riemannian metric, F has to be
(locally) represented by a vector space. At least in the vicinity of some Φ̄ we can approximate
field space by its associated linear space TΦ̄F. In quite generality, the background field method
is a way to cover field space in terms of these local vector spaces in such a way that derived
concepts will be independent of the specific covering. Then the full apparatus of differentia-
tion and superposition applies and a smooth notion of integration can be derived. The local
diffeomorphism equivalence is provided by the exponential map.
Definition 1.3.17 — Background field covering. Let M be a smooth (infinite Fréchet)
manifold. The collection of ordered pairs UBFC ≡ {(p j,U j)} with #(UBFC) < ∞ is a back-
ground field covering of M :⇔
⋃
M = U j with U j ∈ τM ({U j} is an open cover of M)
(p j ,U j)∈UBFC
∀ p : expχMj p : Bj inj (M)(eTp M)⊆ Tp M →U j isomorphismp jj j
When M ≡ F one refers to UBFC as the background field covering of field space and the
tangent vectors ϕ̂ ∈ TΦ̄M at some background field Φ̄ are called fluctuation fields, while Φ̂
is referred to as dynamical field.
Hence, the existence of a partition of unity and geodesic completeness of field space is cer-
tainly important in defining a suitable background field method. A detail account on the nec-
essary properties is presented in [37]. If field space is paracompact, then any finite open cover
U≡ {Ui} gives rise to a partition of unity that ultimately provides a background field covering
of F. This results in a partition of unity in such a way that we sum locally over subsets of vector
1.3 Field space 65
spaces which ultimately allows to define the generating functional of a Quantum Field theory.
Very often geodesic completeness can be ensured by metric completeness via the Hopf-Rinow
theorem. For infinite-dimensional manifolds, which is the case for F, this correspondence how-
ever fails in general. Thus, an explicit derivation of the geodesics is usually unavoidable, though
in certain cases other theorems, as e.g. the Cartan-Hadamard theorem, can be exploited. Oth-
erwise, a suitable field space metric G is the starting point to develop a (local) Levi-Civita
connection on F and determine the normal coordinate form of geodesics along with the expo-
nential map and its properties. In what follows, we assume for simplicity that field space can
be covered by a single background field cover denoted UBFC ≡ (Φ̄,F) or [UBFC] ≡ ([Φ̄], [F]),
respectively.
Intrinsic and extrinsic symmetries
To avoid confusion, especially in the case of quantum gravity, it is instructive to distinguish the
symmetries on field space from the symmetries of the coordinate version of a field. We have
seen that [F] allows for a principal bundle construction Fwith the group of gauge transformation
GF as its symmetry group. Physically distinguished fields live in different equivalence classes
[Φ̂] or [ϕ̂] respectively. While for practical purposes it is more convenient to work with a special
representative Φ̂ ∈ [Φ̂], one has to assure that later results do not depend on this particular
choice but hold for the entire equivalence class [Φ̂]. In the case of field space one speaks
about invariance under gauge transformation GF, an intrinsic symmetry of the formalism, and
a specific choice of gauge condition fixes the ‘coordinates’. It is a vertical automorphism such
that it does not affect the domain part of the fields, i.e. the spacetime coordinates.
X .[F] .= XF/GF ≡ {[Φ̂] |Φ̂ ∈ XF} ≡ {{ρRF (Φ̂, f ) | f ∈ GF}|Φ̂ ∈ XF}
with GF ≡ { f ∈ G|BF(ρRF (•, f )) ≡ ρRF (BF(•), f )}
The symmetry on field space is referred to as intrinsic symmetry.
On the other hand we have a spacetime coordinate invariance of the entire formalism imple-
mented in the construction ofM. This symmetry reflects the missing individuality of spacetime
points and hence spacetime is also an equivalence class of smooth manifolds, M ≡ [M]∼ .Diff(M)
Dissolving the abstract notation by choosing a particular ‘coordinatization’ of spacetime is un-
avoidable in practical calculations. Hence, giving an explicit coordinate form of fields, either
[Φ̂](x) or Φ̂(x) is associated to a specific representative M ∈M ≡ [M]∼ . Again, we haveDiff(M)
to ensure that the results are independent of the particular choice of M rendering the formal-
ism diffeomorphism invariant, Diff(M). Thus, different coordinate versions of fields Φ̂(x) and
(Φ̂ ◦ fM)(x) ≡ Φ̂( fM(x)) with fM ∈ Diff(M) should be indistinguishable, however now due to
the symmetry of spacetime. This ‘horizontal’ invariance is referred to as extrinsic symmetry.
Both concepts have to be carefully told apart for they reflect completely different symmetry
principles.
In summary, the space of physical fields is described by an equivalence class with respect to
the group of gauge transformation. The remaining invariance, the extrinsic symmetry, generated
by fM ∈ Diff(M) is entirely due to spacetime and does not further specify physical fields, but it
labels spacetime points:
X[F](M)≡ {[Φ̂]([x]) |Φ̂ ∈ XF} ≡ {{ρRF (Φ̂◦ fM, fF) | fF ∈ GF ∧ fM ∈ Diff(M)}|Φ̂ ∈ XF}
Under a simultaneous choice of ‘coordinates’ the spacetime invariance reduces to the stabilizer
subgroup of the field with respect to the induced group action of Diff(M). Both intrinsic and
extrinsic symmetries have to be clearly distinguished, since especially in quantum gravity it
may otherwise cause some confusions about invariance requirements.
66 Chapter 1. Physical Content
R Consider for example the space of Lorentzian metrics on TM and select the equivalence
class containing the Minkowski metric: [η ]∼ . On field space all metrics ρRF (η , f[F] F) ≡
f−1∗F ◦η are equivalent. As a representative of this equivalence relation we may select η
and make sure that any field transformationwith respect to fF does not alter the result. The
moment we dissolve the equivalence class of smooth manifolds that constitute spacetime
we have to additionally ensure the invariance of spacetime coordinates while retaining
the field space representative η . This exactly reduces to an invariance under the stabilizer
subgroup of η w.r.t. pullbacks f ∗M of fM ∈ Diff(M). For the Minkowski metric this is the
famous Poincaré group.
1.4 Operators
A profound understanding of field space is provided by studying the effect of operators on
it. This reflects the way we investigated groups by their action on vector spaces, however
in a kind of inverted sense. Representation theory is actually closely tied to operator theory
and in fact a very prominent set of operators are symmetry transformations of fields that link
group theory and functional analysis. It is therefore not astonishing that we will meet several
concepts of representation theory in this broader concept again. From a physical perspective,
transformations are the natural way to implement physical constraints and interpretations into
the mathematical description. The reason is that rather than obtaining the full content of a field
at once, our observations project out only part of the information, usually those visible in their
effects and interactions with the environment. Most of the material presented here is a brief
summary of ref. [3, 26, 28, 42, 43].
For our purpose, an operator maps a field to a new element of field space and is thus an endo-
morphism in the respective category. The more general definition allows for several extension
and can be described as follows:
Definition 1.4.1 — Operators. Let X and Y be sets possibly equipped with further struc-
ture.
A function T : X → Y is an operator on X to Y :⇔
T is a morphism (structure preserving map) in the respective category of X and Y and
Dom(T )≡ X is the domain of T .
Usually we will focus on special operators that are endomorphisms of field space, hence mor-
phisms where target and domain space coincide. Depending on which structural part of field
space the operator acts on, it has to preserve the vector space, smoothness, or/and topological
properties and will be thus denoted as a linear, differentiable, or continuous operator.
1.4.1 The spectral theorem
Fortunately, we usually have linear- and smooth-properties at least locally at our disposal and
are thus mainly focused on linear differentiable operators in what follows. At this point the
virtue of the background field method becomes apparent, for it diffeomorphically relates a non-
vector space with a local patch of a linear space. Operators then act on the tangent space
rather than on field space itself, which in the case of flat field spaces turns out to be a trivial
identification. However, in the general case one has to pay special attention to the range of
validity and to regions of overlap. In the remainder of this section we will consider a (local)
Hilbert space as part of field space, that adds an inner product to the list of features we have at
our disposal:
Definition 1.4.2 — Hilbert space. Let V ≡ (X ,◦V ,F ⊆ C,◦FV ) be a vector space and
1.4 Operators 67
( ) ( )
· , · : X×X → F a map. The ordered pair HV ≡ (V, · , · ) is a Hilbert space :⇔V V
( ) ( )
∀ ∗x,y ∈ X : x , y = y , x (conjugate symmetry)
( ) V V( )
∀x ∈ X : x , x ≥ 0 ∧ x , x = 0 =⇒ x= eV (positive-definite)V
∀x,y,(z ∈ X , ∀α ∈ F : ) ( ) ( )
(α ◦FV x)◦V z , y = α · x , y + z , y (linear in first argument)V √ V V( )
∀x,y ∈ X : dV : (x,y) 7→ x◦V (−y) , x◦V (−y) (V,dV ) metric completeV
Thus, a Hilbert space is a complete inner product space.
( )
Notice that the inner product · , · is a sesquilinear semi-positive definite form and in the
V
real case bilinear.
Once the target and domain space of an operator exhibits a Hilbert space structure, or more
general are densely embedded in such a rich structured space, we have many possibilities to
characterize operators using the respective scalar products. In fact the inner product on a Hilbert
space already classifies the full set of dual vectors, i.e. linear operators from the Hilbert space
to the underlying field K. This deep connection is provided by representation theorems as the
Riesz or the Lax-Milgram representation theorem, stating that
( )
∀v ∈V : Tv : Dom(Tv)⊆V → C with T .v .= • , v is linear operator ∧V
∀T : Dom(T )⊆V →( C lin)ear with Dom(T )≡V, ∃̇vT ∈V :
w , vT ≡ T (w) ∀w ∈ Dom(T )V
In order that vT is unique we have to require that the domain of T is dense in the Hilbert space.
In what follows we will consider two Hilbert spaces HV and HW over the associated vector
spaces V andW , respectively. The corresponding scalar products provide a way to measure the
spreading effect of an operator in the target space. Therefore, we define the operator norm of
T : Dom(T )⊆V →W via the scalar products of domain and target space:
( ) ( )
‖T‖ .Op .= inf{c≥ 0 | T (v) , T (v) ≤ c · v , v ∀v ∈V}W V
If ‖T‖Op < ∞ we say that T is bounded which in fact is equivalent to T being continuous
with respect to the topologies defined by both scalar products. Bounded operators form a very
important class of transformations, for they avoid infinities by re(stricting its )image(to a )‘finite
size’ region. For instance, in the case of an isometric operator T (v) , T (v) = v , v we
W V
obtain ‖T‖Op = 1.
Unfortunately, there are many important examples of unbounded operators that play an
important role in mathematics and physics. Therefore, consider the subspace of Dom(T ) where
the operator T : Dom(T )⊆V →W is bounded and denote it
( )
Dombd(T )≡ {v ∈ Dom(T ) | T (v) , T (v) < ∞}W
On this subspace we can uniquely identify an adjoint operator T ∗ : Dom(T ∗) ⊆V →V that is
naturally related to T for it represents T in the target space. The adjoint operator is implicitly
defined for the bounded domain, i.e. for all v ∈ Dombd(T ), by
(
T ∗
) ( )
(w) , v ..= w , T (v) ∀w ∈ {w ∈W |∃v ∈ Dombd(T ) : T (v) = w}V W
Surely, our main concern lies on operators that are endomorphisms, meaning for which
target and domain spaces are subsets of the same Hilbert space HV . This allows a comparison
of T : Dom(T )⊆V →V and its adjoint T ∗ on a direct level, in particular if both coincide. The
following presents a list of definitions depending on the relation between an endomorphism and
its adjoint:
68 Chapter 1. Physical Content
Definition 1.4.3 — Operator classification. Let HV be a Hilbert space. A linear operator
T : Dom(T )⊆V →V is . . .
. . . symmetric (hermitian) :⇔ ∀v ∈ Dom ∗bd(T ) : T (v) = T (v)
. . . essentially self-adjoint :⇔ ∀v ∈ ∗Dombd(T ∗) : T ∗ (v) = T ∗(v)
. . . self-adjoint :⇔ ∀v ∈ Dombd(T )≡V : T ∗(v) = T (v)
. . . normal :⇔ ∀v ∈ Dombd(T ) : T ∗ ◦T (v) = T ◦T ∗(v)
. . . unitary :⇔ ∀v ∈ Dom (T ) : T ∗ ◦T (v) = T ◦T ∗bd (v) = v
. . . orthogonal projection :⇔ ∀v ∈ Dom (T ) : T ∗bd (v) = T (v) = T ◦T (v)
Symmetric or the even stronger restricted class of self-adjoint operators, the latter being always
bounded by the Hellinger-Toeplitz theorem, play an important role as observables in physics.
The reason is the existence of a powerful theorem that provides criteria for operators to be
mutually diagonalizable with real eigenvalues. In its precise form one version of the spectral
theorem states:28
Theorem 1.4.1 — Spectral theorem. Let HV be a Hilbert space and let
N .bd .= {T :V →V |∀v ∈V : T ∗ ◦T (v) = T ◦T ∗(v) with Dombd(T )≡V}
be the set of bounded normal operators on HV .
Then, for any family of commuting normal operators S ≡ {Tj} ⊆Nbd, i.e. for all Tj ∈ S and
Tk ∈ S with Tj ◦Tk ≡ Tk ◦Tj, the following holds true:
∃(Y,ΣY ,µ : ΣY → R) ∧ U : L2(Y,ΣY ,µ ;C)→V :
∀T ∞j ∈ S ∃ f j ∈ L (Y,ΣY ,µ ;C) U∗ ◦Tj ◦U ≡ f j
If Tj is self-adjoint then f j is real valued.
In other words, the family S of operators is simultaneously diagonalizable. This theorem has
several extensions, generalizations and alterations, for example in case of self-adjoint compact
operators there exist eigenvalues and eigenfunctions that span the Hilbert space.
Since being self-adjoint is central in many aspects of operator theory and in the study of
physical observables, it is important to understand when a symmetric operator can be extended
to a self-adjoint one. The relevant information is encoded in the defect space along with its
defect indices, consisting of the eigenspaces of some complex eigenvalue:
defect (z) ..= {v ∈V |T ∗T (v) = z · v} (dim(defectT (z)),dim(defectT ∗(z)))
For a self-adjoint operator purely imaginary eigenvalues are prohibited and thus dim(defectT (ı))=
dim(defect 29T (−ı)) = 0. This is also reflected in the resolvent set ρOp(T ) and the spectrum
σOp(T ) of T defined by
ρOp(T ) ..= {z ∈C |(T − z · idV ) :V →V is bijective} ,
σOp(T ) ..= C/ρOp(T )≡ {z ∈C | ker(T − z · idV ) 6= {eV} ∨ Im(T − z · idV ) 6=V}
The statement that T is (essentially) self-adjoint is equivalent to ker(T − (±ı) · idV ) = {eV} and
Im(T − z · idV ) is (densely) equivalent to V .
28For the details of measure theory we refer to section 2.1.
29It suffices to consider either +ı or (−ı), for dim(defectT (z)) is constant on the upper and lower complex half
plane [28].
1.4 Operators 69
Now, let us go back to a general operator T0 and systematically approach the question when
T0 gives rise to a self-adjoint extension, meaning that it is essentially a restriction of a self-
adjoint operator T , in the sense that
Graph(T .0) .= {(v,T0(v)) |v ∈Dom(T0)} ⊆ {(v,T (v)) |v ∈Dom(T )} ≡ Graph(T )
The necessary and sufficient conditions can be expressed on the basis of the defect indices of
T0. If dimdefectT (ı) and dimdefectT (−ı) are strictly positive or agree, i.e. dimdefectT (ı) =
dimdefectT (−ı), then T0 can be extended to a symmetric or self-adjoint operator T , respectively.
The enlargement of the domain space in order to promote a symmetric operator to a self-
adjoint one, is tightly connected with a suitable choice of boundary conditions. In particular
for the differential operators we encounter later on, linear maps L : Γ (E1)→ Γ (E2), we have to
impose suitable boundary conditions to apply the machinery of the spectral theorem.
For the remainder of this discussion on operator theory, let us consider a specific class of
differential operators, the elliptic ones, for they are of uttermost importance when studying
and understanding quantum gravity. In a sense, they describe some well behaved differential
operator that among all the infinities they are surrounded by, they may have a finite dimensional
kernel. A differential operator can be judged by its principal symbol, defined for some f ∈
C∞(M,R) with d f (p) ∈ T∗pM and p ∈M by
1 ∣
σ(L,d f )s(p) ..= L( fN · s)∣
N! p
whereby N represents the order of L, the highest power of differentiation in local coordinates.
If σ(L,d f ) is a linear isomorphism for any p ∈M and any d f (p) ∈ T∗pM/{eT∗M} we say that Lp
is weakly elliptic. In case of E2 = E1 an operator is additionally strongly elliptic whenever it is
positive, thus fulfills the condition
( ) ( )
∃c> 0 : σ(L,d f )(v) , v ≥ c · v , v ∀v ∈ Γ (E ), ‖d f‖
Γ (E ) Γ (E ) 1 T
∗M = 1
1 1
Such a differential operator L is a kind of generalized Laplacian operator that shares many
features with this famous map. Strong ellipticity is in fact restricted to operators of even order,
though a weakly elliptic odd operator, as the Dirac one, may be composed with itself to fulfill
the stronger condition. While there are many applications of weakly elliptic operators – in
particular the Atiyah-Singer index theorem – the full range of implications, as for example the
discreteness of all eigenvalues, is only guaranteed for the strong elliptic ones. What is even
more important for practical applications, is the existence of the elliptic regularity theorem. It
assures that solutions v of any differential equation of an elliptic operator L, say L(v) = f are
at least as smooth as the contained coefficients and the resulting function f . For symmetric
operators, strong ellipticity further guarantees that there is only a finite number of negative and
zero eigenfunctions, which is of particular importance for the existence of the heat kernel trace
evaluation.
The spectral theorem introduces some severe constraints on the set of possible eigenvalues
of a linear operator. In case of (weakly) elliptic operators we can finally give a classification of
its zero-eigenspace that turns out to be entirely described by topological properties of the base
space. Therefore, consider a Fredholm operator, meaning an elliptic linear map that has finite
kernel and co-kernel, for which the so called analytical index is well-defined:
index(L) ..= dimkerL−dimcokerL with kerL≡ {s ∈ Γ (E1) |L(s) = eΓ (E2)} ,
∧ cokerL≡ Γ (E2)/Im(L)
Whenever M is compact, this analytical property is tied to a topological index that classifies the
base space. This famous relation, that has many descendants as for example the Gauß-Bonnet
70 Chapter 1. Physical Content
theorem, goes under the name of the Atiyah-Singer index theorem [44]. As it stands it provides
an astonishing bridge between topology and analysis with many fruitful results in mathematics
and physics, in particular the study of instanton solutions.
1.4.2 Trace of operators
For the definition of a trace, we restrict to operators that act as endomorphisms on a single,
linear spaceV . In this case there is an equivalence between the set of finite rank endomorphisms
End<∞(V ) and the tensor product space of V and its continuous dual space V ⋆, i.e. V ⊗V ⋆ ≃
End<∞(V ) [45]. Explicitly, this correspondence is induced by a map r : End<∞(V )→ V ⊗V ⋆
with
E 7→ r(E) = vE ⊗βE with r(E)(u) = (βE(u)) · v .E .= E(u) ∀u ∈V
Thereby, vE and βE are implicitly defined by their action on functions and vector fields, re-
spectively. Furthermore, the linear map on the direct product space t : V ×V ⋆ → R given by
(v,β )→7 t(v,β ) = β (v) can be naturally extended to the associated tensor product space V ⊗V ⋆
by means of the universal property of the tensor algebra:30
T : V ⊗V ⋆ → R , with v⊗β 7→ T (v⊗β ) = t(v,β ) = β (v)
Finally, the composition of T and r yields a linear functional on the space of endomorphisms,
the trace:
Tr : End(V )→ R , with E 7→ Tr[E] = T ◦ r(E) = T (r(E))≡ βE(vE) ∈ R (1.10)
The last equality holds for r(E) = vE ⊗βE and provides a straightforward transition to a coordi-
nate formalization. However, as it is apparent from the above coordinate-free construction the
concept of the trace of a finite-rank operator is basis independent. In general the basis indepen-
dence can be guaranteed for all trace class operators for which additionally the trace converges
absolutely. The trace class of bounded operators {T} defined over a separable Hilbert space
consists of those linear maps for which Tr[(T ∗ ◦T )1/2] is finite.
The cyclic property of Tr follows immediately from the previous considerations, for let
E1, E2, and E4 be suitable endomorphisms with r(En) = vE ⊗ βE . For any permutation ofn n
composition we obtain
r(En ◦Em ◦Eℓ) = (vE ⊗βE )◦ (vE ⊗βn n m E )◦ (vm E ⊗βE )ℓ ℓ
= βE (vE ) ·βE (vE ) · (vE ⊗βE ) n,m, ℓ ∈ {1,2,3}n m m ℓ n ℓ
The linearity of the tensor product transfers to the linearity of T which thus produces the cyclic
property of the operator trace:
Tr [En ◦Em ◦Eℓ] = βE (vE ) ·βE (vE ) ·Tr [vE ⊗βE ] = βE (vE ) ·βE (vE ) ·βE (v )n m m ℓ n ℓ n m m ℓ ℓ En
= βE (vE ) ·βE (vE ) ·Tr [vE ⊗βE ] = Tr [Eℓ ◦Eℓ n n m ℓ m n ◦Em]
In case of operators acting on a supermodule over some superalgebra the usual trace Tr has to
be replaced by the super-trace STr for which the commutation rules produce some additional
factors of (−1). This becomes relevant when field space contains fermionic or ghost fields.
30A universal property is a simplified but still unique criterion for the construction of certain mathematical objects,
while the actual explicit construction may be cumbersome. For instance, the tensor algebra of a vector space V is
characterized by the statement that any linear function from the vector space V to some algebra A over the same
field, can be uniquely extended to a homomorphism between the algebras of the tensor space and A.
1.5 Functionals 71
1.4.3 Determinant of operators
In the Hilbert spaces we are most interested in, infinites plague the standard definition of ma-
trices and certain regularization techniques have to be (implicitly) used to generalize these con-
cepts for arbitrary operators. Determinants of operators are no exception to this requirement,
for they consist of the product of eigenvalues which in the infinite case will in general diverge.
There are several procedures to extend this defining property, among which the mathematical
most rigorous one is based on the zeta function:
( ∣ )
det (T ) ..= exp − dζ d ζT (s)∣ , with ζ (s) ..= Tr[T−sT ] =s s=0 ∑ λ
−s
j
λ j∈σOp(T )
Another version of functional determinant is given by the Fredholm definition for a special
class of bounded invertible operators of the form id+T where T is of trace class. Extending the
Hilbert space to its exterior power the associated operator description gives rise to the so called
Fredholm determinant:
∞
det .Fd(id+T ) .= ∑ Tr(r)[T︸ (•)∧︷· ·︷·∧ v(•︸)]
r=0
r-times
In both cases we see the implicit dependence on the trace definition as expected from its finite
dimensional analog. In fact for T being trace class and z ∈C, we have the following identities:
detFd(exp(T )) = expC(Tr[T ]) , logdetFd(id+ z ·T) = Tr[log(id+ z ·T)]
Finally, let us state a representation of the functional determinant that is more convenient for
practical purposes in quantum field theory. It relates to the properties of the Gaußian measure
that we discuss in detail in section 2.1. So assume we have a free theory given by a Gaußian
(S)
measure µG over some vector spaceV based on some positive self-adjoint operator S (as can be
read of from translation transformation). A related (non-normalized) Gaußian(measure )results
from the Radon-Nikodym theorem with respect to the function g(Φ) = exp(− 12 Φ , T (Φ) )TΦ̄[F]
for some positive self-adjoint operator T . The functional determinant is then given by
∫
det(S+T ) (S) (S) 1( )
µG (U)≡ µG exp(− ϕ , T (ϕ) )det(S) ∈ ⊆ 2 TΦ̄[F]ϕ U TΦ̄[F]
Notice that only the relative size of determinants is determined in this way and for U ≡ TΦ̄[F]
(S)
µG = 1. Furthermore, if we choose a S = id and T to be trace class, then this path integral
representation relates to the Fredholm determinant.
1.5 Functionals
Fields unfold their full beauty in combination with a physical theory that introduces interac-
tions. Usually the concrete information about the theory is encoded in an action functional
that transfers the field content into a measurable quantity, for instance a real (or complex) num-
ber.
Definition 1.5.1 — Functionals. Let V ≡ (XV ,◦V ,K,◦KV ) be a vector space over the field
K. A map A is a functional on V :⇔ A : XV →K.
This general definition gives rise to a huge class of different functionals, among which we have
to select the physical ones by some suitable constraints.31 For example, action functionals
31Notice that dual vectors, i.e. elements of V∗, are special functionals in that they additionally have to be linear,
which is not assumed to hold for a generic functional A.
72 Chapter 1. Physical Content
are usually assumed to have an integral representation in terms of a Lagrangian density over
spacetime. However, since we will work in an effective picture later on, we do not have to
impose any restrictions in this direction yet. Rather, we will use the virtue of group theory
again to reduce the number of candidates by symmetry constraints.
Nevertheless, let us consider some basic applications of generic functionals for a moment,
before let group theory infiltrate. The entire mathematical framework is supposed to be establish
on field space [F] or the total space F. In many regards it turns out to be convenient to convert [F]
into a (local) vector space by virtue of the background field method based on some background
field UBFC ≡ (Φ̄,F) or [UBFC]≡ ([Φ̄], [F]) and the respective exponential map. Thus, in terms
of fluctuation and background fields a general functional is given by:32
A : F→K , Φ̂ 7→ χA[Φ̂]≡ A[exp (ϕ̂(Φ̂,Φ̄))]≡ A[ϕ̂(Φ̂,Φ̄);Φ̄] with
Φ̄
A : TΦ̄F×F→K, A[•;Φ̄] : TΦ̄F→K, ϕ̂ 7→ A[ϕ̂;Φ̄]
Notice that ϕ̂ is the tangent vector that generates a geodesic connecting Φ̂ and Φ̄ with respect
to the field space connection χ . In the context of field theory, the set of functionals respecting a
certain symmetry invariance is denoted as theory space:
T RF ≡ {A : F→K | A[ρF (Φ̂, fG)] = A[Φ̂] ∀ fG∈ GF ∀ Φ̂ ∈ F}
As we will see, besides a differential structure, there is a subspace of T that can be described
by a vector space containing all local monomials based on the field content.
The benefit of working with functionals is due to the properties they may inherit from their
well equipped domain and target spaces. In particular, the space of linear functionals turns out
to be a vector space under pointwise evaluation. The smooth inner product space in which the
fluctuation fields live provides several important operations concerning linearity, topology, con-
vergence, and differentiability, while the complex (or real) numbers have an even richer struc-
ture. One aspect that needs many of those properties concerns the dependence of a functional
result on its field argument, the functional derivative.
1.5.1 Functional derivatives
The functional derivative addresses one of the most interesting question about functionals except
perhaps their symmetry properties. It gives a way to measure the sensitivity of a functional A
on changes of the fluctuation fields. While a functional that is constant on the entire field space
is of subordinate relevance, symmetry constraints may require a vanishing functional derivative
of A along a certain group orbit. This establishes the concept of functional derivatives in the
following sense:
Definition 1.5.2 — Functional derivative. Let F be field space principal bundle with χ a
principal connection. Then dvA[Φ̂] is the (covariant) functional derivative of A : F→ C :⇔
{ }
dvA[Φ̂]≡ lim 1ε A[exp
χ χ
(ϕ̂ + v)]−A[exp (ϕ̂)] with ϕ̂ ≡ ϕ̂(Φ̂,Φ̄) ∈ T
→ Φ̄ Φ̄ Φ̄
F
ε 0
The functional derivative is a covariant derivative on the space of functionals. In the background
field language it linearly translates the fluctuation field such that
{ }
dvA[ϕ̂;Φ̄]≡ lim 1ε A[ϕ̂ + v;Φ̄]−A[ϕ̂;Φ̄]ε→0
32We use the convention that arguments of action functionals are written in square brackets. For convenience we
use the same symbol for maps on field space and the maps for the background field method, distinguishing both by
a semicolon A[Φ̂]≡ A[ϕ̂(Φ̂,Φ̄);Φ̄].
1.5 Functionals 73
Since field space constitutes a Riemannian manifold, functional derivations reflect the usual
differentiation of functions on differentiable structures. In particular, the chain rule applies and
for some functional F : T∗ F→ C reads:
Φ̄
dvF[J(ϕ̂)] = ddvJ(ϕ̂)F[J] with v ∈ TΦ̄F
Notice that the interrelation of fields is provided by the transformation properties of operators.
A functional derivative on the other hand is an operation that maps a functional to another
functional. A very important concept is the second functional derivative that gives rise to the
Hessian and possibly to the Hessian operator of a functional:
Definition 1.5.3 — Hessian, and Hessian operator. Let F be the principal bundle associ-
ated to field space with χ [a princip]al connection, d the variation operator, and G the field
space metric. Then, Hessϕ̂ A[ϕ̂;Φ̄] is the Hessian of a functional A : F→ C :⇔
[ ] 1 ( ( ) ( ))
Hessϕ̂ A[ϕ̂;Φ̄] (v,w)≡ dw dvA[ϕ̂ ;Φ̄] +dv dwA[ϕ̂ ;Φ̄] ∀v,w ∈ TΦ̄F2
[ ]
If it exists, then Ĥessϕ̂ A is the Hessian operator associated to A :⇔
[ ] ( [ ] )
Hessϕ̂ A[ϕ̂;Φ̄] (v,w)≡GΦ̄ v , Ĥessϕ̂ A[ϕ̂;Φ̄] w ∀v,w ∈ TΦ̄F
The self-adjoint property of the Hessian operator is some very important requirement in QFT
for instance. Especially in the presence of spacetime boundaries additional assumptions on field
space are needed in order to extract a suitable Hessian operator for the action functional.
From the definition of the Hessian operator convexity properties can be read off, stating that
when the operator is (positive) non-negative an associated real functional is (strictly) convex,
i.e. it fulfills:
A[t · ϕ̂1+TF (1− t) · ϕ̂2;Φ̄]≤ t ·A[ϕ̂1;Φ̄]+ (1− t) ·A[ϕ̂2;Φ̄]
with < replacing ≤ in the case of strong convexity. The main advantage of convex functions is
the uniqueness of a (possible) extremum and the vast material that is available for its differential
calculus. In particular the information preserving Legendre transform is a central tool in QFT
that interrelates the generating functional with an effective action.
1.5.2 Basis invariants
In the usual calculus of functions it is very convenient to introduce a basis in which each and
every function can be expanded. The corresponding algebra is conventionally established in an
infinite-dimensional Hilbert space with some self-adjoint operator, a typical example being the
Fourier basis. Similarly, we can ask whether or not there is an analog basis on the space of
functionals, in particular on theory space TF. For functionals based on some non-local operator,
this is quite difficult to answer, however for the remaining subspace, which we consider in
the sequel, there is a stepwise construction procedure based on Young diagrams at least for
Levi-Civita connections D̄ for some metric ḡ [46]. We have already seen that the irreducible
components of tensors are obtained using permutations and partitions. For the basis invariants
that constitute the integral kernels of A ∈ TF the crucial observation is that
· · · ⇐⇒ ·· ·· · · ( p + 4 )D̄
︸ µ5
D̄
︷︷ µ
R̄
p+︸4 µ1µ2µ3µ4
p-times
Here R̄ is the Riemann tensor with all indices lowered by ḡ. Each tensor that is a combination of
such derivatives of R̄ can be uniquely decomposed into irreducible representations using Young
74 Chapter 1. Physical Content
diagrams. The association of partitions and tensor structures is not as straightforward, but at
least the implications of the group theoretical considerations of the symmetric group about
dimensionality and the number of derivatives is a great guiding principle.
Since we are only interested in the reduction of tensors to invariant scalars of the Rieman-
nian metric ḡ, for each partition a unique tensor structure survives the decomposition procedure
and in fact it is classified by all rows being of even length. In addition, as mentioned for the
general linear group, diagrams with more then d rows vanish identically. Thus, the number of
different basis invariance is sensitive to the spacetime dimension, which has strong implications
for theories of gravity especially in d = 2.
Keeping these simple rules in mind one can deduce the possible invariants of particular
order in the derivatives which actually give rise to a basis for the local part of TF. For more
details we refer to [46].
R As an example, let us consider the decomposition of R̄µ1µ2µ3µ4 ≡ combined with
D̄ν5 D̄ν6 R̄ν1ν2ν3ν4 ≡ :
· = + + +
whereby all contributions with at least one odd row length have been neglected. De-
pending on the dimensionality of M we obtain no, one, three, or four independent basis
monomials contained in R(ḡ)D̄D̄R(ḡ), namely:
R̄D̄2R̄ for d ≤ 2
R̄µν D̄µD̄ν R̄ for d ≤ 3
R̄µν D̄2R̄µν for d ≤ 3
R̄µ1µ2µ3µ4D̄µ3D̄µ4 R̄µ1µ2 for d ≤ 4
Especially for the heat kernel expansion this technique provides a very instructive way to
construct a suitable basis for TF in case of quantum gravity.
2.1 Measure theory
Measurable spaces
Normalization
Observables
Relation between measures
Measures for gauge theories
Interpretation
2.2 Effective action
Schwinger functional
Effective action
Free theory
2.3 Quantum field theories
Quantum field theories
Theories, phases, and anomalies
Perturbation theory
Euclidean quantum field theory
2. QUANTUM FIELD THEORY
In chapter 1 we considered the ingredients our theoretical description of Nature is based on.
While mathematics provide an infinity of different possible structures we have focused on those
which seem to be favored from a physical perspective. Assuming spacetime to be continuous
and equipped with a differential construction reflects our intuitive (local) picture of what reality
seems to be. By this we have fixed the notion of locality in a very particular way. Next, we
introduced fields as the mediators of interactions and we were led in the realm of differential
geometry and fiber bundles. Again from a mathematical point of view there were plenty of op-
portunities however the most conservative one stuck to differential objects based on Lie groups.
The reason for this choice may reflect the beauty of the Universe to possess a set of symme-
tries that may be tied up with our notion of interactions. We always followed the lines of what
seemed most natural and stay close to the path of our intuitive understanding. Generations of
physicists have enlarged those basic concepts to take care of predictions and experiments that
could not be described within this simple ideas.
Still, we have to decide about another fundamental assumption that builds up our description
of Nature: the theoretical framework. The question that we have yet to answer concerns the way
we combine the ingredients to form a consistent theoretical setting that covers the observations
and predicts the unobserved phenomena.
So, let us begin this chapter on theories and rules with some philosophical considerations.
This brings us back to the basic question physics is concerned with, namely ‘what is Nature?’.
Centuries of great thinkers inquired into the origin and cause of the Universe, whereby logic
has played a very important role. Over the time mathematics has emerged as the language of
science, for its ‘objective’ and changeless character. Whether or not Nature speaks in this lan-
guage is a mystery, for us it will be only one more assumption we have to live with. But even
after the restriction to logic a vast regime remains, thus further assumptions are needed. With a
theoretical construction we somehow fix the grammar and the kind of words Nature is allowed
to use and then verify if the sentences give rise to different facets that can be observed. Hence,
once we have mathematical consistency, everything reduces to observations and the value of a
theoretical setup is measured in its reproduction of known and its prediction of unknown effects.
76 Chapter 2. Quantum Field Theory
The guiding principle to obtain suitable frameworks is surely our perception, however the his-
tory of science has revealed that this might be misleading. Nowadays, consistency, simplicity,
and aesthetics of the mathematical description feature prominently in staying on the track, and
here again symmetries stand out.
Observations symbolizes the lighthouses in the construction of theories and lead to a basic
concept of quantum field theory: (re-)normalization. Here, the vicious cycle is that in order to
draw conclusions from observations we have to frame them into a particular theory. Thus, both
concepts are strongly interwoven and only inconsistencies may help to unravel this knot. The
question what observations can tell us about reality and its rules is related to the question of
determinism and probability. While a deterministic philosophy assumes a single observation
contains all information of its underlying principle, the probabilistic image of the world takes
the position that only a (possible infinite) number of identical observations may reveal Natures
secrets. For long it had been the conception that determinism is at the heart of all rules, but
quantum physics severally shook this confidence. It is still an open issue whether the quantum
world is the door to some new deterministic theory or if probabilities take part in Nature’s game.
The lessons we learn from this is rather of practical than of philosophical character and involves
the mathematical theory of probability. This assumption still covers both, deterministic and
stochastic concepts, and will naturally give rise to a statistical description closely related to the
path integral formalism of quantum mechanics. Only at the very end when choosing a certain
configuration and interpret the results one has to take sides of determinism or indeterminism.
To establish such a broadly applicable concept, we will start with a short summary of mea-
sure theory. Subsequently, we will put the notion of theory and observables on a more solid
ground and have a look at systematic constructions of the former. The remaining sections of
this chapter are then devoted to allocate physics to this mathematical framework and introduce
some simplified methods to evaluate observable quantities. With this at hand, we are then pre-
pared to study a powerful and universal tool in the next chapter, the renormalization group.1
2.1 Measure theory
Probability theory has found its way into physics at the latest in shape of the Wiener process
that describes Brownian motions relying on concepts of measure theory. Its greatest success
and the foundation of this formalization at the root of quantum field theory were laid by Dirac
and some years later by Feynman, see ref. [28]. Till now the generalization of the so called path
integrals play a major role in describing the fundamental forces of Nature. The mathematical
aspects of this section can be found for instance in [47] while the relation to physics is based on
[28].
2.1.1 Measurable spaces
The basic common idea regardless of whether applied to condensed matter physics, quantum
field theory, or societal statistical problems is to sum over all possibilities and get out the most
likeliest candidate.
In the present case, we have imposed certain symmetry and further physically motivated
restrictions to construct field space, the set of all admissible global sections on a certain fiber
bundle. It is this space our operators act on and in which we have to search for suitable solu-
tions to deduce real observables. Hence, we have to somehow weight portions of field space
differently depending on their physical importance or likelihood. Measure theory establishes
1The conceptual program of this chapter is mainly based on a lecture which Maximilian Demmel and myself
delivered in the winter term 2014/2015 at the University of Mainz. It was a real inspiration and full of fruitful
discussion for which I like to thank Maximilian and the audience, giving me this opportunity.
2.1 Measure theory 77
a framework that takes care of the mathematical side of this procedure while on the physical
side we still have to choose among the different possible weighting methods by means of a
consistent theory.
In order to assign a weight or size to each collection of fields we have to endow its subsets
with a suitable algebra that can be related with the real numbers, in particular with its order prin-
ciple. Therefore, we have to endow it with a commutative associative operation that is closed
and invertible and that can be scaled in a certain way. The combination of both requirements
give rise to the definition of σ -algebras:
Definition 2.1.1 — σ-algebras. Let X be a set, 2X its power set. A collection of subsets
Σ ⊆ 2X is a σ -algebra on X :⇔
X , 0/ ∈ Σ (=⇒ existence of inverse & closure)
∀S⊆ Σ, #(S)≤ #(N) : (∪U∈SU) ∈ Σ (closed under countably infinite unions)
∀A ∈ Σ : X/A ∈ Σ (closed under complement)
Elements A ∈ Σ are denoted as measurable sets and (X ,Σ) is called a measurable space.
Cast in this form the required additive and scaling operations are disguised in this more conve-
nient definition. The algebraic construct that reflects addition of real numbers is the symmetric
difference, given by
∀A, B ∈ Σ : A∆B= (A/B)∪ (B/A)
with identity element 0/ and A being its own inverse. Thus (Σ,∆) forms an Abelian group and is
the counterpart of (R,+). On the other hand, ∩ can be regarded as the multiplication of subsets
and (Σ,∩) forms a commutative monoid with identity X , but lacks an inverse. Furthermore,
X/• interpreted as a scalar multiplication of {X} on Σ completes the picture of an algebraic
structure on 2X that models R in a suitable way. We have seen that for a consistent closure of Σ
we have to include X and 0/ . Thus σ -algebras really satisfy the requirements of an algebra.2
In the construction of spacetime we have encountered a similar prescription to introduce
locality into the physics: topology. Topologies and σ -algebras are constructed by endowing
subsets with a specific structure and in fact they share some of their defining properties. First
they distinguish X and 0/ as open and measurable sets, respectively, and secondly they are both
closed under finite intersection (for σ -algebras that is a direct consequence of the remaining
properties). Apart from that, both concepts lay focus on different aspects that fulfill their respec-
tive purposes. While topology allows for any, even uncountably infinite, unions of its elements,
σ -algebras asks only for a countably infinite closure but additionally includes all complements
into its algebraic structure by definition. So in general a topology might not be a σ -algebra,
in fact every Hausdorff space fails to include its complements, and vice versa. Nevertheless,
the fusion of openness and measurability for a certain collection of subsets is a very interesting
and worthwhile problem that is of particular importance in the theory of Lebesgue integration.
The general question is treated in the study of Borel σ -algebras that define measurable sets as
generated from a topology, which is not in general possible [47].
The preparation of those subsets in the way of σ -algebras is only the first step in estab-
lishing a thorough treatment of probability theory. Next, we have to define the root of a the-
ory that allocates those measurable sets a particular value in the real numbers, the measure
2The prefix σ indicates that we have more than a usual algebra of finitary operations, but in fact allow for a
closure of countably infinite unions. This additional restriction takes care of part of the infinities that naturally arise
in the theory of integration.
78 Chapter 2. Quantum Field Theory
itself:3
Definition 2.1.2 — Measures. Let X be a set, Σ ⊆ 2X a σ -algebra.
A function µ : Σ → [0,+∞] defines a measure on Σ :⇔ µ(0/) = 0
∀S⊆ Σ, #(S)≤ #(N) with A∩B= 0/ ∀A,B ∈ S :
µ(∪A∈SA) = ∑ µ(A) (Countable additivity)
A∈S
The ordered pair (X ,Σ,µ) is called a measure space. Given a measurable space (X ,Σ) the
set of well defined measures (with µ(X) = 1) is denoted Meas(X ,Σ) (MeasP(X ,Σ)).
One can extend these non-negative measures to signed or complex ones by Hahn’s decompo-
sition theorem [47], whereby special care has to be taken when treating positive and negative
infinities to preserve the ordering and multiplication principles. Most of the following results
can be extended to signed and complex measures, but we will not mention them any further.
Though important in today’s understanding of physics, due to the seemingly Lorentzian sig-
nature of the gravitational field, in this work we focus on Euclidean theories that allow for a
probabilistic interpretation. The clash between the observational data and this assumption can
be dissolved whenever the Osterwalder-Schrader axioms apply, allowing for a transition from
Euclidean to Lorentzian theories. We will come back to this point in chapter 3 where the full
motivation for the Euclidean treatment is discussed.
The way we constructed measurable sets and the measure itself, allows to endow the class
of all measures on (X ,Σ) with the structure of a linear space, (scalar multiplication, additivity).
For signed and complex measure we can even extend this linear space to a vector space since
then the identity and inverse elements exist.
2.1.2 Normalization
Having established a collection of measurable subsets we encode their relative importance in
a suitable measure. This function takes care of weighting the physically relevant subsets the
most and the irrelevant ones least. Thereby one usually assigns to infinite sized subsets (infinite
cardinality) a finite value in accordance with countably additivity of the measure. This inherited
property of R yields a well-behaved regularization procedure and provides a consistent way to
weigh uncountable infinite sets.
An even deeper regularization is obtained by a probability measure that normalizes the
weight of the entire space to one, i.e. µ(X) = 1. This choice of normalization is a particular
suitable one, for it takes care of another source of infinities by invoking an additional exter-
nal requirement, here the observation that all events (measurable sets) taken together cover all
possibilities that might appear. With this constraint the number of possible measures reduces
significantly, at least if Σ is of infinite cardinality.
In general, we have a series of mathematical conditions our theory, the measure, has to
satisfy. Among them, three additional constraints on µ are of particular importance for the
reflect the integration properties on Euclidean space:
Definition 2.1.3 — Measure properties. Let (X ,Σ,µ) be a measure space. Let G= (Y,◦G)
be a group and ρ : G→ Aut(X) be a free group action on X .
3For this general definition we have to extend the real numbers by positive infinity. The extension of the algebra
is based on the following rules: 0+∞ = ∞+0 = ∞ and 0 ·∞ = ∞ ·0= 0.
2.1 Measure theory 79
The measure µ is denoted . . .
. . . σ -finite :⇔ ∃S⊆ Σ, #(S)≤ #(N) ∧ X = ∪A∈SA : µ(A ∈ S)< ∞
. . . complete :⇔ ∀S⊆ N ∈ Σ ∧ µ(A) = 0 =⇒ S ∈ Σ
. . . ρ-invariant :⇔ ∀g ∈ Y, ∀A ∈ Σ :
[ρ .g(A) .= {ρg(a) |a ∈ A} ∈ Σ] ∧ [µ(A) = µ(ρg(A))]
Probability measures are severely constraint by the simultaneous normalization and regulariza-
tion condition µ(X) = 1.4 However, in most cases we do not have a way to measure the ‘size’
of the full space X , hence to construct a scale relative to the overall size is not feasible. For
several purposes it seems more natural to invoke the weaker form of σ -finiteness on a measure,
that introduces a less strong regularization on µ , but leaves the normalization open. Though the
entire space X may have a countable infinite ‘size’, it is composed of finite subsets that allow for
a successive treatment. It is this condition that we naturally use when defining a (local) length
scale.
Completeness of a measure refers to the concept of null sets, subsets that have no influence
on the integration process, for the do not contribute at all. In physical terms, the contained fields
are excluded in the evaluation and may be denoted as unphysical. Statements that hold true ex-
cept for some null sets of µ , termed µ-almost everywhere, acquire full validity within measure
theory. A thorough treatment of null sets is vital when playing around with measure spaces, for
even though they are invisible on their own, they can have an impact when considering products
of measures. The consistent way to take care of these cases is by completing the measure to
include all subsets of its null sets, or using a complete measure from the outset. Another related
concept is the support of a measure, that is the collection of all sets which have non-vanishing
measure.
The third condition can be thought of as the most extensive one, for it implements symme-
try constraints, a very powerful, multi-faceted, and aesthetic construction. The invariance under
a certain group action severely restricts the class of possible measures on a given σ -algebra
and it will play may be the most important role in forming a sufficiently reduced mathematical
description of Nature. Still, even with very strong constraints from the mathematical side, in
the end one has to build a bridge from theory to reality with the aid of a physical motivated
normalization. This, then hopefully selects a unique theory from all remaining ones, and only
for probability like measures the mathematical regularization may come along with a normal-
ization.
R An important example is the Lebesgue measure on the Borel sets of real numbers, (the
σ -algebra generated from the standard topology on R). It is the natural candidate for
a length (or more generally volume) measuring device, based on three principles: σ -
finiteness, completeness, and invariance under translation (ρa∈R(b ∈ R) = b+ a). These
mathematical constraints leave only a one-parameter family of measures
µL(A= (a,b);λ ) = λ · (a+Rd (−b))
The transition to reality is given by fixing the non-negative parameter λ when assigning
a particular length scale to a standard object.
By introducing a measure we have defined a theory that has to be matched with all observ-
able phenomena it is supposed to describe. One such observation may be sufficient to pin down
the mathematical freedom by the corresponding normalization condition. It is at this step were
4Every finite measure, i.e. 0 6= µ(X) < ∞, can be transformed into a probability measure by an additional
normalization: µ(A)µP(A) = µ( .X)
80 Chapter 2. Quantum Field Theory
theory and experiment meet in order to dissolve the remaining uncertainties of the mathematical
formalism and distinguish whether or not we have followed the right path. Every observation
that contains some notion of size, intrinsically depend on a normalization, and the appearance
of a unit is an indicator for a non pure-mathematical result. We will later see how these nor-
malization, i.e. the mapping of observations to theoretical parameters, can be used as a guiding
principle in deducing fundamental theories.
2.1.3 Observables
Though fields are thought of as the fundamental building blocks that mediate interactions and
form the matter content of the Universe, we usually only observe their effects rather than having
direct access to their shapes. In addition, for observations in the realm of quantum field theory
we choose a measure, and thereby a particular realization of a theory, that judges the influence
of an entire subclass of fields instead of evaluating a single field configuration. Thus, having
a measure spaces that yields a full description of the weighting of field space is crucial, but it
is only half the story. In order to deduce predictions and establish a normalization condition
we have to provide the missing link to observables. These are functions that map a specific
field configuration to an observable value, which is then weighted by the importance of this
field using the underlying measure. It is a kind of very general averaging of functionals on field
space, covered in the concept of measurable functions:
Definition 2.1.4 — Measurable functions. Let (X ,ΣX ,µX ) and (Y,ΣY ,µY ) be measure
spaces. A function f : X →Y is a measurable function :⇔
∀AY ∈ ΣY : f−1(AY )≡ {ϕ ∈ X | f (ϕ) ∈ AY} ∈ ΣX
If (X ,ΣX ,µX ) stands for a theoretical description f is called an observable.
The definition of measurable functions, or observables, mirrors the way we defined continuous
functions in the category of topologies. In fact, if both σ -algebras are generated by a topol-
ogy, the concept of measurable and continuous functions coincide and we have an intuitive
understanding of what observables may look like.
Most of the time the target space will be a real finite-dimensional Euclidean space equipped
with a Lebesgue measure. This is due to our understanding of the world that is basically con-
structed on top of the real numbers and leads to a certain limitations in extracting observables
by experimental devices. We therefore usually stick to the case of f : X → Rd .
Now, the evaluation of an observable on the basis of a certain theory TX ≡ (X ,Σ,µ) pro-
ceeds as follows. First, consider a very simple class of observables, the indicator functions,
given by
{
1 ϕ ∈ A ∈ Σ
χ : Σ×X →{0,1} ⊂ R, χA : X →{0,1}, ϕ →7 χA(ϕ) =
0 ϕ ∈/ A ∈ Σ
As its name suggests, χA indicates whether or not a field is contained in the respective mea-
surable set A ∈ Σ. They can be used as a basis to build more general measurable functions by
exploiting the rich structure of R again. Besides multiplication by a (complex) constant we
may add, subtract, or take the product of indicator functions to obtain an extended class of new
measurable functions, denoted simple:
Definition 2.1.5 — Simple functions. Let (X ,ΣX ,µX ) and (C,ΣC ≡ gen{τC},µL;C) be
2.1 Measure theory 81
(complex) measure spaces. A function f : X → C is a simple function :⇔
+C
∃S≡ {A j} ⊆ Σ, #(S)< ∞ ∧ A j∩Ak=6 j = 0/ , a j ∈ C : f (ϕ) = ∑ χA (ϕ)◦j C· a j
A j∈S
The set of all simple functions on Σ is denoted SΣ, its restriction to non-negative (0 ≤ a j ∈
R) ones S+Σ . It gives rise to a commutative algebra over the complex numbers and can be
extended to describe all measurable functions. Explicitly, given a measure space (X ,Σ,µ), any
measurable function f : X → [0,∞] can be expressed in terms of a sequence ( fℓ)ℓ∈N of simple
functions on Σ:
∀ϕ ∈ X : f (ϕ) = lim fℓ(ϕ)
ℓ→∞
This is a remarkable result that distinguish simple functions as fundamental objects from which
we can deduce an even larger class of functions. In a last generalization matched for our need
to derive intermediate results in field space, we consider any measurable function f : X → Y
on associated σ -algebras ΣX and ΣY , respectively. As long as the target space of f , i.e. Y , is
a normed vector space over the complex or real numbers with a suitable notion of convergence
we can express f in terms of simple real-valued functions, as follows:
( )
+C
(ℓ) (ℓ)
f (ϕ) = lim
→ ∑χ (ℓ)(ϕ)◦RY a j , where a j ∈ Y, ∀φ ∈ Xℓ ∞ A j
j
Finally, to keep sight of our objective to evaluate ‘averaged’ observables we apply the de-
composition of measurable functions into sums of indicator functions. This leads to the well
known concept of integration within the abstract field of measure theory:
Definition 2.1.6 — Integral. Let (X ,ΣX ,µ) and (Y,ΣY ,µY ) be measure spaces, whereby
Y is endowed with a R-vector space structure. Further, let f : X → R be a non-negative
measurable functio∫n.
The map 〈 f 〉 µ | ≡ ϕ∈B dµ(ϕ) f (ϕ) is the integral of f over B ∈ Σ :⇔B
∫ { }+C +C
〈 f 〉 µ | ≡ dµ(ϕ) f (ϕ) ..= sup ∑ µ(A j ∩B)◦C· a j | ∑ χA (ϕ)◦C· a j ≡ g≤ fB ϕ∈ jB g∈S+Σ A j∈S A j∈S
If 〈 f 〉 µ | < ∞ we say f is integrable and an element of L1(X ,Σ,µ ;Y ).B
In the context of physics the integral of an observable is usually denoted as its expectation
value. Since many of the relevant measurable functions also contain negative parts there are
extensions to general real or complex valued functions using a decomposition like f (ϕ) =
f+(ϕ)+(− f−(ϕ)), whereby = f+ and f− have to be non-negative integrable functions. Simi-
larly, functions with values in an arbitrary vector space over C can be decomposed in this form
and are therefore equipped with a notion of integrability.
It turns out that the integral is a linear functional on the space of measurable functions
Ln(X ,Σ,µ ;C), for it linearly maps the vectors f to its scalars, the real (or complex) numbers.
One can show [47] that the integral does not depend on a specific decomposition into simple
functions. Thus, we have found the solid mathematical grounds to treat physical observables
consistently with the underlying theory µ . For a particular operator on field space X the in-
tegral provides an instrument to extract its averaged expectation value that may correspond to
observable data and is thus helpful in verifying the theory.
82 Chapter 2. Quantum Field Theory
2.1.4 Relation between measures
Fields as the natural ingredients for quantum field theory live in what we have called field space
[F]. Equipped with a reasonable σ -algebra Σ[F] we can construct the class of all theories by
defining measures on Σ[F] and giving a physical meaning to certain observable operators on [F].
This leads to a plenitude of different theories among which we have to select the one realized in
Nature. To reduce the number of possibilities one usually invokes physically motivated, mathe-
matical constraints that hopefully leaves only a finite number of free parameters that have to be
fixed by normalization conditions. In one or the other way we need to translate the mathemat-
ical side of measure theory into the language of physics. To this end, we will introduce a very
natural link between mathematics and physics given by the Gaußian measure µG that we define
to be a free, non-interacting theory. Having established the first contact we use this distinguish
element as a reference point to spread a net over the full class of theories. The threats we spin
are based on the Radon-Nikodym theorem that filtrates all σ -finite measures and allows for a
first classification of the space of measures on Σ[F].
Gaußian measure
In order to implement the interface between quantum field theory and measure spaces, we have
to clarify the notion of non-interaction in both cases. From a mathematical perspective, given a
measure space ([F],Σ[F],µ) this corresponds to the question of full independence of the elements
in [F] within the theory µ . The basic idea is to fully decompose the measure in a product of one-
dimensional non-interacting theories without loosing information and retaining the measure
specific structure. While there is a more general description of independence, we only consider
a special class of non-interacting measures, the Gaußian ones, for its status in physics as a free
field theory.
First, let us build new measures from existing ones in the most natural way:
Definition 2.1.7 — Product measure. Let (X ,ΣX ,µX ) and (Y,ΣY ,µY ) be measure spaces.
The measure space (X ×Y,ΣX×Y ,µX×Y ) is a product measure space :⇔
ΣX×Y ≡ gen{(AX ,AY ) |AX ∈ ΣX ∧ AY ∈ ΣY}∫
∀B ∈ ΣX×Y : µX×Y = dµX (x)µY ({y ∈ Y |(x,y) ∈ B})
∫x∈X
= dµY (y)µX ({x ∈ X |(x,y) ∈ B})
y∈Y
We emphasize the relation between µX , µY , and µX×Y by writing µX×Y ≡ µX ×µY .
Notice that the very existence of the product measure results from the Hahn–Kolmogorov theo-
rem, while its uniqueness is in general only valid for σ -finite measures µX and µY . Indeed, as we
attempt to establish a measure theoretical approach to field space, an infinite dimensional mani-
fold that only locally agrees with a suitable vector space, the main focus lays on controlling the
arising infinites. Again a decomposition into an infinite number of constitutions may spoil their
joined properties and in fact the most natural candidate for a measure on field space, an infinite
Lebesgue measure fails to exists at all. For any separable Banach space (a complete normed
topological vector space) of infinite dimension there does not exist a translation-invariant Borel
measure, in particular, the infinite product of Lebesgue measures fails to be of Lebesgue type.
Moreover, any infinite product of the same finite measure µ on (X ,ΣX) is either trivial, infinite,
or a probability measure if and only if µ(X) is less, larger, or equal 1, respectively. Thus, the
natural candidates to extrapolate existing measures to field space are the probabilistic measures,
of which the Gaußian one are particular suitable for our purpose.
For the construction of this class of theories, we have to introduce some notations that relate
2.1 Measure theory 83
measures with each other on a formal level. The first concerns their equivalence:
Definition 2.1.8 — Equivalent measures. Let (X ,Σ) be a measurable space and µ : Σ →R,
ν : Σ → R be two measures on Σ. We say µ ∼ ν , i.e. µ and ν are equivalent :⇔
∀N ∈ Σ : (µ(N) = 0 =⇒ ν(N) = 0) ∧ (ν(N) = 0 =⇒ µ(N) = 0)
That is, µ is absolutely continuous w.r.t. ν and vice versa.
The crucial information is thus encoded in the null sets of µ and ν . Only if the null sets of
two different measures agree both are considered equivalent in this respect which defines an
equivalence class on the associated linear space. This definition is also illustrative from the
physicist’s perspective, since non-relevant fields of a theory should not become relevant in a
different representation.
The equivalence relation∼ interconnects measures on the same underlying σ -algebra. How-
ever, we want to go beyond a single measurable space and study how seemingly different the-
ories turn out to be the image of one another and thus the same. The associated technical term
pushforward measure actually refers to the commonly known concept of a change of variables
in which a suitable transformation of the underlying set is promoted to a transition of entire
measure spaces (in form of a covariant functor):
Definition 2.1.9 — Pushforward measures. Let (X ,ΣX) and (Y,ΣY ) be measurable spaces
and µX : ΣX → R a measure. A measure νY : ΣY → R is the pushforward measure of a
measurable function f : X →Y :⇔ :
∀UY ∈ ΣY : νY (UY ) = µX ( f−1(UY )≡ {x ∈ X | f (x) ∈UY})
One usually emphasizes the relation between µX and νY by writing νY ≡ f∗(µX ).
This is clearly what we have in mind when performing a change of variables in the usual fi-
nite dimensional Lebesgue integration. The tied connection between µ and its pushforward
measures f∗(µ) is also evident by the identification of their respective integrable functions:
g ∈ L1(Y,ΣY , f∗(µX );C) ⇐⇒ g◦ f ∈ L1(X ,ΣX ,µX ;C)∫ ∫
∀g ∈ L1(Y,ΣY , f∗(µX );C) : d[ f∗(µ)](y)g(y) = dµX (x)g( f (x))
y∈UY x∈ f−1(UY )
Now, with these tools at hand we are able to define Gaußian measures in a very general
sense using its properties on the real numbers:
Definition 2.1.10 — Gaußian measure. Let V ≡ (X ,τX ,◦X) be a topological vector space,
V ∗ its dual, andV ⋆ ..= {w∈V ∗ |w continuous}⊆V ∗ the restriction to continuous dual vectors.
Further let (X ,Σ) be a measurable space. A measure µ : Σ → C is a Gaußian measure on
(V,Σ) :⇔
gen{UX ∈ τX} ⊆ Σ ∧ µ(X) = 1 (Borel probability measure)
∀w ∈V ⋆ ∃σ , λ ∈ R : w∗(µ) : R→ R,
∫
exp(−‖x−λ‖2 /2σ 2)
w∗(µ)(UR) = dµL(x) √ R (Gaußian distributions)
x∈UR 2πσ 2
Here µL denotes the Lebesgue measure on (R,ΣR).
While the Lebesgue measure in infinite dimension does not exists, the Gaussian does for it fails
to be translational invariant. For Banach spaces, it follows as a special case of the Cameron-
84 Chapter 2. Quantum Field Theory
Martin theorem that under group translations, ρa :V →V with ρa(v) = v◦V a, for any Gaußian
measure µG there exists a self-adjoint positive operator Sµ :V →V such that for allU ∈ ΣG
∫
ρ ( ( ) ( ) )
µ aG (U)≡ µ 1G(ρa(U)) = dµG(x) exp − 2 a , Sµ (a) + x , Sµ (a) (2.1)G V G V
x∈U
whenever a is a square integrable function ofV , otherwise both measures are not equivalent and
no such relation can be established. Thus, for any Gaußian measure there is a certain admissible
operator Sµ implicitly defined by translations and vice versa. Notice however, that the zeroG
operator is excluded for then the corresponding measure would be translational invariant, thus
ill-defined in infinite dimensions. Furthermore, it is important to keep in mind that the Gaußian
measure is only defined on the subspace V ⋆ of dual vectors.
One of the great advantages of Gaußian measures is the possibility to characterize even
infinite dimensional measures with the aid of a positive semi-definite, symmetric, bilinear form
q on V ⋆ (see for example [48]):
∫ ( )
dµG(x)e
iw(x) = exp 1C − 2q(w,w) (2.2)
x∈X
The full power of this relation will become clear when studying the effective action approach
to quantum field theory, but it should be noted here that eq. (2.2) plays a very fundamental role
in understanding physics. Notice, that every positive bilinear q form can be uniquely identified
with a positive self-adjoint operator T : V ⋆ → V ⋆q , see ref. [28], and thus eq. (2.2) can be
rewritten in terms of the usual inner product on V ⋆, i.e.
∫ ( ( ) )
dµG(x)e
iw(x) = expC − 12 w , Tq(w) ⋆ with T
∗
V q
= T
∈ qx X ( )
and w , Tq(w) ⋆ ≥ 0 ∀w ∈V ⋆V
The Gaußian measure is the ideal prototype for a free theory. If there are no dominant con-
tributions that result in non-equally distributed effects, everything smears out and what remains
is a free Gaußian measure (the statement of the central limit theorem). The natural construction
of free theories is thus built on Gaußian distributions which provides the missing link between
physics and the mathematical theory of measures:
Definition 2.1.11 — Free theory. Let V ≡ (X ,τX ,◦X ) be a topological vector space, V ∗ its
dual, and as above V ⋆ ..= {w ∈ V ∗ |w continuous} ⊆ V ∗. Further let Sym2+(V ) denote the
space of symmetric, positive semidefinite, bilinear forms on V ∗.
The measure space T ≡ (X ,Σ,µ) is called a free theory :⇔
µ is a Gaußian measure
∫ ( )
∀w ∈V ⋆, ∃q ∈ Sym2+(V ) diagonalizable : dµ(x)eiw(x) = exp 1C − 2q(w,w)
x∈X
Thus, a free theory is described by a Gaußian measure that can be split into a complete
product of Gaußian measures on the eigenspaces of an operator on V ⋆.
Under this choice the basis of the (linear) observables is fully uncorrelated and thus the asso-
ciated rays are independent of each other. Whenever a theory is found to be described by a
Gaußian measure we will denote it as a free theory and interpret the underlying fields as non-
interacting.
2.1 Measure theory 85
Radon-Nikodym theorem
We have given the class of Gaußian measures a physical interpretation as free, non-interacting
theories. In a sense, we have normalized the space of measures with a very convenient as-
sumption. From this distinguished elements onwards we can establish further relations between
the mathematical objects and physical explanations. Basically, there are two possibilities to
propagate through the space of theories.
The first method concerns a transformation of the underlying field space as in the case of
integration by substitution. When this change of ‘coordinates’ is based on isomorphisms in the
category of measure spaces, i.e. measurable, bijective functions, it allows us to identify indis-
tinguishable theories. It was for this reason that we had to set up a more elaborated definition
for free theories in order to assure that simple transformations of coordinates do not cloak their
free characters.
The above discussed deformation of field space allows to connect different measure spaces
via pushforwards, and in fact identify them. A second method that is more relevant to actual
get an understanding of the physical content of a theory relates measures in the same σ -algebra,
the famous Radon-Nikodym theorem:
Theorem 2.1.1 — Radon-Nikodym theorem. Let (X ,Σ) be a measurable space, µ : Σ →
[0,∞) a σ -finite positive measure, and ν : Σ →C a σ -finite (complex or signed) measure with
∀A ∈ Σ : µ(A) = 0 =⇒ ν(A). Then the Radon-Nikodym theorem states:
∫
∃g ∈ L1(X ,Σ,µ ;C) : ν(A) = dµ(ϕ)g(ϕ) ∀A ∈ Σ (2.3)
ϕ∈A
The function g is unique µ-almost everywhere and is called the Radon-Nikodym derivative,
sometimes denoted g= dνdµ .
This generalization of the Jacobi-determinant is the central statement on which we will build our
entire construction and deduction of theories in what follows. Especially we can now establish
a connection of Gaußian measures to more general theories ν , whereby the Radon-Nikodym
derivative g reflects the deviation from a free theory and thus contains the contributions of
non-trivial interactions. A sophisticated theory usually involves a sufficient amount of these
correlations that renders the evaluation process of observables arbitrarily difficult. Exploiting
the Radon-Nikodym theorem we have a way to study those interactions as fluctuations of a free
theory which is thus the origin of perturbation theory. In addition, we obtain the possibility for
transitions from positive to complex measures that may help understanding the Euclidean path
integral approach in the light of standard quantum field theory. As if that were not enough, also
for non-perturbative approaches it provides a mechanism to encode the structure of field space
in an effective description on which the functional renormalization group technique heavily
depend on and thus lays the ground for this thesis. To obtain an effective description we have
to introduce the Fourier transform of a measure, also known as the characteristic functional.
Fourier transform of a measure
We have introduced the framework of measure theory in a general way without referring to
specific properties of the underlying set [F] on which we build the σ -algebra and define a theory.
The introduction of the concept of Fourier transforms however requires [F] to be endowed at
least with an additional vector space structure over the complex numbers and thus we have to
make an exception to this rule. In order to model field space in what follows, this vector space
is usually infinite dimensional that brings in a certain amount of complications and the notion
of basis and topologies have to be reconsidered thoroughly. However, we will present a rather
incomplete picture and only give a mere motivation and analogy for the given definition. For a
86 Chapter 2. Quantum Field Theory
more detail account on the appearing subtleties we refer to [28, 47–49].
The general purpose of a Fourier transform is to analyze a vector space in the light of its dual
space. In most cases the full information of a function is also encoded in its Fourier transform
and both can be reconstructed from the other in a unique way. By definition maps a measure
subsets of field space to specific weights that are related to the relevance of the contained fields.
This relevance has to be physically motivated even though there are additional mathematical
constraints that reduces the possibilities. In real life, we are equipped with a certain amount of
observable data and the objective is to deduce the underlying theory, the measure, that matches
those findings. But rather than guessing a measure by brute force we can revert the description
and use the analysis of expectation values to successively shape a class of suitable theories.
The Fourier transform of a measure is the initial step for this procedure and formally defined
by
Definition 2.1.12 — Fourier transform of measures. Let ([F],Σ,µ) be a measure space
and [F] a (infinite-dimensional) vector space over C with dual [F]∗ = { f : [F]→C | f linear}.
Let z ∈C be a complex number. A functional µ̂ : [F]∗ →C is the Fourier transform of µ :⇔
∫
µ̂A( f )≡ dµ(ϕ) expC(z f (ϕ)) , ∀A ∈ Σ, ∀ f ∈ [F]∗
ϕ∈A
The subscript A indicates the range of integration and is omitted in case of A≡ X . For A≡ X
it is further custom to denote Z( f ) ..= µ̂X( f ) as the generating or characteristic functional
which is more common, especially in physics literature.
We have encountered an example of a Fourier transform already in eq. (2.2) when stating
a particular transformation property of the Gaußian measure. Notice, that each dual vector
f ∈ [F]∗ yields a map µ̂•( f ) : Σ →C that defines a measure on ([F],Σ). Whenever µ is a σ -finite
positive measure the Radon-Nikodym theorem applies since by virtue of the properties of the
exponential map, the Radon-Nikodym derivative is a nowhere vanishing measurable function,
and thus µ̂•( f ) is a σ -finite complex measure. The standard Laplace and Fourier transform
are recovered by the choice of z = −1 and z = ı, respectively. In the more general context, it
is difficult to extract the full set of properties for those Fourier transforms. Nevertheless, we
will mention some of them along with their additional assumptions and the range of admissible
z-values.
There is an intensive study of separable Hilbert spaces containing only functions with com-
pact support. The dual space – on which µ̂ depends on – is a special set of linear, continuous
functionals on field space: the distributions. In this case the uniqueness and existence of the
Fourier transform with z ≡ ı is guaranteed and in fact the above relation can be inverted using
Minlos’ theorem [28]. This states that given a functional w : Y → R on the set of compactly
supported functions Y = {J : Rd → C} satisfying
w(0Y ) = 1 (normalized)
∀cr ∈ C, ∀Jr ∈ Y : ∑ c̄s · cr ·w(Jr− J̄s)≥ 0 (positivity)
r,s
w :Y → R is continuous (continuity) (2.4)
there exists a unique Borel probability measure µ on the space of associated distributions with
∫
w(J) = dµ(ϕ)eıϕ(J) ∀J ∈ Y
ϕ∈X
Notice that positive definiteness of w(J) only implies that w(0Y )≥‖w(J)‖R ≥ 0 holds true, with
0Y the zero map. Therefore, the additional condition of normalization, w(0Y ) = 1, ensures that
2.1 Measure theory 87
µ is a probability measure. The requirements stated in eq. (2.4) are properties that are fulfilled
by any Fourier transform of Borel probability measures as long as the conditions of the Minlos’-
or more elementary the Bochner theorem are satisfied. In the context of Euclidean quantum
field theory these constraints build the basis of the Osterwalder-Schrader axioms [50, 51]. The
lesson we can learn at this point is that the Fourier transform is a useful method to characterize
measures over a certain σ -algebra and fully encode the theories information into a functional
of its dual space.
This close connection that is established between a vector space and its continuous dual
space can also be generalized to locally compact groups using the Pontryagin duality theorem
[52]. For any such topological group G ≡ (X ,◦G,τX) that is locally compact and Hausdorff,
there exists a one-parameter family of measures µ (G)H (•;λ ) on the Borel sets ΣG of (X ,τX) that
satisfies:
∀g ∈ X , ∀A ∈ Σ : µ (G)G H (g◦G A;λ ) = 〈A〉( ;λ ) (left-G invariant)
∀C ∈ ΣG compact : µ (G)H (C;λ )< ∞ (compactly finite)
∀U ∈ τX : µ (G)H (U ;λ ) = inf{µ
(G)
H (C;λ ) |C ⊆U ∧C compact} (inner regular)
∀A ∈ ΣG : µ (G) (G)H (A;λ ) = inf{µH (U ;λ ) |A⊆U ∈ τX} (outer regular)
This measure that we have encountered previously in studying group representations is called
the left Haar measure of G. In the context of group theory the Fourier transform provides a
natural correspondence between the group and its dual group G⋆, i.e. the set of characters
{χ : G → {eız |z ∈ R}} that forms a locally compact Abelian group. The Pontryagin duality
ensures that this dual group contains all information of G on basis of G⋆⋆ ≃ G. In other words,
applying the Fourier transform on the Haar measure of a group results in the character group
⋆
equipped with its (up to scaling unique) Haar measure µ (G )H without loosing information. Thus
⋆
the Fourier transform of µ (G )H yields the original measure we started with [52].
The generalized Fourier transformations occupy an important place in the practical treat-
ment of problems in physics and related subjects, because it has the feature to convert differential-
in algebraic equations and vice versa. In a broader context it allows to interchange a vector space
with its dual one that in combination with measure theory provides a way to study field spaces
and the associated theory by investigating the space of observables. In particular for quantum
field theories this translates into the concept of Schwinger functionals and effective actions that
carry the entire information of the underlying measure in terms of its Fourier decomposition.
And finally, we will see how the framework of renormalization group exploits the properties of
Fourier transforms to determine algebraic (integral) objects as the measures by a (functional)
differential equation.
2.1.5 Measures for gauge theories
All the described techniques in this section are at the very basis of QFT where the measure is
defined over an infinite dimensional manifold, field space. This general formalism of probability
theory plays a crucial role in understanding the fundamental interactions of Nature. Together
with the symmetry construction of fiber bundles it gives rise to a universal language in which
those theories seem to be describable. Basically, the transition from the mathematical concepts
to physics is provided by some change of vocabulary, namely interpreting the partition function
as the theory under investigation, the integration variables as dynamical fields and its associated
expectation values as observables of certain fundamental processes. This slight modification
of notation has to be supplemented with some measure theoretic construction that usually only
become relevant when non-trivial field spaces are considered, thus in particular in the case of
quantum gravity.
88 Chapter 2. Quantum Field Theory
We will start by introducing a well known technique for integration on manifolds: the parti-
tion of unity. While for flat field spaces this amounts to a trivial identification of F and TΦ̄F for a
single arbitrary background field Φ̄, in the general case we have to look for some geodesic cover,
a background field covering, that yields a diffeomorphic equivalence between local regions and
their tangent spaces via a (non-trivial) exponential map. Next, we establish a link between a
theories defined for the physical fields, i.e. elements of [F], and their counterparts defined on
F, the space of all mathematically possible fields. The route we are going to follow describes
the general idea of Becchi, Rouet, Stora and Tyutin in a geometrical frame. One explicit con-
struction, suitable for a large class of theories, is the Faddeev-Popov method for which we state
some general conditions under which it becomes admissible even in the case of spacetimes with
non-vanishing boundary.
Partition of unity
In general, field space will be a smooth (infinite-dimensional) manifold that is only locally
diffeomorphic to a vector space. However, the definition of a Fourier transform requires a vector
space on a global scale. In fact, there is quite a number of reasons to rather use the tangential
vectors than the fields itself, for example to invoke operator constraints. But surely the most
important motivation to have a vector space at our disposal is the local existence of Fourier
decomposition, a very powerful tool that is especially relevant in the construction of a generating
functional. Fortunately, there is at least a local notion of vector spaces attached to field space and
we have an intuitive understanding how to formally identify those regions using the background
field covering 1.3.2. In this program we select one element of field space, construct its tangential
space and determine the patch of base space that we can diffeomorphically identify with this
local vector space. By increasing the number of basis elements we finally cover the entire field
space that is then represented by {(Φ̄,UΦ̄,TΦ̄[F])} with ∪Φ̄UΦ̄ = [F]. Then, on each local patch
the full formalism of measure theory including the definition of Fourier transforms is at our
disposal.
Nevertheless, on difficile task remains, namely the recombination of those pieces into a
global object, while assuring no over-counting of fields occurs. On method to obtain a global
object from a collection of local ones is described by the gluing lemma. It provides a method
that relies on a set of smooth functions { f j} defined over some chart of [F]. If all functions
agree on regions of overlap, then there exists a global smooth function that piecewise coincides
with the f j’s. While this is a strong statement, it is far to restrictive for the present case: we
have to find a background field covering that on all regions of overlap fully agrees. Fortu-
nately, there exists another strategy which makes the full agreement of functions f j on large
portions of field space expendable while ensuring a consistent recombination, the partition of
unity.
Definition 2.1.13 — Partition of unity. LetM ≡ (XM,τM,A) be a smooth paracompact man-
ifold and {U j ∈ τM} j∈A be an open cover of XM. Furthermore, let {ς j : XM → R} j∈A be
a collection of smooth functions. The set of ordered pairs {(U j,ς j)} is called a smooth
partition of unity onM :⇔ :
∀ j ∈ A : supp (ς .j) .= {p ∈ XM |ς j(p) 6= 0} ⊆U j
∀ j ∈ A, ∀ p ∈ XM : 0≤ ς j ≤ 1 ∧ ∑ ς j(p) = 1
j∈A
∀ p ∈ XM : #({ j ∈ A | p ∈ supp (ς j)})< ∞
In the finite dimensional case, smooth manifolds are always paracompact which guarantees the
existence of a partition of unity for any open cover of XM. However, in the case of infinite
dimensions, the paracompactness property of the cover is very important in order to let the sum
2.1 Measure theory 89
∑ j∈A ς j(p) converge [53]. Assuming the existence of a smooth partition of unity, we can split
a measure µ on field space into small portions covered by the support of a suitable measurable
partitioning function:
∫
µ([F]) = ∑ dµ(p)ς j(p)
j∈A p∈U j
Apart from its obvious application to measures on [F], it is also relevant for some approximation
techniques in the perturbative treatment of QFT.
Maybe the most prominent application in QFT is the background field method. Hereby, the
cover is such that it coincides with a background field covering for a set of background fields Φ̄
and the open sets are generated by the exponential map, i.e.
∫
µ([F]) = ∑ dµ(expΦ̄ (ϕ̂))ς j(ϕ̂)j
j∈A ϕ̂∈TΦ̄ [F]j
This defines a very general prescription to obtain a local vector space formalization of field
space on the level of measure theory. For the later discussion we usually omit the partition of
unity but assume that there is a single background field Φ̄ from which the entire field space can
be constructed using the exponential map. Whenever this is known to be only approximated
true, one has to come back to this general construction and define a suitable cover with the
prescribed techniques.
Measure theory on field space
In the mathematical treatment considered so far, we have seen that equivalence relations take a
very prominent role in reducing the number of redundancies due to some symmetry principle.
For instance, spacetime represents an entire equivalence class of diffeomorphically identical
smooth structures consistent with a certain point set and topology. This superfluous data re-
duced by the equivalence relation of diffeomorphisms can be traced back to indistinguishable
constituents (points) of spacetime and thus to the symmetry under relabeling of ‘coordinates’.
Similarly, at the very heart of the definition of physical fields via the fiber bundle construction
used in gauge theory, symmetries are to the fore. Redundancies are generated by the group of
gauge transformations and only unrelated global sections of some associated bundle give rise
to different physical objects. While on an abstract level, for instance in the general measure
theoretical implementation of QFT, working with the space of physical indistinguishable fields
[F] is unproblematic, practical calculation are usually based on coordinates, both in spacetime
and in field space. Thus the formalism has to be presented on the basis of some particular rep-
resentative to make the mathematical objects usable for computations. Nevertheless, though
we might introduce some coordinates at intermediate steps of the evaluation, the final results
should be independent on a specific choice and thus it has to be invariant under the symmetry
principle the equivalence relations are based on. It is actually one of the severest problems in
modern field theories to define a systematic and still practical suitable transition from [F] to F.
In a moderate form this issue is also present in quantum mechanics, where physical states
are associated with an infinite number of mathematical elements combined in what is called a
ray in a Hilbert space. There are different ways to implement the symmetry of physical ‘reality’
into the mathematical formalism that contains redundancies in this sense. In case of quantum
mechanics the generic way is the insertion of an arbitrary phase factor into the description
while ensuring that all observable results are independent on a specific phase choice (still phase
differences are in principle observable). In a sense this can be compared with the Faddeev-
Popov method, where so called ghost fields generate the gauge equivalent objects and the final
observable results are required to be independent on these fields. In principle, it is possible to
90 Chapter 2. Quantum Field Theory
work on an abstract level and keep the formalism such that it can be consistently defined using
equivalence classes only. From the perspective of quantum mechanics this requires a well-
behaved operator calculus on some projective space, which however is far more involved from
a practical viewpoint. Thus, it is not astonishing that most of the explicit calculations ignore
the abstract description and employ a specific choice of ‘coordinates’. Nevertheless in the end,
the validity of the results has to be checked by verifying the invariance under the associated
symmetry transformations.
Once a fiber bundle description of the underlying space of equivalence classes, [F], exists,
there is a general treatment to relate measures of the (full) mathematical- with the (restricted)
physical-field space. For field space, such a fiber bundle description was established in subsec-
tion 1.3.2, whereby the physical sector [F] represents the base space of a principal bundle Fwith
structure group G, the group of gauge transformations. Hereby, P =MπP×ρRG denotes the un-P
derlying principal bundle of the interaction symmetry group, for instance SU(3)×SU(2)×U(1)
for the Standard Model of particle physics. Thus, there is a natural, in general non-linear, em-
bedding of [F] in F given by global sections, χ : [F]→ F∈Γ (F), which in the physical literature
goes under the name of gauge fixing conditions. For generic field spaces the existence of global
sections is however not guaranteed at all, but in fact an indication for a trivial bundle. In the
non-trivial case local sections have to be combined with suitable transition functions, as usually,
which is at the heart of Gribov copies [54].
In order to understand the necessity of BRST invariance for gauge field theories it is crucial
to distinguish between the group of gauge transformations acting on a single physical field, and
the group of transformations acting on the set of gauge fixing conditions:
G≡ { f ∈ Iso(Γ (P))} ≡ { f ∈ Iso(P) |πP ◦ f = π ∧ ρRP P ( f (u),g) = f (ρRP (u,g))}
Ggf ≡ { f ∈ Iso(Γ (F))} ≡ { f ∈ Iso(F) |π RF ◦ f = πF ∧ ρF ( f (Φ̂),g) = f (ρRF (Φ̂,g))}
On the field space F the structure group G acts on each element individually such that Φ̂ and
fG(Φ̂) share the same equivalence class and hence are physically indistinguishable. The invari-
ance under G is important in the construction of an action functional that should satisfy
S : F→ R , Φ̂ 7→ S[Φ̂] with S[ fG(Φ̂)] = S[Φ̂] ∀ fG∈ G
This group is associated to the properties of the interaction bundle P rather than field space F
and transforms sections on P to other sections on the same bundle.
Concerning gauge fixing conditions, which are by definition sections on F, i.e. χ ∈ Γ (F),
in the general case there exist gauge transformations which do not respect the smooth structure
on F, for instance
fG◦χ ∈/ Γ (F) in general
Thus fG keeps the notion of a physical field intact but in principle it destroys the concept of
gauge fixing conditions.
On the other hand, Ggf is explicitly constructed such that it preserves the notion of gauge
fixing conditions, as well as the equivalence class of physical fields. To understand the scope of
application for Ggf let us have a look at the usual way gauge fixing conditions appear in physical
theories:
(χ)
mBRST : Γ (F)×MeasP(F,ΣF)→MeasP([F],Σ[F]) with µF 7→ mBRST(χ ,µF) = µ[F]
The map mBRST is a certain – yet to defined – method that reduces the measure, or generating
functional, from field space to the physical sector. A possible realization may assume the form
2.1 Measure theory 91
of the Faddeev-Popov method, i.e.
∫ ∫
≡ (χ)µF dµG(ϕ̂;Φ̄)ρ(ϕ̂;Φ̄) 7→ dµG(ϕ̂ ;Φ̄)ρ(ϕ̂ ;Φ̄)Nδ (χ)δ (χ([ϕ̂ ]))≡ µ
∈ ∈ [F]ϕ̂ TΦ̄F ϕ̂ TΦ̄F
Notice that Nδ (χ)δ (χ([ϕ̂ ])) represents the invariant delta distribution on field space, with
Nδ (χ) being the Jacobian determinant associated to this hyperplane embedding. Depending
on the explicit implementation method mBRST, the resulting theory on the physical sector will
depend on the gauge fixing condition and thus µ[F] generates different theories for different em-
beddings of [F] in F. However, a suitable map mBRST has to give rise to a unique theory in the
(χ)
physical sector, which corresponds to µ[ ] ≡ µ[F] being independent on χ . On the level of mF BRST
the invariance under transformations of the gauge fixing condition reads:
!
mBRST(χ ,µF) = mBRST( fG (χ),µF)gf
Hence, a systematic procedure to define mBRST should be based on the underlying fiber bun-
dle construction and emerges from purely differential geometry considerations. This program,
known as BRST-invariant gauge fixing of a theory, goes back to independent works of Bec-
chi, Rouet, Stora [55] and Tyutin. Since then, anti-BRST as well as BRST transformations
have been understood in the light of Chevalley-Eilenberg cohomology [56] of the Lie algebra
associated to the adjoint bundle of F. In this geometrical framework BRST transformations are
simply related to the coboundary operator of this cohomology, and are denoted sBRST. This is a
very important fact, for it tells us that a successive application of BRST operations vanishes,
i.e. sBRST ◦ sBRST = 0, which is also known as Wess-Zumino consistency condition. For details we
refer to [57, 58] which provides a mathematical overview.
For our purpose it suffices to know that the structure of the principle bundle F gives rise to a
nilpotent operation sBRST. In order to give a precise meaning to mBRST that can be easily extended
to the generating functional and its derived concepts, let us decompose field space F into its
vertical and horizontal components.
TΦ̄F≡ VΦ̄F⊕HΦ̄F≃ LieAlg(G)×TΦ̄[F]
This split corresponds to the phase representation of Hilbert rays, now in a more general setting.
In particular it is based on a (non-trivial) field space connection χF : TF → LieAlg(G) that
defines the horizontal space by its kernel. Basically, we have encountered many of the currently
discussed aspects already in subsection 1.3.2 but with a different emphasis. Central for the
following discussion is the decomposition of the fluctuation fields into physical ones and those
that generate the Lie algebra of gauge transformations. Explicitly it states:
ϕ̂ ≡ ρRF∗(ϕ̂,C)
Furthermore the connection defined on G is called the Faddeev-Popov ghost η and represents
the Maurer-Cartan form, thus η(C) = C holds true for all C ∈ LieAlg(G). Ghost fields vio-
late the spin-statistic theorem and are as such unphysical quantities. They are represented by
anticommuting vector fields assuming values in LieAlg(G). Thus the direction of physically in-
distinguishable and distinguishable fields is nicely separated, which also extends to the measure
spaces, i.e.
µT F(UF ≡ (U[F],UC))≡ µT [F](UΦ̄ Φ̄ [F])×µLieAlg(G)(UC)
For the reduction to the physical sector, we have to apply a suitable projection mBRST which
should be BRST invariant for this implies invariance under Ggf. The nilpotent character of
92 Chapter 2. Quantum Field Theory
the BRST transformation allows for any insertion that can be expressed as the image of sBRST.
Furthermore, we have to ensure that ρ(ϕ̂;Φ̄) is in the kernel of sBRST, which in fact is naturally
satisfied whenever ρ is gauge invariant. Hence, two theories that differ by a total derivative w.r.t.
sBRST are BRST equivalent and therefore agree on the physical sector:5
∫ ∫
µF ≡ dµ(ϕ̂ ;Φ̄)≃ dµ(ϕ̂ ;Φ̄)d[C] sBRST(Ψ(ϕ̂ ,C;Φ̄)) with ϕ̂ ≡ ([ϕ̂ ],C)
ϕ̂∈TΦ̄F ϕ̂∈TΦ̄F
Now, in order to cancel the gauge orbit integral by means of a suitable implementation of gauge
fixing conditions, we consider a specific, sBRST-exact insertion:
ρBRST(ϕ̂ ,C;Φ̄;χ) with sBRST(ρBRST(ϕ̂ ,C;Φ̄;χ)) = 0
Besides these geometric construction yielding the physical- and ghost-fields by the fiber bundle
formalism, there is an analytic part of the BRST method introducing the anti-ghost fieldC and
the so-called Nakanishi-Lautrup field B ∈ C∞(M). Therefore, let us assume we have a gauge
fixing condition χ ∈Γ (F) that – consider as a hyperplane embedded in field space – corresponds
to the a function
F: F→ C∞(M) with ker(F[Φ̄](ϕ̂)−B)≡ Im(χ)
Here the artificial Nakanishi-Lautrup field B appears in the formalism, representing the right-
hand-side (RHS) of the hyperplane equation F[Φ̄](ϕ̂) = B. It allows for a smooth implementa-
tion and a more flexible gauge fixing construction. Now, we have to adapt B and later C̄ such
that neither BRST invariance nor other important properties are lost, in particular we require
!
sBRST(B) = 0 =⇒ sBRSTF[Φ̄](ϕ̂) = 0 for ϕ̂ ∈ χ([ϕ̂])
Notice that the constraint for the gauge fixing condition is only satisfied for fields on the hyper-
plane, while in general sBRSTF[Φ̄](ϕ̂) 6= 0.
Supplementary to B we introduce the anti-ghost field C̄, an anti-commuting dual vector
field that in a sense represents the (unphysical) partner of C. It acts as a Lagrange multiplier
that installs the hyperplane condition into the generating functional. The BRST transformation
is chosen such that it linearly relates to B, i.e. sBRST(C̄) = B. In total, assuming [F] to describe
only the gauge (interaction) sector, i.e. neglecting matter fields for a moment, we have the
following set of rules for the four types of fields:
sBRST(ϕ̂) = dη +Lη ϕ̂ ∧ s 1BRST(η) =− 2 [η , η ]LieAlg(LieAlg(G))
sBRST(C̄) = B ∧ sBRST(B) = 0 (2.5)
As long as the Lie algebra LieAlg(G) closes and thus fulfills the Jacobi-identity, the action of
sBRST on the field content is indeed nilpotent. Hence, mBRST is defined as
∫
mBRST(µ) ..= dµ(ϕ̂ ;Φ̄)d[C]d[C̄]d[B] sBRST(ρBRST([ϕ̂ ],C,C̄,B;Φ̄)) (2.6)
([ϕ̂],C,C̄,B)∈TΦ̄F×UC̄;B
∫
Here ν(U)≡ U d[C]d[C̄]d[B] ρBRST([ϕ̂ ],C,C̄,B;Φ̄) defines a suitable measure for the supplemen-
tary (unphysical) fields.
It should be mentioned that there is also a geometric construction involving the anti-BRST
transformations, where the roles of C and C̄ are interchanged. Invariance of mBRST under both
5In the language of differential geometry, the set of physically equivalent theories is a sBRST-closed but not exact
form.
2.1 Measure theory 93
BRST- and anti-BRST transformations is very closely related to the background field method
and its associated background Ward identities [59].
Before we are going to present the Faddeev-Popov method as a specific example, let us
make a final remark on how to control the BRST invariance of the theory on the level of the
expectation values. Since on the effective side all fields are integrated out, we need source terms
for every BRST transformation thus the additional term
( )
exp Kϕ̂(sBR(ST ϕ̂)+KC(sBRSTC)+KC̄(sBRSTC̄)+K)B(sBRSTB)
≡ exp Kϕ̂(sBRST ϕ̂)+KC(sBRSTC)+KC̄(B)
In the last step we inserted the trivial BRST transformations of C̄ and B. One then incorporates
this kind of Fourier transformation of the BRST field transformations into the generating func-
tional and hence the variation w.r.t. the BRST sources K• gives rise to expectation values of the
corresponding operation sBRST(•). On the level of the effective action, which will be introduced
in the next section, the BRST symmetry is intrinsically present in form of the Slavnov-Taylor
identity:
[ ]
∑ Tr (dδK Γ[ϕ̂;Φ̄|K])(dφ δφ Γ[ϕ̂;Φ̄|K]) = 0
φ∈{ϕ̂ ,C,C̄}
There is no summation over B due to the trivial BRST transformation and thus its vanishing
source term.
Before the advent of the concept of BRST invariance and its geometrical interpretation,
Faddeev and Popov introduced a very successful method to treat functional integrals for gauge
theories [60]. It is in fact a special case for mBRST that is suitable for a large class of QFT and
given by
(∫ √ ( ))
ρ FP dBRST([ϕ̂ ],C,C̄,B;Φ̄)≡ exp d x ḡ C̄F[Φ̄](ϕ̂)− 12 ᾱC̄B
M
Under the application of sBRST the exponential reduces to a sum over the familiar gauge fixing-
and ghost-functionals given by
∫ √ ( )
S [ϕ̂,C,C̄;Φ̄]≡+ ddx ḡ − 1gf 2 ᾱ(B)
2+BF[Φ̄](ϕ̂)
∫M √ ( )
Sgh[ϕ̂;Φ̄]≡− ddx ḡ C̄ sBRSTF[Φ̄](ϕ̂)
M
The ghost term depends on C̄ as well as on C, due to sBRST. It represents the functional de-
terminant over anti-commuting fields which is associated to the Jacobian of the hyperplane
embedding, i.e. Nδ (χ). On the other hand the gauge fixing term can be associated to a
smeared delta distribution which becomes more explicit once we perform the Gaussian inte-
gration d[B]exp(Sgf):
∫ √ ( )
S [ϕ̂;Φ̄]≡+ ddx ḡ − 1gf 2 ᾱ
−1(F[Φ̄](ϕ̂))2
M
Thus, for a set of field space configurations an explicit implementation of mBRST is given by
∫ ( )
m (µ) .BRST .= dµ([ϕ̂ ],C;Φ̄)d[C̄] exp −Sgf[ϕ̂ ;Φ̄]−Sgh[ϕ̂,C,C̄;Φ̄] (2.7)
([ϕ̂],C,C̄)∈TΦ̄F×UC̄
It is BRST invariant and corresponds to a unique measure on [F], as required. In the following
sections of part I we will implicitly assume, without emphasizing, that we can freely transit
between measures on F and [F] based on some appropriate mBRST.
94 Chapter 2. Quantum Field Theory
In the presence of a non-vanishing spacetime boundary, reference [61] lists additional re-
quirements that have to be satisfied in order to apply the Faddeev-Popov method. The first
necessary ingredients are gauge invariant boundary conditions for field space and the ghost
sector. Furthermore, the given boundary conditions should be such that F[Φ̄](ϕ̂ +LCϕ̂) = 0
gives rise to a unique solution for C w.r.t. any ϕ̂ . Once these requirements are fulfilled, the
Faddeev-Popov method can be self-consistently implemented even in the case of ∂M 6= 0/ .
2.1.6 Interpretation
We have started this chapter by leaving the interpretation of the chosen probabilistic way open
and we will continue to do so. Nevertheless, it is very instructive to emphasize the universal
character of this measure theoretical approach and see how the different philosophical concepts
emerge.
Obviously, this method is best suited for a stochastic treatment of Nature, where events take
place with a certain probability. However, already at this step we meet determinism in the shape
of effective theories and upon implementing experimental data. No matter what the sources of
uncertainties are, whether they are intrinsically present in the laws of Nature or if they are due
to a lack of experimental precision, the formalism does not care. Mathematics on its own is
free from any interpretation and the true genius of natural science is precisely to link numbers,
letters and formula with real life. Uncertainties nowadays play a very important role, at least
since the advent of quantum mechanics that gave them a prominent place even in the theoretical
description. The future generations of physicists have to figure out if this feature is merely an
artifact of an effective description or truly imprinted in the laws of physics.
Surely, there are methods more appropriate to study deterministic theories. It is yet very
convenient to recover it within the same tool used to study generically different approaches, es-
pecially for comparison reasons. On the basis of measure theory the transition of a probabilistic
to a full deterministic setting is carried out by simply restricting the class of admissible mea-
sures. While in the general case we will invoke only symmetry constraints and some plausible
regularization conditions, determinism requires a very special class of measure spaces where
only a single field contributes non-trivially. This corresponds to a measure comparable to a
delta distribution over the realized field. Hence, one may suggest that the more a single field
dominates a measure on field space, the closer we come to a deterministic description.
In this section we have met a very powerful tool that may help us in understanding the
foundations of Nature. Covering both, deterministic and general probabilistic theories, measure
theory provides a universal approach that takes care of experimental and theoretical uncertain-
ties on the same footing. In the remainder of this chapter we will see how this prescription
naturally fits in the beautiful concept of quantum field theories.
2.2 Effective action
The Fourier transform we considered in the end of section 2.1 provides a very practical tool to
investigate the space of theories that culminates in the effective action formalism of quantum
field theories. It relates the (tangent of) field space to its dual space and this connection can
be extended to express a theory, defined by a suitable measure, in terms of its field expectation
values. This is a more natural construction that reflects the actual process of deducing theories
from observable data. On the level of expectation values we are able to address questions in a
more instructive way which are obscured when considered from the perspective of probability
measures.
2.2 Effective action 95
2.2.1 Schwinger functional
Once we decided about a particular theory by means of a probability measure the main focus
lays on the evaluation of observables to relate theory with experiments. These measurable func-
tions are naturally defined over field space or in case of the background field method over the
respective tangent space. When evaluated for the corresponding theory, we obtain its expecta-
tion values given by
∫ ∫
〈 f (Φ̂)〉 ..µ = dµ(Φ̂) f (Φ̂) ∨ 〈 f (ϕ̂)〉 ..µ = ∑ dµ(ϕ̂ ;Φ̄) f (ϕ̂ ;Φ̄)
Φ̂∈[F] Φ̄ ϕ̂∈TΦ̄[F]
Instead of evaluating every observable separately, we can draw on a set of basis functions and
build all measurable functions on top of them. On such collection contains the monomials of
fields that constitutes a large class of observables, in particular the polynomials. Each basis
element is thus a power of the dynamical or fluctuation field with respect to the pointwise
product of fields, i.e.
Φ̂, Φ̂2 ≡ Φ̂⊗ Φ̂, Φ̂3, · · · ∨ ϕ̂ , ϕ̂2 ≡ ϕ̂ ⊗ ϕ̂, ϕ̂3, · · ·
Measurable functions form an associative, commutative algebra under pointwise and tensorial
operations, and as such their expectation values can be recovered by a suitable linear combina-
tion of field monomials, which are classified by their degree of homogeneity n and also denoted
n-point functions. The most significant role is played by those of degree one, which naturally
relate the dynamical or fluctuation fields with their expectation valued counterparts:
∫ ∫
Φ ..= 〈Φ̂〉µ ≡ dµ(Φ̂)Φ̂ ∨ ϕ ..= 〈ϕ̂〉µ ≡ ∑ dµ(ϕ̂ ;Φ̄) ϕ̂
Φ̂∈[F] Φ̄ ϕ̂∈TΦ̄[F]
They describe the most likeliest field configuration for the respective theory defined by the
measure µ on which they depend. For each µ there is a unique field ϕ and a variation of this
expectation value is equal to a change of the underlying theory. This is the aforementioned
duality between expectation values and measures which allows us to attack the construction of
suitable theories from a different perspective.
To this end, let us start with a particular field space [F] and define a suitable σ -algebra, most
preferable of Borel type to have a coincidence between measurable and continuous functions.
For this specified measurable space ([F],Σ[F]) the set of probability measures is given by
MeasP([F],Σ[F])≡ {µ : Σ[F] → R |µ([F]) = 1 ∧ µ measure}
This space is more then just a simple set, but is equipped with algebraic operations that are
related to averaging processes. Every measure µ ∈ MeasP([F],Σ[F]) defines its associated ex-
pectation field Φ ≡ 〈Φ̂〉µ in a unique way yielding the total set of all possible Φ in ([F],Σ[F]):
〈[F]〉 ..= {Φ ≡ 〈Φ̂〉µ |µ ∈MeasP([F],Σ[F])}
The linearity of integration and of field space assure that every Φ is a scalar multiple of a
specific element on [F], i.e. for every µ there exists an element in field space Φ̂µ and a (positive
real) number z ∈ R such that Φ ≡ 〈Φ̂〉µ = z ◦R·[F] Φ̂µ . Whenever [F] is closed under this scalar
multiplication, necessarily it has to be defined over the field R, the expectation fields 〈[F]〉
occupy a subspace of [F], i.e. 〈[F]〉 ⊆ [F].
R Notice that in the case of [F] ⊆ N a probability measure usually produces non-integer
expectation values and thus 〈[F]〉* [F].
96 Chapter 2. Quantum Field Theory
Even though a measure uniquely determines its associated expectation value, we cannot expect
that the converse holds true. There may be an infinity of different measures that result in the
same Φ, hence injectivity is lost. However, under certain circumstances the full variety of ex-
pectation values for field monomials allows to uniquely reconstruct the measure from its observ-
ables. The missing link is provided by the Fourier transform of a measure µ that fully classifies
the underlying theory. As it stands, it relates two measures by a Radon-Nikodym derivative
containing an exponential linear in the fluctuation field, by virtue of a dual continuous vector J.
When the integration is performed of the entire field space the measure specific information are
then encoded in the resulting functional, its Fourier transform. Expending the Radon-Nikodym
derivative in a power series reveals the knowledge contained in the set of monomial expectation
values:
∫ ∫ ∞ zn
Z[J]≡ µ̂T [F](J;Φ̄) = dµ(ϕ̂) expC(z · J(ϕ̂)) = dµ(ϕ̂)Φ̄ ∑ (J(ϕ̂))
n
ϕ∈TΦ̄[F] ϕ∈TΦ̄[F]
∫ n=0
n!
∞ zn
= ∑ J︸⊗·︷·︷·⊗ J︸ dµ(ϕ̂) ︸ϕ̂ ⊗·· ·⊗ ϕ̂0 n! ϕ∈T [F] ︷︷ ︸n= Φ̄
n-times n-times
∞ zn
= ∑ J︸⊗·︷·︷·⊗ J︸ 〈ϕ̂ ⊗·· ·⊗ ϕ̂〉n! µ
n=0
n-times
The critical point in this demonstration is the interchange of integration and summation which
may yield indefinite intermediate results. Nevertheless, it is instructive to note that the Taylor-
expansion of the exponential can be used to deduce expectation values of field monomials. In
this light, we can extract any n-point function from the generating functional Z[J] by means of
taking n-times the derivative with respect to J and then setting J = eT∗ [F] to get rid of the higher
Φ̄
order terms. The basis of this expansion is the Euler-MacLaurin formula yielding
∣
︸dδJ ·︷·︷·
∣
dδ︸J Z[J;Φ̄]∣ = z
ℓ δJ⊗·· ·⊗δJ 〈ϕ̂ ⊗·· ·⊗ ϕ̂〉µ
J=e ∗ ︸ ︷︷ ︸T [F] ︸ ︷︷ ︸
ℓ-times Φ̄ ℓ-times ℓ-times
For the Gaußian measure the interchange of Taylor-expansion and integration turns out to
be well behaved due to the particular nice structure of its Fourier transform:
( ( ))
µ̂G[J;Φ̄] = expC − 12G
−1 J , AΦ̄ J
A self-adjoint, positive AΦ̄ corresponds to a free theory, for which we can directly compute its
n-point functions. Therefore, let us start taking the respective number of functional derivatives
of the generating functional:
∣ { ( ) ( )∣ ℓ! − 1 ℓ −1 ⊗(ℓ/2)· · · ∣ (ℓ/2)! 2 G δJ , AΦ̄ δJ for ℓ ∈ 2Nd︸δJ ︷︷ dδ︸J Z[J;Φ̄] =J=eT∗ [F] 0 , for ℓ ∈ 2N+1
ℓ-times Φ̄
Since AΦ̄ is self-adjoint and for ℓ even, n-point functions are given by
( )ℓ ( )
〈ϕ̂ℓ〉 1 ℓ! ⊗(ℓ/2)µ = G−1 • , A • for ℓ ∈ 2NG 2z (ℓ/2)! Φ̄
As a consequence the propagator, the 2-point fun(ction, whi)ch is a measure for the relation of two
fields, assumes the form: 〈ϕ̂ ⊗ ϕ̂〉 = z−1G−1µ • , AΦ̄ • , while the mean field 〈ϕ̂〉µ = eTΦ̄ [F]G G
vanishes.
2.2 Effective action 97
A more instructive way to present the n-point functions is the associated Schwinger func-
tional. We have seen that even in the free theory there appear relations between the non-
interacting fields due to the exponential properties. To extract the connected propagator along
with the higher connected n-point functions we choose a representation of Z[J;Φ̄] that trans-
forms the multiplicative exponential in an additive construction:
Definition 2.2.1 — Schwinger functional. Let ([F],Σ[F],µ) describe a probability theory.
Assume Φ̄ defines a suitable background field covering. Then the functional W [•;Φ̄] :
T∗ [F]→ C is the Schwinger functional of µ :⇔
Φ̄
W [J;Φ̄]≡ 1 ln µ̂(J;Φ̄) = 1 lnZ[J;Φ̄]
z z
The connected n-point functions are implicitly defined by
· · · ∣
∣
d . ℓ−1︸δJ ︷︷ dδ︸J W [J;Φ̄]∣ .= z δJ⊗·· ·⊗δJ 〈ϕ̂ ⊗·· ·⊗ ϕ̂〉
con
µ
J=e ∗ ︸ ︷︷ ︸T [F] ︸ ︷︷ ︸
ℓ-times Φ̄ ℓ-times ℓ-times
The logarithm acts on a real positive or complex number, depending on the type of Fourier trans-
form and the measure µ . Now, in the transition from the generating to the Schwinger functional
we get rid of an enormous number of redundant information and retain only the more elementary
building blocks that in their mutual combination will reproduce the original n-point functions.
Therefore, this condensed construction is very helpful for systematically understanding the full
theory µ . The mechanism of reduction becomes apparent when considering the connected n-
point functions and their relation with the original ones. For instance, the connected propagator,
likewise defined by the second variation ofW and a subsequent evaluation at J = eT∗̄ [F], assumes
Φ
the form:
∣ ( ( ) )∣
〈ϕ̂ ⊗ ϕ̂〉con = z−1 ∣d d W [J;Φ̄]∣ = z−2 ⊗2 ∣W • • d•d•Z[J;Φ̄]− d•Z[J;Φ̄] ∣
J=eT∗ [F] J=eT∗ [F]
Φ̄ Φ̄
= 〈ϕ̂ ⊗ ϕ̂〉µ −〈ϕ̂〉µ ⊗〈ϕ̂〉µ
Notice, since the logarithm is an isomorphism on [0,1]⊂R we are not loosing any information
when moving to the Schwinger functional.
The amount of superfluous data in the generating functional is best illustrated on the basis
of a free theory for which the Schwinger functional is given by
( )
W [J;Φ̄] =− 1 −1 −12z G J , AΦ̄ J
Hence, the infinite number of non-vanishing n-point functions is only due to a single connected
expectation value, the propagator while all higher correlation functions are identically zero:
( )
〈ϕ̂ ⊗ ϕ̂〉con ≡ 〈ϕ̂ ⊗ ϕ̂〉 = z−2 −1W µ G • , AΦ̄ •G
We are now going to proceed to yet another representation of the Fourier transform in which
the theory specific information are cast into a more natural form. While there is no further
reduction of information, we merely replace the sources J by their associated expectation values.
2.2.2 Effective action
In a first step we have transfered the measure specific information into its characteristic func-
tional, which is the Fourier transform of the measure. Under certain condition, this transition
is an isomorphism and thus preserves all information of the underlying theory. It turns out to
be convenient to rephrase the Fourier transform into the Schwinger functional that reduces the
98 Chapter 2. Quantum Field Theory
amount of data by retaining only the connected building blocks. We are now going to absorb the
content ofW [J;Φ̄] into a better accessible object that fits more our needs: the effective action. In
this case the space of measures comes to the fore, which turns out to be the basic ingredient for
the functional renormalization group. For this purpose we have to introduce a suitable method
to transfer the full information of a functional into another one defined over the dual space.
Legendre-Fenchel transform
The transition from the dual vectors J to a functional of expectation fields is given by the
Legendre-Fenchel transform.
Definition 2.2.2 — Legendre-Fenchel transform. LetV be a smooth (infinite dimensional)
vector space and V ⋆ the associated continuous dual space. A functional B : V → R is the
Legendre-Fenchel transform of a functional A :V ⋆ → R :⇔ :
B[b]≡ LFT(A)[b] = sup {a(b)−A[a]} ∀b ∈V (2.8)
a∈V ⋆
Two functionals that are related by a Legendre-Fenchel transform possess a close mutual link
which, under a set of conditions, is so strong that one can uniquely reconstruct one from the
other. In this case both contain the same information and for them the Legendre-Fenchel trans-
form defines an involution. The class of functionals that fulfill these requirements is suitable
to reformulate the Schwinger functional in terms of the effective action and thereby replace the
sources, elements J ∈ T∗ [F], by the more physically accessible expectation fields, i.e. fields
Φ̄
like ϕ ∈ TΦ̄ 〈[F]〉.
Definition 2.2.3 — Effective action. Let ([F],Σ[F]) describe a measurable space and denote
MeasP([F],Σ[F]) the space of probability theories. Let µ ∈MeasP([F],Σ[F]) be a probability
measure and W [•;Φ̄] the corresponding Schwinger functional with Φ̄ defining a suitable
background field covering. Then, the functional Γ : TΦ̄ 〈[F]〉 → C is the effective action of
W :⇔
{ }
Γ[ϕ ;Φ̄]≡ LFT(W [•;Φ̄])[ϕ ;Φ̄]≡ sup J(ϕ)−W [J;Φ̄]
J∈T∗ [F]
Φ̄
The associated n-point functions are called one-particle irreducible correlation functions and
are given by
∣
· · · ∣d d . ℓ−1 1PI
︸δϕ ︷︷ δϕ
Γ[ϕ ;Φ̄]∣ .= z δ︸ ϕ=eT 〈[F]〉 ︸
ϕ ⊗·︷·︷·⊗δϕ︸ 〈ϕ̂ ⊗·· ·⊗ ϕ̂〉
Φ̄ ︸ ︷︷
µ ︸
ℓ-times ℓ-times ℓ-times
In what follows, we work out the hidden connection between a functional and its Legendre-
Fenchel transform along with the requirements we need for an information preserving transition.
This entire formalism is not specific to the effective action approach, but is a general property
of the Legendre-Fenchel transform that is here only exemplified for the physical quantities
of interest. For convenience we are going to abbreviate W ≡W [•;Φ̄] and Γ ≡ Γ[•;Φ̄] in the
following derivation.
Thus, let us start with a Schwinger functional W : T∗ [F] → C and consider its Legendre-
Φ̄
Fenchel transform LFT(W )≡ Γ :TΦ̄ 〈[F]〉→C. In general the supremum may be not contained
in the co-domain of W . However, in case that W is second order differentiable and strictly
convex, meaning that its Hessian is positive definite, the supremum condition reduces to the
Legendre transform evaluate at a unique global maximum ofW . We denote the position of the
2.2 Effective action 99
supremum Jm ≡ ϑ(ϕ) ∈ T∗ [F] to indicate that it depends on the value of ϕ and write:Φ̄
LFT(W )[ϕ ] = sup {J(ϕ)−W [J]}= Jm(ϕ)−W [Jm] = (ϑ(ϕ)) (ϕ)−W [ϑ(ϕ)] ≡ Γ[ϕ ]
J∈T∗ [F]
Φ̄
with dwW [ϑ(ϕ)] = w(ϕ) ∀w ∈ T∗Φ̄[F] (2.9)
Whenever W is convex, the second variation of (J(ϕ)−W [J]) is negative and thus ϑ(ϕ) de-
scribes a maximum. While in general W may not fulfill the strong convexity condition, its
Legendre-Fenchel transform is at least convex, as can be seen as follows: First note that
supx∈X⊆Y{ f (x)} ≤ supx∈Y{ f (x)} holds true that further implies
sup{ f (x)+g(x)} = sup { f (x1)+g(x2)}
x∈X (x1,x2)∈{(x,x) |x∈X}
≤ sup { f (x1)+g(x2)}= sup{ f (x)}+ sup{g(x)}
(x1,x2)∈X×X x∈X x∈X
Hence, with ϕ ≡ (1−λ )◦R·T 〈[F]〉 ϕ1+T 〈[F]〉 λ ◦R·T 〈[F]〉 ϕ1 for all ϕ1, ϕ2 ∈ TΦ̄ 〈[F]〉 and λ ∈ [0,1]Φ̄ Φ̄ Φ̄
we can exploit the linearity of J and artificially insert a zero 0= λW [J]−λW [J] to obtain:
Γ[ϕ ] = sup {J(ϕ)−W [J]}= sup {(1−λ )J(ϕ1)+λJ(ϕ2)− (1−λ )W [J]−λW [J]}
J∈T∗ [F] J∈T∗ [F]
Φ̄ Φ̄
= sup {(1−λ ) [J(ϕ1)−W [J]]+λ [J(ϕ1)−W [J]]}
J∈T∗ [F]
Φ̄
≤ (1−λ ) sup {J(ϕ1)−W [J]}+λ sup {J(ϕ2)−W [J]}= (1−λ )Γ[ϕ1]+λΓ[ϕ2]
J∈T∗ [F] J∈T∗ [F]
Φ̄ Φ̄
Thus Γ[(1−λ ) ◦R·T 〈[F]〉 ϕ1+T 〈[F]〉 λ ◦R·T 〈[F]〉 ϕ1] ≤ (1−λ )Γ[ϕ1]+λΓ[ϕ2] proves the convexityΦ̄ Φ̄ Φ̄
of the effective action, or the Legendre-Fenchel transform in general.
Under the above assumptions of strict convexity and differentiability of the Schwinger func-
tional, LFT(W ) has a very important property that is crucial in successively relating W with
the effective action Γ. If we perform a second Legendre transformation, now on Γ, the solution
to the associated supremum condition again depends on the evaluation value, i.e. ϕm = γ(J) ∈
TΦ̄[F]:
LFT(Γ)[J] = sup {J(ϕ)−Γ[ϕ ]}= J(ϕm)−Γ[ϕm] = J (γ(J))−Γ[γ(J)]
ϕ∈TΦ̄[F]
with dvΓ[γ(J)] = J(v) ∀v ∈ TΦ̄ 〈[F]〉 (2.10)
The constraint that results from the supremum condition for the Legendre transform is exactly
the effective field equations stating that the maximum of Γ is positioned at the source, i.e.
dvΓ[γ(J)] = J(v) for all v ∈ TΦ̄ 〈[F]〉 and J ∈ T∗ [F]. Next, we replace Γ by its explicit formΦ̄
given in eq. (2.9) that results in
LFT(Γ)[J] = J (γ(J))− ((ϑ(ϕ)) (ϕ)−W [ϑ(ϕ)])ϕ=γ(ϑ )
= J (γ(J))− (ϑ(γ(ϑ))) (γ(ϑ))+W [ϑ(γ(ϑ))]
with dvΓ[γ(J)] = J(v) ∧ dwW [ϑ(ϕ)] = w(ϕ) ∀v ∈ TΦ̄ 〈[F]〉 , w ∈ T∗Φ̄[F]
The important observation is that ϑ and γ are not independent but in fact inverse operations of
each other. This can be seen by evaluating the defining condition of γ(J) using both parts of eq.
(2.9), linearity of dual vectors, and the chain rule.
J(v) = dvΓ[γ(J)] = dv ((ϑ(ϕ)) (ϕ)−W [ϑ(ϕ)])( ϕ=γ(J) )
= (dvϑ(ϕ)) (ϕ)+ (ϑ(ϕ)) (v)− dwW [ϑ(ϕ)]|w=dvϑ (ϕ) b=γ(J)
= ((dvϑ(ϕ)) (ϕ)+ (ϑ(ϕ)) (v)− (dvϑ(ϕ)) (ϕ))ϕ=γ(J) = (ϑ(γ(J))) (v)
100 Chapter 2. Quantum Field Theory
In the second last step we simplified the equality using dwW [ϑ(ϕ)] = w(ϕ) for w = dvϑ(ϕ).
Since J(v) = (ϑ(γ(J))) (v) holds for all v∈TΦ̄ 〈[F]〉we conclude J= (ϑ ◦γ)(J) which confirms
the inverse behavior of ϑ and γ considered as functions TΦ̄ 〈[F]〉→T∗ [F] and T∗ [F]→TΦ̄ 〈[F]〉,Φ̄ Φ̄
respectively. Along the same lines one establishes the reversed relation, so that we obtain:
ϑ ◦ γ = idT∗ [F] ∧ γ ◦ϑ = idTΦ̄〈[F]〉 (2.11)Φ̄
On a deeper level this connection implies a similar relation between the associated Hessian
operators of a functionalW and its Legendre-Fenchel transform Γ, in case they exist. To reveal
this missing link we go back to eq. (2.11) and determine its Hessian with the aid of dvΓ[ϕ ] =
(ϑ(ϕ)) (v):
[ ]
Hessγ(J) Γ (v1,v2) =
1
2 [dv2dv1Γ[ϕ ]+dv1dv2Γ[ϕ ]]ϕ=γ(J) (2.12)
= 12 [(dv2ϑ(ϕ)) (v1)+dv1 (ϑ(ϕ)) (v2)][ ( ( ) ) (ϕ=γ(J() ) )]
= 1 −1 −12 G G • , dv2ϑ(ϕ) , v1 +G G • , dv1ϑ(ϕ) , v2 ϕ=γ(J)
At this stage we have to assume that bothW and Γ give rise to a we(ll behaved H)essian operator
in order to proceed. It suffices to assure that the operator v 7→G−1 • , dvϑ(ϕ) is self-adjoint
to obtain:
[ ] ( ( ) )∣
Hessγ(J) Γ (v
−1 ∣
1,v2) =G G • , dv2ϑ(ϕ) , v1( ( ) )∣ϕ=γ(J)
=G G−1 • , dv1ϑ(ϕ) , v ∣2 ϕ=γ(J)
If this is the case, the Hessian operator associated to Γ is simply given by
[ ] − ( )∣Ĥess 1 ∣
[ ]
γ(J) Γ v=G • , dvϑ(ϕ) ( Ĥessϕ=γ(J) γ(J) Γ : TΦ̄ 〈[F]〉 → TΦ̄ 〈[F]〉 ) (2.13)
Along similar lines and with an equivalent assumption on the Hessian ofW we obtain:
[ ] ( )∣ [ ]
Ĥessϑ (ϕ) W w=G • , d ∣wγ(J) ( Ĥess ∗ ∗J=ϑ (ϕ) ϑ (ϕ) W : TΦ̄[F]→ TΦ̄[F] ) (2.14)
Both equations (2.13) and (2.14) can be reverted due to the non-degeneracy of the field space
metric, yielding
∣∣ ( [ ] )dvϑ(ϕ) =G • , Ĥess Γ v (2.15a)
∣ J=γ(J)
γ(J)
∣ − ( [ ] )dwγ(J) =G 1 • , Ĥess W w (2.15b)ϕ=ϑ (ϕ) ϑ (ϕ)
In the final step we go back to eq. (2.11), take its functional derivative, and insert the appropriate
equation of (2.15):
∣ ∣ ∣
w= d ∣w J = dwϑ(γ(J))∣ = d ∣J=ϑ (ϕ) J=ϑ (ϕ) dwγ(J)ϑ(ϕ)( [ ] ∣ ) ϕ=γ(J);J=ϑ (ϕ)
=G • , Ĥess Γ d γ(J)∣γ(J) w
( [ ] ( J=ϑ (ϕ) [ ] ))
=G • , Ĥessγ(J) Γ G−1 • , ĤessJ W w (2.16)
Similarly, the second equality of eq. (2.15) produces the inverted result:
∣ ∣ ∣
v= dv ϕ∣ = d ∣ ∣ϕ=γ(J) vγ(ϑ(ϕ)) = d γ(J) (2.17)( [ ] ϕ=γ(J)
dvϑ (ϕ)
∣ ) J=ϑ (ϕ);ϕ=γ(J)
=G−1 • , Ĥessϑ (ϕ) W dvϑ(ϕ)∣
( [ ] ( ϕ=γ(J[)
=G−
] ))
1 • , Ĥessϑ (ϕ) W G • , Ĥessϕ Γ v (2.18)
2.2 Effective action 101
These are exactly the identities we were looking for, namely relations connecting the Hessian
operators of a functional and its associated Legendre-Fenchel transform. Except for some fac-
tors of the field space metric both are their mutual inverse operators, i.e.
[ ] ( ( [ ])−1 )
Ĥess Γ =G−1b • , Ĥessϑ (ϕ) W G (2.19a)
[ ] ( ( [ ])−1 )
Ĥessa W =G • , Ĥess −1γ(J) Γ G (2.19b)
The Legendre-Fenchel transform thus not only establish the bridge between the Schwinger
functional and the effective action, but also provides the link between the connected and the
irreducible propagators. In case of a strictly convex and differentiable W [J;Φ̄] the correspon-
dence is in fact an isomorphism and thus all information of the Schwinger functional is also
contained in Γ[ϕ ;Φ̄]. If both allow for a self-adjoint Hessian operator than the property of Leg-
endre transforms ensures that their propagators are indeed the inverse of each other which has
further implications on the relation of higher n-point functions.
The advantage of working with the effective formalism instead of using connected corre-
lation functions becomes apparent when systematically approaching the underlying theory or
when using non-perturbative approximations. While the Schwinger functional is concerned
with field space itself, the effective action depends on the expectation values and may be in-
terpreted as a functional of theories for a specific field space. Varying the expectation value
for fixed sources corresponds to altering the measure and thereby the theory itself. It turns out
to be of great benefit to work in the effective picture whenever the theory rather than the field
space is the objects of interest. This reflects the distinction between [F] and 〈[F]〉 meaning that
perturbation of fluctuations are something fundamentally different than perturbations of expec-
tation values. This has to be kept in mind when introducing approximations in one or the other
description.
2.2.3 Free theory
At this point it is very instructive to exemplify the described techniques for a free theory. By
definition each non-interacting theory is classified by a linear, positive, self-adjoint operator
A ∗Φ̄ = A over field space, for which the generating functional can be cast into the followingΦ̄
form:
( − ( )) 1 ( )Z[J;Φ̄] = exp − 1G 12 J , AΦ̄J =⇒ W [J;Φ̄] = lnZ[J;Φ̄] =− 1 −12 G J , Az z Φ̄J
Notice that the Schwinger functional is bilinear in the sources and symmetric due to the self-
adjoint property of AΦ̄. As stated above, the effective theory is defined by the Legendre-Fenchel
transformation of the Schwinger functional and explicitly given by
{ } { − ( )}Γ[ϕ ;Φ̄] = sup J(ϕ)−W [J;Φ̄] = sup J(ϕ)+ 1G 12 J , AΦ̄ J
J∈T∗ [F] J∈T∗ [F]
Φ̄ Φ̄
Due to the positive definiteness of AΦ̄ the Schwinger functional W [J;Φ̄] is strictly convex and
surely differentiable, so we can evaluate the supremum by determine its global maximum:
( )
0= dv ϑ(ϕ)−W [ϑ(ϕ);Φ̄] = dvϑ(ϕ)−dvW [ϑ(ϕ);Φ̄]
The maximum function ϑ maps to a dual vector which is linear in the expectation fields and
thus its functional derivative is simply given by
( ( ))
d −1vJ(ϕ) = v(ϕ) =G v , G • , ϕ
102 Chapter 2. Quantum Field Theory
In the last step used the isomorphism provided by the field space metric to artificially rewrite
the real number v(ϕ). The second contribution originates from the functional variation of W ,
i.e.
d W [J;Φ̄] =− 1 G−
( ) − ( ) ( ( ) )1 v , A J − 1 G 1v 2 Φ̄ 2 J , AΦ̄ v =− 1 G−12 v , A + A∗ Jz
− ( )
z z Φ̄ op Φ̄
=−z 1G−1 v , AΦ̄ J
Here we used the fact that AΦ̄ is self-adjoint and the field space metric symmetric in its argu-
ments. Combining both results to deduce the maximum yields
0= v(ϕ)+ z− −
( ) ( ( )) ( )
1G 1 v , AΦ̄ ϑ(ϕ) =G
−1 v , G • , ϕ + z−1G−1 v , AΦ̄ ϑ(ϕ)( ( ) )
=G−1 v , G • , ϕ + z−1AΦ̄ ϑ(ϕ)
The field space metric is a non-degenerated symmetric bilinear form and the above condition
holds for all v ∈ TΦ̄ 〈[F]〉. Hence the second argument has to be the identity 0 ≡ eT 〈[F]〉 whichΦ̄
further implies
ϑ(ϕ) =−z A−
( ) ( )
1G • , ϕ ⇐⇒ ϕ =−z−1G−1 • , AΦ̄ ϑ(ϕ) (2.20)Φ̄
This reveals a special feature for a free theory, namely the linear relation between the maximum
field ϑ and the mean fluctuations ϕ . While ϑ(ϕ)(v) is a dual vector and thus linear in v, the
explicit dependence on ϕ is in general not linear. Next, we evaluate the supremum condition
for Γ[•;Φ̄] using the relation between ϑ and ϕ :
∣
Γ[ϕ ;Φ̄] = J(ϕ)∣
( )
+ 1 G−1 ϑ(ϕ) , A ϑ(ϕ)
(J=ϑ (ϕ) (2z )) Φ̄( )
=G−1 ϑ(ϕ) , G • , ϕ + 1 G−12 ϑ(ϕ) , Aϑ(ϕ)( ) z− ( )=− 1G 1 ϑ(ϕ) , Aϑ(ϕ) + 1 G−12 ϑ(ϕ) , Aϑ(ϕ)z ( ) z
=− 1 G−12 ϑ(ϕ) , Aϑ(ϕ)z
This results in a bilinear expression in ϑ which is by eq. (2.20) also linear in ϕ . We conversion
to the bilinear form of Γ[•;Φ̄] in ϕ is found to be
( )
Γ[ϕ ;Φ̄] =− 1 G−12 ϑ(ϕ) , Aϑ(ϕ)z
− ( − ( ) ( ))=− z G 1 A 1G • , ϕ , AA−12 G • , ϕ( Φ̄− ( ) (
Φ̄ ))
=− z2 G
1 G • , ϕ , A−1G • , ϕ
( ( Φ̄ ) )
=− z −1 −12 G ϕ , G • , A G ϕ (2.21)Φ̄
This confirms the general inversion relation between the Hessian operator of W [•;Φ̄] and the
respective one of Γ[•;Φ̄] found in eq. (2.19), here explicitly given by
[ ] ( ( [ ])−1 )
Ĥess W [•;Φ̄] =−z−1J AΦ̄ =G • , Ĥessϕ̂ Γ[•;Φ̄] G−1
For a free theory, the bilinearity of the Schwinger functional implies:
( [ ] ) [ ] ( )
Γ[ϕ ;Φ̄] = 12 G ϕ , Ĥessϕ̂ Γ[•;Φ̄] ϕ with Ĥess Γ[•;Φ̄] ≡−zG
−1
ϕ̂ • , A−1GΦ̄
Depending on the form of the field space metric this Hessian operator is diagonal whenever AΦ̄
is diagonal. When G contains non-diagonal[element]s it is still non-degenerated and a change of
basis in TΦ̄ 〈[F]〉will yield a diagonal Ĥessϕ̂ Γ[•;Φ̄] . Thus, for any free theory, the one-particle
and connected propagators are diagonalizable and in fact the only non-vanishing connected n-
point functions.
We are now going to leave the more technical side of field theory and solidify the physi-
cal part of quantum field theory. We will see how the mathematical considerations yield to a
straightforward description of probabilistic theories and how it provides a deep understanding
of the physical concepts.
2.3 Quantum field theories 103
2.3 Quantum field theories
The beginning of the 20th century witnessed the birth of a new era in physics that shook science
to its very foundations. Hundred years later, our description and understanding of Nature has
undergone a severe change in which generations of physicists and mathematicians have re-built
a seemingly solid basement for future investigations. This transformation that took place at the
turn of the century culminated in works of Planck and Einstein and from then on resulted in
deep insights of the nature of particles and fields that constitute our Universe, culminating in
the formulation of the Standard Model of particle physics in the late 60’ s. One can say that
this time span of natural science was the golden age of particle physics where we gained re-
markable precisions between experiments and theory on these energy scales. Furthermore, the
inconsistencies and problems that arise in the Standard Model of particle physics can be mainly
consolidated to a single interaction, gravitation, and thus some of them disappear whenever
gravitational effects are negligible. Nevertheless, the great success of recent decades provides
only a kind of affirmation that there is some hidden truth in what we have derived. The incorpo-
ration of gravitational effects is surely one of the most exciting questions future generations of
scientists have to answer. Fortunately, the scales at which gravity becomes relevant or dominates
the structure of the Universe are two-sided: On one hand quantum effects of the gravitational
force are assumed to strongly participate in the fusion of interactions at the Planck scale, thus at
very high energies. On the other hand we observe its importance on cosmological scales where
all other interactions have smeared out and what is left is a (effective) gravitational theory well
described by General Relativity. This dual importance in describing infrared and ultraviolet
phenomena may help us to deduce a fundamental theoretical description of gravity by means of
cosmological data and it seems that with the advent of the new millennium the era of cosmology
revives in order to understand the origin of the small gap that remains in the (possibly) dense
blanket of our today’s knowledge. As usually, smallness is only relative and time will tell what
is hidden on the next, deeper level.
We have prepared the mathematical grounds to present the framework of quantum field the-
ory in the light of symmetry principles and measure theory. The missing mathematical rigor
in some parts of this presentation is partially due to focus on the essential concepts relevant
for physics and partially because some of the interesting constructions that appear in physics
are yet to be put on solid mathematical grounds. Therefore, the following discussion has to
be understood on a formal level. In this section we will continue to translate measure theoret-
ical concepts into the language of quantum field theory, state its Euclidean axioms, perform
symmetry analysis, and present some perturbative techniques to evaluate observables within
this framework. Finally, we conclude with a short summary of the Standard Model of particle
physics before introducing the main, non-perturbative tool for later calculations: the Functional
Renormalization Group Equation. For references we refer to [24, 25, 28].
2.3.1 Quantum field theories
Quantum field theory is an attempt to reconcile the theory of special relativity with quantum
mechanics, which is indicated by its incorporation of the physical constants c and h̄, closely
related to the two very successful theoretical descriptions. Within the formulation of gauge the-
ory, QFT had its greatest success so far in the shape of the SM that has not only recovered the
classical observations in its low energy effective limit, but has also made remarkably precise
predictions that led to new discoveries and a deep understanding of Nature. Before its formula-
tion in terms of gauge theory, there had been several attempts to extend the classical world to
be emerging from the laws of quantum physics. In this process it is very difficult for scientists
to not go astray due to the large amount of possibilities of how to extend a certain mathematical
formalism. There are several distinct routes one can pursue, which all lead to the same classical
104 Chapter 2. Quantum Field Theory
limit but are fundamentally different beyond this classical regime.6 In order to deduce the ‘right’
mechanism or strategy one has to distinguish between the various approaches by verifying or
falsifying their predictions using experiments, or by revealing flaws in the mathematical descrip-
tion. Nowadays, we meet the same problem at the heart of constructing a theory of quantum
gravity, however with the difference that so far there is no real experimental guiding line that
helps in singling out certain theories. Overcoming the mathematical issues that appear when
applying the standard methods of QFT to gravity turns out to be a severe challenge, and all
approaches that either suggest cures on the level of the theory itself or extend the general frame-
work still struggle with their own flaws emerging due to their particular generalization ideas.
Thus, in comparison to the strong- and electroweak forces there is so far no fully satisfactory
program that describes a quantum version of the classical gravitational interaction. The usual
procedure that provides a systematic way to construct a quantum field theory from a classical
Hamiltonian or Lagrangian formalism is denoted quantization. Though it is surely not the way
Nature works, i.e. generating the more fundamental laws from effective ones, it at least ensures
that all experiments of the ‘classical world’ are also described by its quantum theory. This
is one of the pillars a solid theory is based on, the verification and prediction of observations.
The other one is its mathematical consistency, which is a fundamental necessity as long as one
assumes Nature to be written in this language.
From the various, under certain assumptions equivalent, description of quantum theories,
in the sequel we are following the route of the generating functional approach to QFT. We
basically differentiate between two approaches: the ‘deductive’ and ‘inductive’ strategy. Both
invoke the same mathematical machinery, namely measure theory, and in addition both rely on
some spacetime manifold over which a suitable field space in accordance with a set of symme-
tries is constructed. What differs is the perspective – or direction – from where the methods to
deduce results are applied. In the ‘deductive’ approach one sets up a theory, a measure space
([F],Σ[F],µ), over some fundamental fields right at the beginning and walks through all the pre-
scribed steps to obtain the associated observables as expectation values, that can be measured
in certain, especially adapted experiments. The difficulty resides in the fact that usually there
is a large class of theories which fulfill the mathematical and physically motivated restrictions.
Thus, finding the physically realized measure out of all possible ones is a laborious task involv-
ing a large number of calculations whereby observables from the theory have to be compared
to real experiments. The more mathematical constraints are available the less trial and error is
needed.
On the contrary, in the ‘inductive’ approach one assumes a familiarity with the observable
data and starts out with the expectation fields Φ ∈ 〈[F]〉 from which one tries to infer the un-
derlying theory by reverting each step towards a finally admissible measure space ([F],Σ[F],µ).
This second approach stays closer to the experimental side and has to assume much less at the
very beginning, at least if it is based on real physical data. Its difficulty however is the pro-
cess of guessing the right measure by its produced results. This is a quite complex task since
different measure spaces may produce the same expectation values for a a set of observables.
Hence, only the full knowledge of all experimental outcomes mays reveal the right, realized
theory consisting of field space [F] and a measure µ .
While the two strategies start from opposite directions and have in principle different partial
objections, in practice there is not a clear separation. Instead one employs deduction and induc-
tion techniques alternately to ultimately find the fundamental theory underlying Nature. This
synthesis is found in the renormalization procedure in perturbative methods, for instance. Be-
fore we continue to discuss the mixed forms of both strategies suitable for practical applications,
6This already happens on a classical level, where different theories describe the same so far observed phenomena
successfully, for instance in the theory of gravity.
2.3 Quantum field theories 105
we present their uncombined versions first.
The ‘deductive’ approach
Starting from a theory and deducing the experimental observables as predictions is surely the
ultimate dream of physicists. Therefore, we need at least some mathematical constraints that
guide us on this difficult journey. Nevertheless, in the end we have to utilize experiments to
uniquely distinguish between the different mathematical constructions. In our setting based on
measure theoretical concepts, the ‘deductive’ path proceeds along the following lines:
A. Spacetime Define a set of admissible spacetimes as smooth manifolds over some topol-
ogy, possibly with additional constraints (e.g. compactness,. . . ):
spacetimes ≡ {[M]∼ |M smooth manifold + constraints}Diff(M)
In this step one has to choose a cardinality of spacetime, a topological class, and a differ-
ential structure to uniquely specify the associated equivalence class of smooth manifolds.
B. Symmetries Select a symmetry, a (Lie) group G ≡ (XG,◦G), that should be preserved
by the theory. It may consist of several basic groups that are combined by means of a
(semi-)direct product. For a choice of spacetime, construct the class of principal fiber
bundles:
PBs(M,G)≡ {[P=MπP×ρRG]∼P FB }
Thus, in this second step we choose the symmetry group G and the underlying differential
structure by means of an equivalence class of PBs(M,G).
C. Field space For each class of principal bundles we can construct the set of compatible
principal connections that constitute the gauge fields:
[C]≡ {{ f ∗Gω | fG∈ G}|ω : TP→ g principal connection }
Here and in what follows Grepresents the group of gauge transformations.
Additionally, define a set of representations ρ j of G on some vector spaces Vj with V ≡
V1×V2× . . . to obtain the matter sector:
[FM]≡ {{ fV ◦φ ◦ fG| fG∈ G∧ fV ∈ Iso(V )}|φ : P→V
with φ(ρRP (•,g)) = ρ(g−1,φ(•))}
Next, we have to transform the space [F], being defined as a suitable combination of
[C] and [FM], into a Fréchet manifold F sliced by the respective diffeomorphism group,
which is usually equipped with a field space metric G. Therefore, it is convenient to
employ the background field covering to express F in terms of its tangent spaces. In this
setting, we can apply geometric methods to field space, in particular we can establish a
principal bundle construction based on all mathematical fields with the group of gauge
transformation as its structure group. In the non-trivial case, which appears in theories of
QG the corresponding principal connections and smooth structures are also non-trivial.
Hence, in this step we choose the field content of our theory by fixing a connection and
determine a number of representations.
D. Measure space We use the topology of [F] to define the associated Borel σ -algebra Σ[F].
The class of theories is then given by the set of all measures on ([F],Σ[F]):
Meas([F],Σ[F])≡ {([F],Σ[F],µ) |µ measure on ([F],Σ[F])}
106 Chapter 2. Quantum Field Theory
Usually it is appropriate to restrict to measures that are Radon-Nikodym transforms of
some Gaußian measure µG, i.e.
{ ∫ }
([F],Σ[F],µ) ∈Meas([F],Σ[F]) |µ(U ∈ Σ[F])≡ µG(u)ρ(u)
x∈U
Then, normalizing the Gaußian measures such that µG([F])≡ 1 holds true determines the
notion of a free theory, see [62]. Fixing the measure corresponds to a choice of theory.
In case of gauge theories, we usually replace the physical field space by the full mathe-
matical space F using a gauge fixing independent description, for instance by requiring
BRST invariance of the measure, see subsection 2.1.5 for details.
E. Observables To extract observables from a theory it is convenient to introduce the Fourier
transform of the measure and therefore express field space as a patchwork of its tangent
spaces by means of a background field covering UBFC ≡ {(Φ̄ j,U j)} and a partition of
unity {(U j,ς j)}.
∫
Z[J;UBFC] = µ([F])
−1 · ∑ dµ(expΦ̄ (ϕ̂))ς j(ϕ̂) expC(zJ(ϕ̂))j
(Φ̄ ,U )∈ ϕ̂∈UU jj j BFC
To be consistent, observables should not depend on the choice of UBFC. In what follows,
we assume that [F] can be covered by a single tangent space at Φ̄, say, and thus we obtain
a simplified expression for the generating functional:
∫
Z[J;Φ̄] = µ([F])−1 · dµ(exp
∈ [F]
(ϕ̂)) expC(zJ(ϕ̂)) (2.22)
ϕ̂ TΦ̄[F]
Notice that we have formally normalized the theory with the factor µ([F])−1 in order to
describe a probability setting, i.e. Z[0;Φ̄] = 1 and thus a vanishing Schwinger functional
for source-free constellations, W [0;Φ̄] = ln(Z[0;Φ̄]) = 0. If we proceed to the effective
action, Γ[ϕ ;Φ̄] = supJ{J(ϕ)−W [J;Φ̄]}, we have to ensure that the Schwinger functional
is strictly convex to retain all information during the transition. Notice, that the effective
action is a functional of the expectation fields ϕ rather than the quantum analogs ϕ̂ , which
in principle can live in different spaces.7
One then selects a set of observables, represented by some functional on F, and evaluates
them using functional variation with respect to the sources. This establishes the link to
the experimental side of physics where predictions derived from the underlying theory
have to be tested. Hence, the final step consists in testing a number of experimental
observables against theoretical predictions:
Obs≡ {O : [F]→ R |O measurable }
Likewise, one can work on the level of Γ[ϕ ;Φ̄] directly and relate its vertices to observ-
ables.
We have seen that at several points we have to make a decision where physical input is needed
to distinguish between the mathematical concepts. Thus, this ‘deductive’ method is already far
from being purely deductive.
7In particular, this appears for probability theories for which F is not closed under scalar multiplication of R.
For instance, the expected number one rolls with a fair dice is Φ ≡ 〈Φ̂〉dice = 7/2 which is a fraction rather than an
integer.
2.3 Quantum field theories 107
The ‘inductive’ approach
The somewhat opposite direction is the ‘inductive’ method, whereby initially one observes Na-
ture and then deduces the mathematical structure hidden behind the curtain of phenomena. Even
though some steps seem to be equivalent to the ‘deductive’ approach there is a subtle difference
as we will see.
A. Spacetime In fact, the first step actually coincides with the ‘deductive’ one. We define
the set of admissible spacetimes, being smooth manifolds over some topology that may
fulfill additional constraints:
spacetimes ≡ {[M]∼ |M smooth manifold + constraints}Diff(M)
Again one decides about the cardinality of spacetime, the topological class, and the dif-
ferential structure when selecting an equivalence class of smooth manifolds.
B. Symmetries The order in which we present the ingredients of the theory in both cases, in
the ‘deductive’ or ‘inductive’ approach, already indicates the central role of symmetries in
determining the laws of Nature. There is however a difference in the interpretation of the
postulated symmetries. While the former describes invariance of the fundamental theory,
the symmetries we introduce in the present context concern observables and appear on
an effective level. The two symmetry concepts may not agree and symmetries we deduce
from observing Nature may differ from those actually realized. On the one hand the
averaging procedure may smear out certain impurities thus giving rise to new symmetries
on macroscopic scales. On the other hand, the fundamental theory can exhibit symmetries
only for some relevant subclasses of fields, for instance the classical sector, while on other
measurable sets they disappear. The overall sum, the integral, can result in observables
that cloak the fundamental invariance and the so-called anomalies arise. Usually, we
expect symmetries to survive the construction of expectation values and assume that the
fundamental symmetries are also indirectly visible in the effective action. Given a choice
of spacetime and the symmetry G〈[F]〉, we construct the class of principal fiber bundles:
PBs(M,G〈[F]〉)≡ {[P=MπP×ρRG〈[F]〉]∼P FB }
An element of PBs(M,G〈[F]〉) defines the differential structure. In this second step we
have to select the symmetry group G〈[F]〉 of expectation fields.
C. Effective field space Once a specific class of (observable) symmetries G〈[F]〉 ≡ (XG,◦G)
is fixed the field space is recovered in the usual way by fiber bundle techniques. We
start by defining the set of all principal connections compatible with a choice of P ∈
PBs(M,G〈[F]〉).
[〈C〉]≡ {{ f ∗Gω | fG∈ G}|ω : TP→ g principal connection }
Like their quantum analogs, they can be understood as gauge fields that mediate the
interactions, however here they have a different interpretation as expectation fields. Again,
the physical field content is obtained by an equivalence class of principal connections
generated by the group of gauge transformations, now with respect to G〈[F]〉 rather than
G.
Similarly, the space of matter fields on the level of expectation values, is defined in an
analogous way to its quantum counterpart. We first of all choose a set of representa-
tions ρ j of G〈[F]〉 on some vector spaces Vj with V ≡V1×V2× . . . and then consider the
collection of smooth sections as matter fields:
[〈FM〉]≡ {{ fV ◦φ ◦ fG| fG∈ G∧ fV ∈ Iso(V )}|φ : P→V
with φ(ρRP (•,g)) = ρ(g−1,φ(•))}
108 Chapter 2. Quantum Field Theory
Dropping the discrimination between matter and gauge fields and introduce the combined
space 〈[F]〉, we proceed along the same lines as previously, however with a different in-
terpretation for the underlying fields. First, the extended space 〈F〉 is transferred into a
Fréchet manifold sliced by the respective diffeomorphism group. It is then usually pos-
sible to equip 〈F〉 with a field space metric G, whereby the background field method is
applied in order to diffeomorphically relate 〈F〉with its tangent spaces. The resulting prin-
cipal bundle structure can then be studied concerning its (non-)triviality and a connection
on field space has to be selected in order to define field variations.
Hence, in this step we choose the field content of our theory by fixing a connection
and determine a number of representations. We further establish a differential geometry
description of 〈F〉.
D. Observables In this setting, observables are maps from the space of expectation fields to
the real or complex numbers:
〈Obs〉 ≡ {〈O〉 : 〈[F]〉 → C | 〈O〉 smooth }
At least the subspace of convex functions is closely related to the concept of (convex)
observables in the deductive approach. Given a probability theory µ , then Jensen’s in-
equality provides a relation between a convex (quantum) observable O ∈ Obs and an
associated convex element 〈A〉 ∈ 〈Obs〉:
〈O〉 (ϕ)≡ 〈O〉 (〈ϕ̂〉µ)≤ 〈O(ϕ̂)〉µ
The equality is satisfied only in case of linear, convex functions A, thus in particular we
have ϕ ≡ 〈[ϕ̂ ]〉µ for a probability theory.
E. Effective Theories Next, we have to establish a theory describing the properties and
phenomena of observables. Instead of integrating or averaging over the fundamental
degrees of freedom, we consider a prescription build on the space of expectation fields
〈[F]〉 which matches the observables. Therefore, we start with the effective action Γ[ϕ ;Φ̄]
and ‘deduce’ its structure from the observed phenomena. Usually, however, one starts
with a very general ansatz for Γ[ϕ ;Φ̄] based on the symmetry G〈[F]〉 and then matches the
coefficients or couplings with experimental results:
∣∣
dδϕ · · ·dδϕ Γ[ϕ ;Φ̄]∣ ≡ O(ϕ )︸ ︷︷ ︸ Expϕ=ϕExp
ℓ-times
An effective action for which all n-point functions can be simultaneously diagonalized
describes a free theory.
F. Reconstruction of the measure If the effective action is based on a fundamental theory
described by the integral representation à la Feynman, then we should be able to revert
the steps from Z[J;Φ̄] to Γ[ϕ ;Φ̄]. Therefore, we have to ensure that Γ[ϕ ;Φ̄] is convex in
order to be the Legendre-Fenchel transform of a Schwinger functionalW [J;Φ̄]:
{ }
WΓ[J;Φ̄] = LFT(Γ[•;Φ̄])[J;Φ̄]≡ sup J(ϕ)−Γ[ϕ ;Φ̄]
ϕ∈TΦ̄〈[F]〉
Whereas the possible convexity of W [J;Φ̄] is an important, even though not necessary
feature that ensures all information is available on the level of the effective action, the
convexity of Γ[ϕ ;Φ̄] is mandatory. For any Γ[ϕ ;Φ̄] which derives from a Schwinger func-
tional W [J;Φ̄] convexity is an inevitable requirement one has to impose. All deviations
from Γ[ϕ ;Φ̄] being convex result in some ignorance of the fundamental theory towards
certain observable effects and hence the fundamental nature of the theory is questionable.
2.3 Quantum field theories 109
In the next step,WΓ[J;Φ̄] has to be converted into the generating functional, which in the
usual case is trivial:
ZΓ[J;Φ̄]≡ exp(WΓ[J;Φ̄])
The consistency of the formalism now requires that if Γ[ϕ ;Φ̄] is a true effective action de-
rived from a fundamental measure µ in the usual way, then ZΓ[J;Φ̄] has to be the Fourier
transform of µ . As such, it should exhibit all the properties a generic Fourier transform
satisfies, in particular those described by the Minlos- or Bochner theorem mentioned in
subsection 2.1.4, i.e.
ZΓ[0;Φ̄] = 1 normalized, ZΓ[•;Φ̄] continuous
∀c ∗r ∈ C ,∀Jr ∈ TΦ̄F : ∑ c̄s · c
∗
r ·ZΓ[Jr− Js ;Φ̄]≥ 0 positive definite
r,s
Notice that a consequence of positive-definiteness are the following two necessary, but not
sufficient, conditions: ZΓ[0;Φ̄] > 0 and |ZΓ[J;Φ̄]| ≤ ZΓ[0;Φ̄]. Any admissible effective
action Γ should be related to a ZΓ which satisfies the above conditions, i.e. which is the
Fourier transform of a suitable probability measure. Parts of the Osterwalder-Schrader
axioms are related to exactly those requirements.
If ZΓ is well behaved in the above context, then we can apply the inverse Fourier transform
and obtain the underlying measure µ . However, the actual field content is still unknown,
since there might be theories that can be described by the same effective action albeit
based on different measurable spaces. This concludes the second strategy, a route from
observations to a fundamental theory.
A comparison
Even though it is instructive to keep the ‘deductive’ and ‘inductive’ ideas kind of separated,
for practical applications a combination of both approaches is necessary and one encounters
‘smooth transitions’ between them in any construction of a theory. A ‘deductive’ method always
requires experimental input to define the mathematical boundaries and in a sense makes a choice
of what is real and what is pure imagination. Mathematics is usually not the ground to decide
about this, but rather provides insights why the Universe may resolve the way it does. On
the other hand, the ‘inductive’ approach is of no need if it is purely for the sake of describing
observed experiments without producing independent predictions. At a certain point one has to
leave either of these approaches and start working in the opposite direction to gain a true insight
into the very foundations of Nature. The functional renormalization group method is actually a
remarkable example where one starts by some very basic experimental input on the level of the
effective action and derives a mathematically consistent theory which then is used to produce
new predictions from fundamental principles.
We have seen that the relation in the path integral approach is given by the transition from
fundamental fields ϕ̂ to its expectation values ϕ ≡ 〈ϕ̂〉µ and from a measure µ to an effective
action Γ. As should be emphasized again, the space of expectation fields will in general differ
from the one describing the actual degrees of freedom we averaged over. It may even affect the
boundary conditions the fields satisfy. Hence, one has to be particularly careful which side of the
theory one tries to interpret, after or before integration. Approximation schemes applied to the
effective action rather concern variations of the measure itself, while perturbation approaches of
Z[J;Φ̄] rather address small deviations of the underlying field space. We will put some special
emphasis on this point when considering perturbation theory in the measure-based context.
110 Chapter 2. Quantum Field Theory
2.3.2 Theories, phases, and anomalies
No matter if one starts with the ‘deductive’ or ‘inductive’ framework, there is a general proce-
dure to construct theories, quite similar to fitting curves to observable data. The initial ingredi-
ents are given by a symmetry group G and a concept of spacetime M, which might be discrete
or continuous (here we prefer the latter case). Notice that this choice is usually guided by our
observations and thus inductive in nature.
Once these basic conditions are settled, we determine the most general smooth functional
A : F→ R compatible with the invariance conditions and the spacetime structure:8
A[σ(g,Φ̂)] = A[Φ̂] and A[ fM∗Φ̂] = A[Φ̂] (2.23)
Here, σ : G×F → F stands for the G-representation on F and fM denotes an element of the
diffeomorphism group acting on Φ̂. The solution to this equation is by no means unique and
additional initial conditions have to be chosen. In the theory of differential equations, parame-
ters usually substitute the remaining freedom and likewise we can define a family of functionals
that solve eq. (2.23), each of its elements denote an action functional:
S : F×Λ → R , with S[σ(g,Φ̂);{u}] = S[Φ̂;{u}] and S[ fM∗Φ̂;{u}] = S[Φ̂;{u}]
Here, Λ represents the space of free parameters. The more supplementary mathematical con-
straints we can physically justify, the less parameters u ∈ Λ appear. Each left-over parameter u
has to be fixed by experimental data in the way a generic family of functions has to be matched
to an experimentally observed curve. This amounts to imposing initial conditions that are com-
patible with the phenomena we experience in Nature.
Similar to the family of action functionals, we can define a quantum field theoretical version
based on a family of probability measures satisfying certain symmetry and diffeomorphism
properties:
µ : ΣF×Λ → R , with µ(•;{u}) ∈MeasP(F,ΣF) and further constraints
A particular example is the Lebesgue measure for which a single parameter ‘survives’ the math-
ematical constraints and ultimately has to be fixed by a normalization condition. In the general
case, where we have a possibly large number of free parameters that define the most general the-
ory compatible with the invariance requirements, we have to evaluate observables O : ΣF → R
each representing a particular experimental setting. For any free parameter, we need at least one
observation OExp to fix the initial conditions, i.e.
∞ ∣
! 〈 〉 ≡ (ℓ)OExp = O µ ;{u} ∑ αO ·dJ · · ·
∣
d Z[J;Φ̄;{u}]
︸ Exp ︷︷ JExp
∣
ℓ=0 ︸ J=0
ℓ-times
Notice the dependence of the RHS on the free parameters {u}, which are in general not observ-
ables but can be inferred from the set of observable equations. In order to have full predictivity
of the theory there should be only a finite number of independent parameters {u}. Hence, either
we have strong enough mathematical constraints at the very beginning, or we have to impose
relations between the different parameters in order to reduce their number to be finite. If this
is the case, a probability theory is denoted renormalizable. The origin of this word is found in
perturbation theory, which is discussed in the next subsection. A concept of renormalizability
that extends the perturbative notion is later given in the context of Asymptotic Safety.
8In the following we consider the measure-based description and refer for its effective action counterpart to the
discussion of the Functional Renormalization Group in chapter 4.
2.3 Quantum field theories 111
R Notice that besides consistency conflicts that appear on a mathematical level, observations
can falsify the assumptions on G andM, retrospectively.
( j)
Now, let us assume we are given a set of experimental observations {OExp} that should match
with a certain class of theories, µ(•;{u}), compatible with a symmetry group G and spacetime
M, and its number should be sufficient to fix all free parameters {u}. Then, if possible, we
can use the standard methods of fitting curves to data, which usually defines a kind of iterative
procedure, as for instance the Gauß-Newton algorithm. In general, this amounts to a sequence
in the space of probability measures MeasP(F,ΣF) that ultimately converges to the possible
real theory. Initially, one has to choose any allowed set of parameters {u(0)}, evaluate the
predictions for the experimental results 〈O〉µ ;{u(0)} and apply a suitable algorithm to reduce the
error by adjustment of {u(0)}:
• { (0)} −a−lg−or−it−h→m • { (1)} −a−lg−or−it−h→m · · · −a−lg−or−it−hmµ( ; u ) µ( ; u ) → µ(•;{uExp})
The convergence and number of iterative steps of this series depends on the starting point {u(0)}
and hence physically reasonable choices of these initial values, as for instance results from the
classical theory, may severely reduce the work.
Notice that in each step, the choice of {u( j)R } fixes a measure, i.e. µ( j)(•) ≡ µ(•;{u( j)})
and thus gives rise to new expectation values and observable results. The procedure of
( j)
adjusting (or normalizing) the theory w.r.t. {OExp} is referred to as (re-)normalization in
certain contexts.
Besides this experimental matching, there is another very useful approach that addresses the
symmetry properties ofMeasP(F,ΣF) depending on the set of parameters. The characterization
of the family of admissible theories nicely demonstrates the significance of studying symmetry
phases and phase transitions. W∫e first focus our attention on the density ρ(Φ̂;{u}) and after-
wards consider the full measure dµG ρ(Φ̂;{u}). While the general constraint imposed on the
class of admissible theories is given by the group action of G, for certain measures, for instance
the trivial one ρ(Φ̂;{u}) = 1, the system will exhibit further invariances. Explicitly, let us cate-
gorize the space of probability densities ρ(Φ̂;{u}) under a particular symmetry group H . The
parameter space Λ is given by different choices for ~u ≡ {u} and for the density function the
order space will be the (quantum) field space, i.e. S≡ F. Depending on the constraint-functions
f ( j), we investigate different properties of the probability densities concerning their symmetry
behavior. Typical examples address the transformation of the density itself and/or solutions to
the field equations:
f (0)
!
(ϕ̂) = ρ(σ(h,Φ̂);{u})−ρ(Φ̂;{u}) = 0 ∀h ∈H
!
f (1)(ϕ̂) = dvρ(Φ̂;{u}) = 0 ∀v ∈ TΦ̄F
The space of solutions to these constraints is sensitive to the particular choice of~u≡{u} and will
in general be only a subspace of S ≡ F, denoted Sol(~u). The properties of Sol(~u) characterize
the phase in which the corresponding theory lives in. Important choices of H are the conformal
group, or O(N), where N denotes the number scalar fields, for instance. The first condition
f (0) specifies the subspace Sol(~u) for which the associated density ρ(Φ̂;{u}) is conformally
invariant. If for a particular choice of ~u the full space satisfies this criterion, i.e. Sol(~u) ≡
S ≡ F holds true, then the density is conformally invariant, irrespectively of its argument. In
another example, based on the cyclic symmetry group H ≡ Z2, let us consider the following
112 Chapter 2. Quantum Field Theory
two theories, specified by, say, ~u(a) = (2,1,−3,0,0 · · · ) and ~u(b) = (2,0,−3,0,0 · · · ) that are
special cases of the general family of polynomial functionals (here (•)u≥3 = 0):
ρ(Φ̂;~u(a) ≡ (a) (a) (a)) u0 +u Φ̂+u Φ̂21 2 =⇒ Sol(~u(a)) = {eF}
(b) (b)
ρ(Φ̂;~u(b))≡ u0 + 0 · Φ̂+u 22 Φ̂ =⇒ Sol(~u(b)) = F
Here, we assumed that F is vector space and eF denotes its identity element. Thus, we observe
that both theories live in different phases. In a similar way one can study the properties of the
solutions of the field equations and classify theories according to their solution spaces.
In most approaches, the density is given by a classical analog of the action functional and
thus is invariant under classical symmetries, i.e.
( )
ρ(Φ̂;~u)≡ exp −S[Φ̂;~u]
Nevertheless, the quantum theory includes a measure µG and may violate the invariance prin-
cipal on the quantum level. That is, while the density exhibits a certain symmetry, the full
measure does not:
∫ ∫
µG(Φ̂)ρ(Φ̂;~u) 6= µG(Φ̂)ρ(Φ̂;~u) while ρ(σ(h,Φ̂);{u}) ≡ ρ(Φ̂;{u})
σ(h,U) U
In the transition from a classical theory to its quantum version the breaking of a classical sym-
metry is referred to as the appearance of an anomaly. This phenomenon may occur for any
probability density, not necessarily restricted to exponential types. A very famous example is
the conformal anomaly, where in addition to the failure of the quantum theory to exhibit con-
formal invariance, the trace of the energy-momentum tensor has a non-vanishing expectation
value.
2.3.3 Perturbation theory
The reason why we put so much emphasis on symmetries and group theoretical aspects in the
introduction is essentially the important status of these concepts in identifying suitable theories.
It is one of the most instructive guiding principals that helps in deducing the laws of Nature
from experiments. In the study of the renormalization group, symmetries again play a crucial
role. They will identify the behavior of entire universality classes under the renormalization
group flow and thus provide the foundations of what follows.
No matter from which direction we hunt the fundamental theory of Nature, whether we start
from some theoretical assumptions or from certain deductions of observations, it is always the
aim to reduce the amount of input as far as possible. Symmetries surely play a central role,
due to the powerful constraints they introduce, nevertheless at a certain point we have to make
a connection to reality in order to uniquely determine the right theory. The famous example
we have already encountered is the Lebesgue measure which is defined on the basis of some
mathematical constraints that seem plausible for measuring lengths. The result is a family of
measures parametrized by a single variable λ , i.e.
µL(A= (a,b);λ ) = λ · (a+Rd (−b))
This remaining freedom still needs one more physical information we have to add to our de-
scription to uniquely determine the correct measure. In fact, in this case it corresponds to fixing
a physical unit, meaning that we have to perform a measurement and allocate a specific value
to this length. This normalization allows to express all other sizes in terms of multiples of this
prototype unit (also compare the definition of the international standard units).
2.3 Quantum field theories 113
The same ideas apply to the concepts of quantum field theory. At the very outset, we collect
a set of mathematical requirements that we assume to be physically relevant. The most general
form of theory that respects the demands involves a set of free parameters. While on paper we
have a clear separation of ‘inductive’ and ‘deductive’ methods, in praxis the applied techniques
are similar to each other. In both cases invoking (for instance symmetry) constraints, we are left
with a family of theories; each member being a suitable candidate from a mathematical point of
view:
∫
Z[J;Φ̄;{u}] ≡ dµG(ϕ̂ ;Φ̄) ρ(ϕ̂;Φ̄;{u}) or Γ[ϕ ;Φ̄;{u}]
ϕ̂
In this way we can make the construction systematic in either of the approaches and start with
mathematical assumptions before incorporating experimental results. To recognize the hidden
symmetries that underlie the laws of Nature requires deep insight and is in fact either some
fortunate coincidence or true stroke of genius. Once restricted to a class of suitable theories, the
next step involves evaluating observables. In this general setting the actual value is determined
up to the free parameters {u} we introduced previously, thus every allocation to experimental
data will fix at least one of them.
∫
〈O(ϕ̂)〉µ ;{u} ≡ ∑ dµ(ϕ̂ ;Φ̄;{u})O(ϕ̂ ;Φ̄) or 〈O(ϕ̂)〉1PI∈ µ ;{u}Φ̄ ϕ̂ TΦ̄[F]
This process, known as normalization, may also come along with mathematical restrictions
due to regularity aspects. This bridge between theory and experiment can be summarized by
u ≡ u(OExp(ϕ̂)), i.e. indicating the dependence of the free parameters on certain observable
data.
While the general techniques are similar on the level of Z and Γ, the interpretation, in
particular of approximation schemes, will differ severely. As long as the theories are treated on
an exact level, we have the chain of equivalences on which the entire effective action approach
is based on:
µ(•;Φ̄|{u}) ⇐⇒ Z[J;Φ̄;{u}] ⇐⇒ W [J;Φ̄;{u}] ⇐⇒ Γ[ϕ ;Φ̄;{u}]
Likewise the expectation values given by their respective functional derivatives contain the same
information, though in different formats. However, the exact level is far from being reached in
almost any interesting QFT and hence a generation of physicists derived a number of different
approximation techniques that resolve certain obstruction, however by paying the price of loos-
ing certain information. Depending for which purpose they are made, these methods either work
on the level of the generating functional or the effective action. It is crucial to understand the dif-
ferences between those schemes whenever interpretation or validity issues are concerned. For
instance, employing perturbation methods on Z[J;Φ̄;{u}] in which one modifies the measure
while field space is untouched, may correspond to small perturbations around a fixed expecta-
tion field Φ = exp (ϕ) in Γ[ϕ ;Φ̄;{u}]. Similarly, small fluctuations around Φ on the side of
Φ̄
the effective action may still belong to a full integration over [F], however evaluated only over a
very small class of different measures instead of all possible ones. We will see that theories con-
verging to a free field theory can be usually well described within perturbation approaches of
Z[J;Φ̄;{u}], while otherwise additional tools are in need, sometimes more appropriately stated
for the ‘dual’ object Γ[ϕ ;Φ̄;{u}].
The necessity of approximation schemes is seen as soon as one leaves the formal descrip-
tion of QFT and works on explicit calculations to deduce observable quantities of Nature. Apart
from the Gaußian measure there are only very rare and exotic cases for which the functional
integral is understood and analytically solved. In particular the generating functional, i.e. the
114 Chapter 2. Quantum Field Theory
Fourier transform of the measure, is in most discussions only a formal object. Thus, over the
years a full landscape of different approximation techniques have emerged which partially re-
veal the hidden aspects of possibly fundamental theo(ries. Almo)st exclusively, exponential types
of probability densities are studied, ρ(Φ̂;~u) ≡ exp −S[Φ̂;~u] , whereby the exponent usually
mirrors the classical action functional. Thus, one sometimes refers to it as the path integral
quantization of classical theories. However, in general the relation will differ and only in a cer-
tain limit the classical theory is recovered by its quantum version, whereby S denotes the bare
action. The reconstruction of S by observable data is a very difficult task and involves the steps
described in when discussing the ‘inductive’ approach.
In what follows, we address perturbation theory in a very broad sense, where the probability
density is approximated in a particular manner to allow for a partial integration. This almost
exclusively corresponds to reducing the theory to a Gaußian integral, for its distinguished status
in functional integration. Hence, perturbation theory deals with a sequence of measures that
ultimately should converge towards the true theory:
−a−lg−or−it−h→m −a−lg−or−it−h→m · · · −a−lg−or−ithmµ(0) µ(1) −→ µ (2.24)
The idea is to start with a measure µ(0) that is as close as possible to the true theory µ while
being as simple as possible, meaning close to Gaußian. In certain settings, both requirements
are mutually exclusive since∫the theory turns out to be highly non-trivial. So, let us assume
we are given a theory µ = dµG ρ(Φ̂) then its crudest approximation is setting ρ(Φ̂) = 1,
which corresponds to the associated free theory. In this case we can compute in principal any
correlation function and observable, though the results may have little to do with the actual
theory µ . Nevertheless, there might be regions on field space U0 ∈ ΣF where ρ(Φ̂) = 1 is
approximately fulfilled, thus
∫ ∫
µ(U0) = dµG ρ(Φ̂)≈ dµG ≡ µG(U0) if ρ(Φ̂)≈ 1 ∀ Φ̂ ∈U0
U0 U0
In the above sequence (2.24) the measure µ(0)(E ∈ ΣF)≡ µ(E ∩U0) may correspond to a start-
ing point of the perturbative algorithm.9 One then extends U0 to a larger region U0 ⊂U1 ∈ ΣF.
However, in order to be reliably describing µ one also has to adapt the probability density, by
expanding ρ(Φ̂)≈ 1+ f (Φ̂)+ · · · , for instance in a Taylor series w.r.t. the fields. One arrives at
∫
µ(1)(E)≡ dµG · (1+ f (Φ̂))≡ µG(E)+ 〈 f (Φ̂)〉
∩ µG(E)E U1
This algorithm can be pursued until the computational abilities are reached or the theory seems
to converge. The latter is a quite difficult and rarely appearing case, where the fundamental
theory turns out to be almost Gaußian, i.e. free. However, depending on the explicit expansion
scheme and the initial region U0 one may have a well defined and proper description of the
theory under certain conditions where the above approximation is valid. For instance, expanding
in momentum modes may show that the theory is close to trivial in the ultraviolet (UV) while on
large scale structures it is highly non-Gaußian. In the following we present some examples of
approximation schemes: the method of steepest descent, and perturbation in small parameters
or couplings.
R There is a very important difference concerning the perturbation technique and the Renor-
malization Group approach, though both define a sequence of measures by means of a
filtration on the σ -algebra. The former is restricted to a region in field space that is well-
described by a free theory, while the latter ultimately extends to the full space and rather
replaces the functional integral by a systematic sequence on the space of measures.
9In fact E ∩U0 is an element of ΣF by the properties of σ -algebras.
2.3 Quantum field theories 115
Method of steepest descent
While equally in their pattern, the three approximation techniques differ in their interpretation
and, depending on the theory µ , in their range of validity. The method of steepest descent
(saddle point approximation) searches for a particular convenient region U0 in field space in
which one expands the probability density around its maximum dvρ(Φ̂max) = 0 for all v:
(2)
ρ(Φ̂) = ρ(Φ̂max)+d ρ(Φ̂ )+O((Φ̂− Φ̂ )3)(Φ̂−Φ̂ max maxmax)
Notice that while the first term is a constant under the functional integral, the linear term van-
ishes due to the maximum condition and what remains are corrections quadratic in the deviation
of the maximum position. Hence, for this approximation to work, we choose Φ̂max ∈U0 and
restrict the integration measure such that O((Φ̂− Φ̂ 3max) ) are subordinate:
∫ ∫
(2)
µ(E) = dµG(Φ̂)ρ(Φ̂)≈ ρ(Φ̂max) dµG(Φ̂)d − ρ(Φ̂max)+O((Φ̂− Φ̂max)
3)
Φ̂∈E Φ̂∈E (Φ̂ Φ̂max)
The more pronounced the maximum of ρ , or likewise the minimum of S, is, the better the range
of possible validity.
Parameter expansion
A similar technique involves a small parameter that factors the additional term in an exponential
type probability density, say ρ(ϕ̂;Φ̄) = exp(−λV [ϕ̂ ;Φ̄]). As long as V [ϕ̂;Φ̄] is comparably
small w.r.t. λ the generating functional can be expanded in powers of λ :
∫ ( ) ∞ ℓ
Z[J;Φ̄] = dµG(ϕ̂;Φ̄) exp −λV [ϕ̂;Φ̄]+ J(ϕ̂) = ∑ (−1) ℓ ℓℓ! λ 〈V [ϕ̂;Φ̄] 〉µG
TΦ̄F ℓ=0
Notice that the expectation values are computed for the Gaußian measure. This method is
appropriate if λ V [ϕ̂ ;Φ̄]≪ 1 and thus the series converges rapidly enough. If this condition is
only fulfilled for some subset of F the range of validity has to be reduced to this region.
Expansion in h̄
We should mention another technique that not directly relates to the above approximation
schemes but involves a constant of Nature. Here, one reestablishes Planck’s constant in the
description and expands the theory in powers of this constant h̄. Assume we are given a proba-
bility density of exponential type, then the Schwinger functional in units of h̄ is given by
∫ ( )
exp(− 1W [J;Φ̄])≡ d[ϕ̂ ] exp − 1S[ϕ̂;Φ̄]+ J(ϕ̂)
h̄ h̄
TΦ̄F
Converting this measure using the Legendre transform and the expression for the effective action
one obtains ∫ ( )
exp(− 1Γ[ϕ ;Φ̄])≡ d[ϕ̂ ] exp − 1S[ϕ̂ ;Φ̄]+d
h̄ h̄ (ϕ̂−ϕ)Γ[ϕ ]
TΦ̄F
Finally, one expands the effective action in a series in h̄, i.e. Γ[ϕ ;Φ̄] = S[ϕ ;Φ̄]+∑∞ ℓℓ=1 h̄ Γℓ[ϕ ;Φ̄].
On the left-hand-side (LHS) this is straightforward, however on the RHS we have to compute
the expectation value of the reduced density, which may need further simplifications. Now,
comparison of the coefficients for each order in h̄ results in
Γ[ϕ ;Φ̄] = S[ϕ ;Φ̄]+ h̄2TrlnS
(2)[ϕ ;Φ̄]+O(h̄2)
Notice that compared to the other methods, this technique is more concerned with the effective
action and thus the expectation fields.
R Under certain circumstances this method can be referred to as loop expansion, though in
general they do not agree [63, 64].
116 Chapter 2. Quantum Field Theory
Perturbative renormalizability and Regularization
The problem with the above approximation techniques descends from the subtlety to split a full
measure into parts and evaluate possibly undefined quantities. From a mathematical point these
systematic expansions are difficult to control and in general on each intermediate step infinities
may appear.10 Hence, all the prescribed perturbative methods have to be understood as an
asymptotic expansion which might not converge to the full theory, but nevertheless provides
insight into its structure and properties.
The failure of perturbation methods or of a naive choice for the fundamental theory which
results in divergences is usually cured by introducing a regulator scheme, that spots the defi-
ciency of the description and cuts off the problematic terms in a suitable manner. In a sense it
remedies the artificially introduced divergences by modifying the perturbation method such that
potential infinities are replaced by a finite cutoff Λ. In the limit of removing the regulator Λ the
divergences reappear. The advantage of this procedure is control over the problematic terms and
a finite description on each step of the evaluation. While the regulator is constructed to elimi-
nate infinities by means of a finite upper or lower bound on intermediate steps, one assumes that
for the full theory this regulator is superfluous and in fact only an indication of either a flawed
evaluation technique or an insufficient initial theory. Though general results may depend on Λ,
it is desirable that predictions of the regulated theory do not depend on the particular regulariza-
tion scheme employed. Hence, regularization should be understood as a mathematical trick to
compensate for impurities of the applied (perturbation) technique, for instance by means of a
suitable adaption of the range of integration. As such it should be understood to be mandatory
to regularize the theory in each step of the perturbative expansion.
Once, the theory is regularized and cured from infinities, one has to bridge the mathematical
theory with physical observables and thereby fix the free parameters. This procedure is known
as normalization and provides the actual testing ground for theories, which are so far only
mathematical objects based on some (physically motivated) symmetry principle.
R Exactly this point, where theory meets experiments and mathematical objects, like expec-
tation values, are related to observable quantities, describes the appearance of physical
units, or scales, in the mathematical formalism. Experiments are comparative processes,
where some quantity is measured with respect to another. Remarkably, there are only
seven basis units defined in the International System of Units, hence there are (theoreti-
cally) at most seven comparativemeasurements necessary to fix a quantity uniquely. How-
ever, most of the basis units are in fact related by some constant of Nature and thus are
not independent.
In the perturbative approach normalization actually corresponds to a (re-)normalization11 al-
gorithm, as can be understood as follows. Let us take a family of measures that fulfill some
symmetry based constraint and thus contains a number of free parameters that have to be spec-
ified by experimental input, hence Z[J;{u}]. Since this quantity and its expectation values are
in general not computable, one relies on a perturbative treatment and has the sequences of gen-
erating functionals, or likewise measures, now suitably regularized by a regulator Λ:
Λ algorithm Λ algorithmZ(0)[J;{u}] −−−−−→ Z(1)[J;{u}] −−−−−→ ·· · −
a−lg−or−it−h→m Z[J;{u}] (2.25)
At each level we consider a different class of theories and thus a different set of expectation
values and observable predictions. If we want to match these theories to experiments in order
10Even a well defined probability theory µ may suffer from those infinities when some approximation scheme is
applied to evaluate its correlation functions.
11We write (re-)normalization in this context to distinguish the concept from the one appearing in the Renormal-
ization Group approach.
2.3 Quantum field theories 117
to reduce the number of free parameters, we have to compute specific correlation functions and
evaluate them for certain physical conditions that match the experimental setting:
∞ ∣
! 〈 〉 ≡ (ℓ) (ℓ) ∣OExp = O µΛ ;{u} ∑ αO ·dJ ZΛ [J;Φ̄;{u}]∣ (2.26)( j) Exp ( j)
ℓ=0 J=0
For, say, n free parameters in the theory we need at least n such equations and therefore n
experiments. Then, in principle we can solve for the unknowns and fix the theory completely, at
least if n is finite. This normalization of the theory is a physical process that actually addresses
Z[J;{u}]. Applied to some approximate form of Z[J;{u}], say of level j, we denote the set of
parameters that solves n equations of the type (2.26) by {u( j)(OExp;Λ)}. At level j, the unique
(approximated) measure is thus given by
Λ (r) !Z( j)[J;{u (r)( j)(OExp;Λ)}] with OExp = 〈O 〉µΛ ;{u} r ∈ 1, · · · ,n( j)
whereby now, there is no remaining free parameter. Remarkably, the matching of finite physical
values usually results in absorption of the cutoff dependence into the normalization procedure.
If we perform the same steps on the next level of the perturbative expansion, the theory,
thus the expectation values, and therefore the solutions {u( j+1)(OExp;Λ)} will differ in gen-
eral from {u( j)(OExp;Λ)}. This explains the notion of (re-)normalization since the previous
normalized, or physically matched theory ZΛ( j)[J;{u( j)(OExp;Λ)}] gets replaced by a new one
ZΛ( +1)[J;{u( j+1)(OExp;Λ)}] with a (re-)normalized set of parameters. The crucial requirementj
one imposes on a perturbatively renormalizable theory is that the number of free parameters
converges to a finite value, thus at a certain order in the expansion one expects that no new
terms become relevant but rather that all effects can be absorbed in the already present coef-
ficients. This corresponds to studying a general family of measures and (re-)normalizing the
parameters only:
ZΛ
a
[J;{u(0)}]−−
lg−or−it−h→m Λ { algorithm algorithmZ [J; u(1)}]−−−−−→ ·· · −−−−−→ Z[J;{uExp}]
For instance, the initial theory may start with all {u} vanishing and then during the process
one switches on the contributions from certain invariants and observes their influence on the
results.
Definition 2.3.1 — Perturbative (re-)normalizability. Let (F,ΣF) be a measurable space
and µ(•;{u}) be a family of probability measures. Then we say that µ(•;{u}) is perturba-
tively renormalizable w.r.t. the algorithm A :⇔
◮ The sequence ({u(ℓ)(OExp;Λ)})ℓ∈N0 converges to {uExp} within the validity of the per-
turbative method A. Hereby, {u(ℓ)(OExp;Λ)} is defined by
ZΛ[J;{u(0)}]−−−
A−→ ZΛ A[J;{u(1)}]−−−−→ ·· · −−
A−→ Z[J;{uExp}]
◮ The number of free parameters in the theory µ(•;{u}) converges to a finite value, i.e.
lim #{u(ℓ)(OExp;Λ)} = n< ∞
ℓ→∞
While the first criterion ensures that the chosen perturbation technique defines a valid approx-
imation for µ , the second refers to the predictivity of the theory, stating that perturbatively
(re-)normalizable theories only need a finite number of experiments to be uniquely determined.
118 Chapter 2. Quantum Field Theory
R Notice that even though a theory may be found to be perturbatively non-(re-)normalizable
it is not in general doomed to failure. Perturbative non-(re-)normalizability can be an
artifact of the applied perturbation technique which might be incompatible with the un-
derlying theory. We will encounter a generalization of this perturbative concept in the
context of the Renormalization Group.
2.3.4 Euclidean quantum field theory
Einstein’s theory of Special Relativity is based on a Minkowski spacetime and since all fun-
damental interactions are nowadays constructed over these Lorentzian spacetimes the general
requirement of Lorentz invariance is at the roots of QFT. Furthermore, Einstein’s later gen-
eralization to General Relativity locally embeds the concepts of special relativity and thus the
classical theory of gravitation also relies on a pseudo-Riemannian geometry. However, in the
functional integral approach to study field theories the non-compactness of the Lorentz group
is found to be a problematic aspect of Lorentzian theories that leads to some very undesirable
difficulties in a mathematical treatment of the Feynman integral.
A remedy to this obstruction, at least on a technical level, was introduced by Schwinger us-
ing analytical continuation to complex time. This procedure is known as Wick rotation tL → ıtE
whereby the vacuum correlation functions get replaced by their Euclidean version that usu-
ally fulfill better controlled differential equations.12 On the level of functional integrals, this
amounts to converting the Feynman-Kac integral by a Wiener (probability) measure, for in-
stance, and the ordinary Fourier transform z= ı is substituted by its Laplace counterpart z=−1.
If the density of the measure was previously given by ρ = eıSL it assumes a non-oscillating
form ρ −SE = e in the Euclidean framework and thus one has to ensure that the action functional
is bounded below. The great advantage of this method is founded in the invariance require-
ments of spacetimes, where analytical continuation replaces the non-compact Lorentz group by
its compact Euclidean counterpart. This basically explains the interest in Euclidean Quantum
Field theory (EQFT), in particular for a better controlled mathematical treatment of functional
integrations [28].
In the fifties Wightman proposed a set of axioms which translate the general idea into a
rigorous prescription [65]. Some twenty years later the Osterwalder-Schrader axioms presented
a catalog of criteria that specify the condition under which an EQFT describes a well-behaved
Lorentzian theory [28, 50, 51]. For a given probability theory µ its generating functional Z[J]
is required to fulfill the following axioms:
A. Analyticity The generating functional Z[J] has to be analytic.
This ensures analytic properties of the Minkowski theory.
B. Regularity The generating functional Z[J] and its correlation functions have to be regular.
This severely restricts emergence of local singularities in the Minkowski case
C. Euclidean invariance Let GE be the group of Euclidean transformations. Then one re-
quires the theory to be invariant under GE, i.e. Z[σE(gE,J)] = Z[J] for all gE ∈ GE.
This corresponds to invariance under the Lorentz group.
D. Reflection positivity Let Θ− denote the time reflection operation sending t to Θ−(t) =−t.
Then, the Euclidean theory should be positive w.r.t. time reflection, i.e.
M .jℓ .= Z[J j−Θ−(Jℓ)] for which Mφ = λφ =⇒ λ ≥ 0
Thus M is a positive matrix having only non-negative eigenvalues.
This ensures positivity of the inner product in the Hilbert space the Minkowski theory acts
on.
12 If one replace the time coordinate (at least locally) with a complex version τ = t+ ıT a compactification of the
complex direction leads to a thermodynamical interpretation of the description, where T represents the temperature
of the system.
2.3 Quantum field theories 119
E. Ergodicity Denote Θs the operation of time translations. Then the ergodicity condition
has to be satisfied:
∫ tE
lim 1 dsΘsO(ϕ̂(x))Θ→ −s
≡ 〈O(ϕ̂)〉 ∀O ∈ Obs
tE ∞
tE µ0
Thus, time averaging of expectation values corresponds to averaging over field space.
This corresponds to the uniqueness of the vacuum in the Minkowski case.
Hence, the above requirements imposed on the Euclidean theory ensure the validity of the
Wightman axioms in its Minkowski counterpart [28].
While this prescription is well suited for flat spacetimes, it already appears that for general
curved geometries modifications are in need. In particular the reflection positivity and ergodicity
make explicit use of a (global) time-like Killing vector field which does not exist in the general
case. Still the mathematical formalism of functional integration can be promoted to general
spacetimes; the Wightman- and Osterwalder-Schrader axioms need a generalization in terms of
a covariant formalism that describes the continuation from Euclidean to Lorentzian theories in
this more general setting. We briefly comment on this issue in chapter 3 and refer to ref.[26, 66,
67] for more details.

3.1 General Relativity
3.2 General aspects of Quantum Gravity
Amotivation for theories of Quantum Gravity
Field space of Riemannian metrics
Background Independence
Boundaries in Quantum Gravity
Euclidean Quantum Gravity
3.3 Theories of Quantum Gravity
String theory
Loop Quantum Gravity
Causal Dynamical Triangulation
Experimental verification
3. QUANTUM GRAVITY
On the path towards a deep understanding of Nature physicists are confronted with two theoreti-
cal concepts, the Standard Model of particle physics and General Relativity, each outstanding in
its predictivity and precision. In quest of a unified theory that intertwine the quantum nature of
matter with the geometrical construction of spacetime, the search still continues. Decades of re-
search have brought deep insights into both concepts, have revealed their deficiencies and their
mathematical foundations. Many physicists nowadays believe that a solid theory of quantum
gravity may cure most, or even all, of the so far encountered issues. Promoting GR to a quan-
tum level is a very sophisticated task, since most of the methods available in the construction of
previous quantum field theories turned out to be inapplicable.
One of the reasons may be that, considered as a gauge theory, the set of mediating gauge
fields defines a non-trivial field space with non-vanishing curvature, hence the simple vector
space identification has to be replaced by the geometric background field method. Thereby,
Background Independence, the requirement that the formulation should be independent on any
– possibly prescribed – background geometry, becomes central to all constructions of quantum
theories of gravity [39, 68, 69].
In addition, it turned out that from the perspective of conventional perturbative approaches
quantum gravity seems to be non-renormalizable [70]. While GR provides an excellent descrip-
tion of classical gravitational physics it seems to be only an effective description on our energy
scales. The task is thus to extend and generalize the classical theory to fit into the framework of
quantum field theory while recovering the GR limit at sufficient low energy scales. Due to the
lack of experimental results, a large variety of approaches towards a quantum theory of gravity
are pursued by different communities all sharing the same ultimate goal of a unified language
(not necessarily theory) in which Nature can be described.
In this chapter, we are going to present the general setting of GR in a nutshell along with
the necessary implementations for spacetimes with non-vanishing boundary. Next, we take a
look at the reasons why most physicists believe that finding a solid theory of quantum gravity is
the most important task in theoretical physics. Besides of aesthetic arguments, there are some
severe problems in having non-unified descriptions for the SM and gravity, most of them known
122 Chapter 3. Quantum Gravity
since many decades and steadily at the top of the list of unsolved problems in physics. After
briefly comment on some general aspects of Quantum Gravity and the importance of boundaries
in cosmology, we briefly present an incomplete summary of some very prominent approaches
to a possible fundamental theory of gravity. For more details on this subject and a thorough
introduction to the various approaches we refer to [26, 39, 68, 69, 71–76].
In the next chapter, we will put the scene for studying the effective average action approach
to QG based on the FRGE. This in particular requires a characterization of field space and the
underlying symmetry principle, whereby we restrict to the metric setting, the field content we
considering later on in a particular bi-metric truncation.
3.1 General Relativity
Gravitation is one of the fundamental interactions of Nature that we experience very pronounced
in our every-day life. Considered a bit closer its actual influence seems to be restricted to verti-
cal accelerations and cosmic phenomena, while most of the effects we perceive with our senses
are mainly due to electromagnetism. The conception of the world before the discovery of elec-
tromagnetism was based on Newton’s laws of mechanics, where there was a general setting
describing changes of momenta with respect to four axioms, introducing the concept of forces.
This construction was highly successful in explaining our macroscopic world. however it re-
mained an effective description which was already apparent due to the number of different
forces which did not derive from first principle, but rather were derived purely from observa-
tions. This theoretical framework has survived generations of physicists and is still capable of
explaining most of the phenomena around us.
But, the moment scientists left the region of perception and dug deeper into the structure
of matter, new unseen features of Nature emerged which did not fit into the general description
of Newton. And in fact, the explanation of light, a particular expression of this newly discov-
ered interaction, had already undergone many conceptual changes over the centuries before it
was apparently consistently described within Maxwell’s equations. What this story and many
others tells us, is that the ignorance of either experimentally or theoretically perceiving Nature
sets natural limitations on the range of validity for our current model of the Universe. Even
though we have a very powerful and extremely predictive theory with the Standard Model of
particle physics, we can only be sure about its validity on the energy scales we have tested it.
The same holds true for GR, which has so far explained all deviations of observations from
classical mechanics, however we can only be sure about its precision within the scales accessi-
ble by todays experiments. Hence, observations are an extremely valuable treasure that helps in
understanding the world. Whenever experimental data is unavailable, it is important to check
the inner consistency of theories while at the same time one has to invent new techniques to
make significant observations accessible.
The history of gravitation has experienced phases of stagnation and also sudden progress,
as for instance with the invention of better telescopes that opened the cosmos to a deeper search
for the fundamental nature of the Universe. This interplay between experiments and theories
will once answer the question , concerning the gravitational interaction. From the experimental
perspective the current (classical) theory of gravity, i.e. General Relativity, is remarkably pre-
cises in its predictions and explanations and we rely on them whenever we make use of a GPS
device, say. On the other hand, there are theoretical inconsistencies that are difficult to cure
within the theory itself and there are also aesthetic mathematical indications that there might be
a theory beyond GR. However, we postpone this subject to the next section and first consider
the famous theory of GR.
In 1905, Albert Einstein started to revolutionize the concept of space and time by combin-
ing them into a unified geometrical object, denoted spacetime. His theory of SR challenged
3.1 General Relativity 123
Newton’s perspective of the nature of the Universe and with this the classical conception of the
world. By changing the assumption on the symmetry properties of space and time had remark-
able consequences which however became only relevant when velocities close to the upper limit,
given by the speed of light, are reached. In the spirit of this conceptually new picture of space-
time, Einstein worked for ten years on translating Newton’s law of gravity into the language
of SR. In 1915, after ten years of effort, Einstein finally proposed a new theory of gravitation,
General Relativity [77], that in fact provided also a generalization of SR on global scales, while
retaining its local validity. In Einstein’s understanding, the geometry of spacetime is best de-
scribed by a (pseudo-)Riemannian manifold (M,g) and the gravitational interaction is related
to the corresponding Levi-Civita connection. A property of (pseudo-)Riemannian manifolds is
the existence of normal coordinates in which the metric becomes flat and thus for the respective
signature of g Special Relativity is recovered in the local vicinity of every point on M. The de-
viation from g being completely flat, is encapsulated in the Riemann curvature tensor, and thus
on global scales one may encounter non-trivial effects which are not described by SR. This is a
mathematical consequence of Einstein’s equivalence principle that loosely speaking equals the
trajectory of an observer in a homogeneous gravitational field with one under uniform accelera-
tion. The geometric formulation of the gravitational interaction is described by Einstein’s field
equation, which related the curvature of spacetime’s geometry to interaction with matter:
Gµν +gµνΛ = 8πG T µνN (3.1)
( )
The LHS contains the gravitational contribution, where Gµν ≡ Rµν − 1gµν2 R denotes the Ein-
stein tensor. On the RHS the energy momentum tensor T µν is coupled via Newton’s constant
GN to the geometry of spacetime and thus gives rise to an interaction of matter fields and the
metric g.
Equation (3.1) defines a non-linear differential equation with T µν representing a source
term and Λ a free parameter, the cosmological constant, which in cosmology is related to the
vacuum energy of the Universe. Thus, solutions to Einstein’s field equation for some matter
content T µν are not closed under superposition since linearity is missing.
A very important consequence of the geometrical description of field equations is a require-
ment imposed on the energy momentum tensor T µν . By definition Gµν is symmetric and by
virtue of the Bianchi identities it is also divergenceless, i.e. DµGµν = 0. Hence, only for
symmetric source terms with vanishing divergence, i.e. T µν = T ν µ and D T µνµ , there exist
solutions to equation (3.1). (Notice that gµνΛ is symmetric and by metric compatibility of D
also divergenceless.) A particular well studied class of solutions are those for vacuum field
content, where T µν = 0 vanishes. For instance in four dimensions vacuum solutions with self-
dual curvature tensor define the class of instantons. Another prominent example is given by the
Schwarzschild metric, which – according to Birkhoff’s theorem – describes the most general
vacuum solution with spherically symmetry and Λ = 0. It has a single free parameter, which
sometimes is referred to as the mass of a black hole situated at the origin.
As on cosmological scales all other fundamental interactions have compensated, what re-
mains is the comparatively weak gravitational ‘force’. Thus, the greatest impact of GR is found
in the depth of our cosmos, in explaining the movement of the planets in our solar system,
gravitational lensing, large scale structures or in short a deep insight into the structure of the
Universe. Constructing a cosmological model that reflects the homogeneity and isotropy prop-
erties of space observed in studying the night sky, the statement of the cosmological principle, it
turns out that these requirements result in the famous Friedmann–Lemaître–Robertson–Walker
(FLRW) metric that includes a time dependent scale factor a(t) for the space foliations:
( )
gFLRW =−dt⊗dt+a(t)2 (1− k · r2)−1dr⊗dr+ r2dΩ2 (3.2)
124 Chapter 3. Quantum Gravity
Here dΩ2 = dθ ⊗ dθ + sin2(θ)dφ ⊗ dφ denotes the spherical part of the metric. Furthermore,
the constant k ∈ {−1,0,+1} represents the topological class of negative, zero, or positive cur-
vature, respectively. In order for gFLRW to be a solution to Einstein’s field equation a suitable
energy momentum tensor must exhibit the same symmetry properties in the spatial part. For
the Friedmann equations one assume matter to be described by a perfect fluid for which T µν
assumes the form:
T = (ρ(t)+ p(t))dt⊗dt+ p(t)g with p(ρ) = w ·ρ
The additional relation between the density ρ and the pressure p, given by p(ρ) = w ·ρ , is a
special case of an equation of state. Depending on the ‘fluids’ properties and main constituents
w will assume different values during the phases of evolution of our Universe. Finally, one can
show that gFLRW defines an analytic solution for Einstein’s field equation, the so-called Friedmann
equations, which describe the time evolution of the scale factor a(t) with respect to the sources
ρ(t) and p(t):
( )
ρ̇(t) =− ȧ(t)3 · (ρ(t)+ p(t)) (3.3a)
( ) a(t)
ä(t)
=− 43πGN (ρ(t)+3p(t))+ 13Λ (3.3b)a(t)
Most of the cosmological considerations are derived from these equations. Matching the free
functions and parameters to observations, the Friedmann metric gives rise to an space-expansion
and probably (re-)collapse of the Universe and it predicts a time-like singularity, the Big Bang.
With this application of Einstein’s theory of GR to cosmology, it plays a central role in theoret-
ical physics also in the beginning of the twenty-first century.
3.2 General aspects of Quantum Gravity
3.2.1 A motivation for theories of Quantum Gravity
The form in which Albert Einstein presented his theory of GR describes the field equations for
a (pseudo-)Riemannian metric that corresponds to the geometry of spacetime and encodes the
gravitational interaction in a non-trivial curvature. Though, extremely successful in its tested
ranges, there are several attempts to reformulate GR in terms of different field content, more
general topological settings, and find a more fundamental principle from which Einstein’s field
equation derive naturally. All of these approaches have to cope with the tremendous success of
GR and should at least reproduce all those results in a particular limit.
Examples of the non-metric field content is the Einstein-Cartan theory, where the gravi-
tational connection is more generally described by a principal connection on a frame bundle
where torsion does not have to vanish and compatibility with a Riemannian metric is only a
special case. Changes of the topology involve a discretization of spacetime and thus will only
yield GR in an effective limit. The third motivation is rather concerned with a general prescrip-
tion that explains the origin of Einstein’s field equations from a more fundamental principle. In
fact there are various theories, which may combine two or even all of these three reformulation
attempts, which are close enough to compete with the experimental predictivity and validity of
GR . Besides this, some approaches even cure some consistency problems intrinsically present
in General Relativity.
Before motivating a theory of Quantum Gravity let us consider a very prominent example to
reformulate GR in terms of a variational principle of an action functional, the Einstein-Hilbert
action here stated for d∫-dimensions:
1 √
SEH[g] = d
dx |g| (R−2Λ) (3.4)
16πGN M
3.2 General aspects of Quantum Gravity 125
∫ √
If one adds a general action functional for the matter content S dMat[φ ;g] = d x |g|LMat the
variational principle w.r.t. the metric assumes the form:
dδg (SEH[g]+S
µν µν µν
Mat[φ ;g]) = 0 =⇒ G +g Λ = 8πGNT (3.5)
√ √
Hereby, T µνhµν = −2 | |
−1
g dh( |g|LMat) directly derives from the matter Lagrangian. This
relation between the Einstein-Hilbert action and GR was proposed by David Hilbert already in
1915, and provides a more elegant way to relate gravity with other matter interaction. How-
ever, generalizing GR usually comes along with new issues. In the present case the variational
principle requires M to be a compact manifold without boundary. Any deviation from these
additional assumptions will spoil the derivation of Einstein’s field equation. In particular for the
important class of compact spacetimes with a non-empty boundary, the variation of the Einstein-
Hilbert action generates boundary terms for both Dirichlet or Neumann field constraints. The
cure to this problem in case of Dirichlet boundary conditions was introduced by York, Gibbons
and Hawking [78, 79], who suggest to add another compensating boundary term to the action
functional, the Gibbons-Hawking-York term:
∫
1
S d−
√
1
GHY[g] = d x |H| K (3.6)
8πGN ∂M
The appearing geometrical objects describe the properties of the embedded boundary ∂M. Usu-
ally, one starts with an embedding hyperplane equation, Σ(x) = 0 whenever x ∈ ∂M. The
derivative of this hyperplane describes deviations from the boundary and points in the normal
direction w.r.t. Σ. It defines the outward pointing normal vector field nµ = ∂ µΣ(x) that can
be used to project tensor fields at x ∈ M tangent to the boundary. One especially important
2-tensor field is the tangent metric Hµν = gµν −nµnν which can be translated into the induced
metric defined on the manifold ∂M. Finally, the trace of the extrinsic curvature Kµν = Dµnν
is the source of the compensating derivatives SGHY[g] that is needed in order to compensate the
obstructing boundary terms in the variational procedure of the Einstein-Hilbert action. It turns
out that there is a fixed ratio for the coefficients of both terms that leads to a full cancellation of
all boundary contributions, namely:
∫ √ ∫ √
ddx |g| R+2 · dd−1x |H| K
M ∂M
The additional factor of 2 in front of the Gibbons-Hawking-York term is mandatory in the
classical theory.
As we have seen, General Relativity combines the geometry of spacetime with theories
of matter and thus is a central piece in the formulation of Nature in terms of fundamental
interactions. If GR is really part of the fundamental theories there are very severe constraints
and requirements it has to pass. This involves consistency, as well as aesthetic conditions in
order to match the quantum field character of the other ‘forces’. While from an experimental
perspective, there is no evidence which suggests a breakdown or an effective nature of GR,
from a theoretical point of view there are quite a number of indications that we are only dealing
with an extremely successful effective description. Therefore, let us consider Einstein’s field
equation again:
Gµν︸ +︷︷g
µνΛ︸ ︸ ︷=︷ ︸ ︸8πG
µν
︷N︷T ︸ (3.7)
geometry relation QFT
There are three main statements in this equality. First of all, the geometric structure of space-
time is fully described by a suitable metric and a constant parameter Λ, which constitutes the
126 Chapter 3. Quantum Gravity
LHS. Second, the matter content, which in our today’s understanding is formulated in terms
of a QFT represent the RHS and is factor by a constant GN. Finally, we have an equality
which establishes a link between spacetime’s geometry and the fundamental matter interaction.
Intertwining different theories has important consequences, in particular if singularities or in-
consistencies appear, since they ultimately affect both sides of the equation.
In our current understanding of Nature, the quantum character of the electroweak and strong
interaction has undergone many experimental tests, so far outstandingly successful, in particular
quantum electrodynamics. Nowadays, the Standard Model of particle physics seems to be fully
confirmed within its range of validity, having found probably all predicted particles and reveal-
ing an extremely accurate coincidence of theory and experiment. This however describes a first,
more conceptual, problem within equation (3.7). While the RHS seems to be best written in
the language of QFT, the LHS is purely classical with no occurrence of h̄, indicating that those
contributions have to be taken care of in matter-matter interactions only. This is partly accept-
able since gravity is special among the fundamental interactions in that it is interwoven with the
geometry of spacetime, but still from an aesthetic point of view and the strong restrictions on
the quantum nature of matter suggest to question this classical to quantum correspondence.
If we further take a look at a list of unsolved problems in physics we will see that many of
the stated issues address the equality of eq. (3.7) where in a general or cosmological setting a
theory of matter is connected to the curvature on spacetime. For instance, the hierarchy problem
addresses the coupling constant GN on the RHS of eq. (3.7). It defines the strength with which
the gravitational interaction effects the matter theory and vice versa. Compared to the other
fundamental interactions, gravity is by magnitudes weaker and only becomes relevant when
all other effects have smeared out, as for instance on cosmological scales. Another interesting
consequence of the equality sign concerns the cosmological constant Λ. There seems to be a
clash between observational data setting an upper limit of |Λ| < 10−52m2 with the theoretical
prediction one obtains when interpreting Λ as the quantum vacuum energy of some QFT in the
matter sector. The discrepancy is such huge that there seems to be no way to justify the severe
fine-tuning of Λ on the basis of the matter content.
Probably the strongest indication that at least some part of eq. (3.7) needs to be improved is
the problem of dark matter and energy. Based on some cosmological model that derives from
Einstein’s field equations and involved techniques to chart the Universe one predicts the matter
distribution in galaxies and beyond. These results however seem to be in sharp contrast with
observations, for instance the flattening of the velocity for galaxy rotation curves which requires
quite an amount of matter at ‘dark’ regions, meaning that the gravitational objects necessary to
account for these effects have to interact very weakly, if at all, with respect to the electromag-
netism, thus photons. From the field content of the Standard Model of particle physics there
is no appropriate candidate that could be in charge of almost 27% of the energy content of the
Universe, while the visible ordinary matter constitutes only about 5%. The missing energy 68%
is also unexplained by the standard concepts and is denoted dark energy, which only interacts
gravitationally. Introducing dark energy seemed to be unavoidable to match the cosmological
observations of an almost flat Universe with an accelerated expansion, with theoretical predic-
tions based on the observed (electromagnetically interacting) energy distribution. Again, one
part of Einstein’s field equations has to be modified in order to account for around 95% of the
gravitational effects. On explanation is given by fields which are not yet described within the
SM. This would suggest a refinement of the RHS of eq. (3.7), i.e. the matter content of the
theory, for instance by introducing new particles. On the other hand, the geometrical side of
Einstein’s field equations could be only an effective description to which further terms should
be added or even for which a completely different formulation is needed. This shows some very
profound issues in the relation of QFT and GR on cosmological scales.
3.2 General aspects of Quantum Gravity 127
Finally, let us focus our attention on the geometrical part of Einstein’s field equation and
consider the properties of solutions for some very general matter sector. It turns out that under
very natural conditions GR will be plagued by singularities, in that geodesic completeness is
lost and thus GR predicts its own breakdown. This was first studied by Penrose and Hawking
who gave certain conditions under which space- or time-like singularities form in GR [80–
82]. Basically, one imposes a set of (causal) conditions on the global properties of spacetime,
assumes the existence of a trapped region where gravity is such strong that light rays can not
escape, and requires some energy condition to be satisfied by the matter content. For instance
the weak energy condition which states that light rays converge caused by gravitational effects,
(in other words the energy of matter is non-negative) is some very natural condition that seems
to be respected during the entire evolution of our Universe, even in a possible inflationary phase.
Combined with some natural causal requirements of spacetime the interior of a black hole for
example will contain a singularity. Thus, rather than some exotic case in which GR experiences
its limitations, singularities appear under very mild and natural conditions indicating that for
strong gravitational effects modifications might become relevant.
Hence, there is some inconsistency within the current form of Einstein’s field equations,
either hidden in our understanding of the QFT character of matter, the geometrical description
of spacetime or in relating both. In any way cosmology and even structural aspects suggest
to improve Einstein’s field equations in one or the other direction. Furthermore, mathematical
aesthetics argue in favor of searching for a quantum theory of gravity to translate this interaction
into the language of the others. Though it might well be possible that we have to further work
on the matter content, one should keep in mind that gravity is only studied in a small window of
scales, where all other interactions have smeared out. It would be a remarkable coincidence that
Einstein found a theory that could be extended to higher and lower energy scales without any
modifications just by observing a small fraction of scales an extrapolate the full landscape of
gravitational physics. It is similar to considering a segment of a function’s graph and guess its
outer shape with almost no guiding principle. So far, we have not even seen which fundamental
phenomena arise from gravitational interactions but only tested and deduced rules from a small
accessible spectrum. The great virtue of gravity is that although high energy physics is still
out of reach, it is possible to similarly test its fundamental structure encoded in the large scale
structures of the Universe and find some further advice if GR is really fundamental or only
effective in nature.
3.2.2 Field space of Riemannian metrics
One of the difficulties in studying quantum theories of gravity in the conventional way, is that the
set of all gravitational fields, denoted field space F, has a non-trivial structure when considered
as a manifold. In this subsection, we briefly comment on this important aspect of Quantum
Gravity which is responsible for some issues encountered in ‘quantizing’ General Relativity.
For further details on the space of Riemannian metrics, we refer to [83] and references therein.
The most conservative assumption on the field content of gravity is the set of Riemannian
metrics for a compact, oriented smooth manifold M. Therefore, consider the spaces of smooth
tensor fields over M endowed with the Whitney C∞-topology, whereby only the second order
covariant tensor fields T(0,2)M≡Γ (T∗M⊗T∗M) will be relevant in what follows. SinceM has
a well-defined orientation, we consider the group action of Gl+(d;R) onT(0,2)M and decompose
this space in the respective irreducible components, i.e. into an anti-symmetric and symmetric
part. The latter is part of the definition of a metric tensor field and in fact defines a smooth
vector space which is globally flat from a geometrical point of view. In order to obtain a (pseudo-
)Riemannian metric that defines an inner product on each tangent space ofM, we have to impose
an additional constraint that selects only the non-degenerate symmetric tensor fields. Thus, for
128 Chapter 3. Quantum Gravity
all p ∈M, we define the following operator:
(Bp(g)) (V,W )≡ (1−|sgn(det(gp))|)+ |gp(V,W )−gp(W,V )| ∀V,W ∈ TpM
The kernel of Bp(g) gives rise to the space of Riemannian metrics, whereby the second term is
redundant if we already restrict to symmetric tensor fields only. The crucial ingredient in this
constraint is the first term that requires a non-vanishing determinant and thus a non-degenerate
bilinear form:
Riem(M) ..= {g ∈ T(0,2)M |g ∈ ker(B)} (3.8)
This space of Riemannian metrics onM is constructed by a highly non-linear operator constraint,
and thus we expect that Riem(M) does not retain the global vector space properties of T(0,2)M
or of its symmetric subspace. In fact, it turns out that Riem(M) defines an open (convex) cone
within the space of symmetric 2-tensor fields Γ (S2T∗M). Though it is in general non-compact,
this subregion is path connected and in case of Euclidean signature it is also simply-connected
[83, 84].
For a given Riemannian metric, one can construct an inner product on T(0,2)M which for
elements of Riem(M) becomes positive definite:
( ) ∫ [ ]
t .
√
1 , t2 .= d
dx g Trg t1 t
T
2 ∀ t1, t (0,2)2 ∈ T Mg
M
In combination with the Fréchet space property of Riem(M) we can endow the space of Rie-
mannian metrics with a Riemannian structure on its own, in particular the above inner product
defines a natural Riemannian metric G on Riem(M). The main difference to ordinary field
spaces is the non-trivial structure of (Riem(M),G) that gives rise to non-trivial geodesics and
the necessity of the background field method. Furthermore, field space exhibits very interesting
aspects related to the Levi-Civita connection associated to G which in the present case results
in a non-vanishing field space curvature. The interesting geometry of metric field space is one
reason why theories of gravitation are far more involved than standard QFTs one encounters
usually.
R Notice that metric fluctuations live in the tangent space to some background field ḡ which
is isomorphic to Γ (S2T∗M), since Riem(M) is an open subset.
Remarkably, in the Euclidean case the exponential map is a diffeomorphism and thus a single
background field ḡ suffices to cover the full field space [84].
The physical field content, also denoted the space of geometries, corresponds to the base
space of Riem(M) considered as a principal bundle with structure group Diff+(M), the orienta-
tion preserving diffeomorphisms onM. Basically, one defines the group action of Diff+(M) on
Riem(M) by pullback and establishes an isomorphism between the space of vector fields and
the Lie algebra of Diff+(M) using the Lie derivative. Then, the space of geometries Riem(M)′/
Diff+(M) satisfies a number of differential topological properties, in particular there exists an
induced Riemannian structure with an induced metric.1 These smooth embedding of the space
of geometries into Riem(M) is guaranteed by Ebin’s slice theorem [40].
There are many subtleties in studying Riem(M), most of them are due to the infinite di-
mensionality of this space.2 Thus, usually, compactness of M is required to ensure a kind of
finiteness and all so far stated results are presented under this particular assumption [83].
1Here, ′Riem(M) denotes the space of Riemannian metrics that admit no non-trivial isometries.
2Notice however that the space of Einstein metrics in Riem(M)/Diff(M) is only finite dimensional.
3.2 General aspects of Quantum Gravity 129
Fortunately, there are extensions to pseudo-Riemannian metrics with compact support and
most important in the current context to compact manifolds with boundary [85]. In this latter
case one has to impose boundary conditions on the fields and the diffeomorphism group, which
in ref. [85] where chosen to be of Dirichlet type. A similar slicing theorem was recovered and
the above concepts could be extended to spacetimes with non-vanishing boundary.
3.2.3 Background Independence
Since in Quantum Gravity it is the geometry of spacetime itself which becomes dynamically,
there is no a priori fixed background in which the field content evolves. Missing the arena a
standard QFT description is placed in, already very basic constructions, as for instance integra-
tion over spacetime points, becomes conceptually difficult. There are several techniques how
to implement this requirement of Background Independence into a consistent and still practical
formalism. There are basically two directions one can pursue at this point:
The first possibility is literally abandon the idea of introducing any background at all and
let spacetime emerge by itself. The advantage of this approach is the intrinsic implementation
of Background Independence. However, at the same time one has to leave the realm of standard
QFT which makes heavily use of a predefined geometry.
In the second case, one actually introduces a background field, ḡ, and applies the standard
techniques, treating the metric fluctuations in the same way as matter fields on a rigid geometry.
If field space allows for a smooth structure, this corresponds to the background field method
[68]. However, in the end, one has to ensure that the explicit form of ḡ does not affect the
obtained results, but in fact can be understood as a mere technical trick to recover the standard
QFT picture on intermediate steps. This describes both, the advantage and disadvantage of
the second approach, where we have all the standard tools of QFT at our disposal, but have to
put effort in re-establishing Background Independence, in this sense requiring that physics is
literally independent on the background ḡ.
3.2.4 Boundaries in Quantum Gravity
Boundaries play a very important role in physical problems, for they represent regions, usually
in spacetime, for which we have a certain amount of information at our disposal. A boundary, B
is essentially a collection of points that defines an interior and possibly an exterior component,
which in general describe conceptually different phases of the problem.
If these points can be combined in a smooth fashion that gives rise to a manifold these kind
of problems are particular interesting in studying differential equations. So, let us assume we
can find an embedding of the boundary, B, in the Riemannian manifold (M,g)where g denotes a
metric tensor field. Usually, this is based on a smooth hypersurface condition Σ(x), with x ∈M,
a straightforward generalization of plane equations.
In order to geometrically classify the embedding defined by ker(Σ)≃ B, it is convenient to
introduce the normal vector field (in our convention assumed to point outwards):
nµ(y)≡ ∂ µΣ(y) with y ∈ ker(Σ)≃ B (3.9)
Moving along the normal vector implies leaving the hypersurface, while in the orthogonal di-
rection to n one explores Σ in more detail. This yields to an decomposition into tangent and
normal vector fields, the former described by the tangential metric Hµν ≡ ḡµν −nµnν which is
simply the projection of the Riemannian metric g onto Σ.
A measure of the deformation Σ experiences within M is given by the extrinsic curvature
tensor, that determines the rate of change in n:
Kµν = Dµnν and K = D̄ µµn (3.10)
130 Chapter 3. Quantum Gravity
Here Dµ denotes the covariant derivative w.r.t. g. As its name suggests, Kµν defines the cur-
vature or bending of the hypersurface as embedded into M. It has to be carefully distinguished
from the intrinsic geodesic structure of B, the intrinsic curvature associated to the Riemann
tensor, Rµνρσ , that is independent of any embedding. From the Gauß-Codazzi equation both
concepts of curvature can be related, for instance as follows
( ) ( )
dR|Σ = d−1R+ K2−KµνKµν +2Dν nµD nνµ −nνD̄µnµ (3.11)
Thus, the four-dimensional geometry of M, given by R, when evaluated on Σ, decomposes into
the intrinsic and extrinsic curvature of Σ and a further total derivative. Thus, without knowing
the topological structure of B, we cannot say which part of the curvature is due to the embedding
and which part is intrinsically present in B.
This geometrical picture of hypersurfaces and thus general notions of boundaries, allows to
study initial value problems in any (smooth) branch of physics. In the usual cases where the
underlying space(time) is flat, R = 0, and B is a subset of the Euclidean space with a trivial
embedding, the curvature of Σ can be completely associated to a non-vanishing Kµν . We then
recover the generic sets of differential equations that have to be evaluated for a specific field
configuration, or boundary condition. Thereby the boundary B or its embedding Σ has usually a
physically significance. For instance B may be associated to the contact layer of two materials
with distinct properties, to be explicit a conductor and an isolator. Or it may reflect the symmetry
property of the configuration, e.g. spheres in an isotropic problem. In any case, the boundary
describes a ‘region of knowledge’, meaning that we have a suitable amount of information on
B which restricts the set of solutions of some (differential) equation, usually in a unique way.
For simplicity let us start with some differential equation ∂ 2x f (x) = J(x), which allows for,
let say, an infinite number of solutions fsol ∈ Sol. If this equation is related to a physical prob-
lem, we have to implement further constraints, either by experimental observation or additional
theoretical requirements, in order to obtain a fully predictive description, i.e. #(Solphys.) = 1.
Depending on the explicit setting, we thus define a hypersurface Σ on M where we have access
to further information, for instance we know that it represents the surface of a conductor, and
then evaluate the class of mathematical solutions fsol ∈ Sol on Σ to see if they satisfy the addi-
tional constraint. If Sol ≡ 0/ , the system is overdetermined and either the underlying physical
law does not described Nature, or there is some inconsistency in the experimental results.
Thus, boundaries are not at all restricted to describe spatial edges of spacetime, but in fact
occur in almost any problem based on differential equations, as an example. It represents a
way to implement physical configurations or observations into the mathematical framework by
feeding the system with an additional amount of data. The imposed boundary conditions can
assume any form, however the most familiar ones are of Dirichlet and Neumann type, where
either the solution itself or its derivative is fixed on Σ. Apart from local generalizations, so called
mixed conditions, there are also non-local forms, where the integrated value of the solution
is fixed, for instance the total electric charge on a surface. In the realm of QFT the most
popular example of boundary conditions is the Casimir effect, where two conduction planes are
‘exposed’ to the vacuum fluctuations of the electromagnetic field and thereby attract each other.
In the context of this thesis, the most interesting applications are found in cosmology. There,
boundaries usually occur either as real spatial regions in our four dimensional world, or as a
Cauchy hypersurface, which describes an ‘instant of time’. A vividly discussed proposal of
Hartle and Hawking [86] suggests to study a kind of quantum ground state of the Universe,
Ψ0(H), which is a sum over all spatially closed geometries, Riem(M)sp.cl. described by metrics
g that give rise to the induced (spatial) metric H on some Cauchy surface Σt :i
∫
Ψ0(H) ∝ d[g] exp(−SE[g]) (3.12)
g∈Riem(M)sp.cl.∧g|Σt ≡Hi
3.3 Theories of Quantum Gravity 131
This describes a boundary value problem related to the Wheeler-DeWitt equation [86] which
then should yield the full evolution of the Universe.
Another important example is found in the correspondence of the statistical mechanics of
black holes and its thermodynamic description, relating the entropy to the area of the horizon.
The latter represents a natural separation between two different ‘phases’, the inner and outer
region and their corresponding sets of solutions.
Studying boundaries is thus a very central aspects in understanding mathematical equations
for physical applications. A theory of Quantum Gravity that incorporates the possibility to treat
those generalized spacetime geometries is thus an essential tool to describe quantum effects in
cosmology and study the evolution and history of our Universe. For further reading on the value
of boundaries in physics we refer to [26].
3.2.5 Euclidean Quantum Gravity
Finally, let us briefly comment on the Euclidean formalization of Quantum Gravity. For flat
spacetime, there is a general framework how to transit from the Lorentzian picture to its Eu-
clidean analog and vice versa. The Wick rotation tL = ıtE, of an Euclidean theory will result in a
proper QFT if the Osterwalder-Schrader axioms are satisfied. The advantage of the Euclidean
formulation in favor of the Lorentzian description is found in the properties of the respective
symmetry group, which is only compact in the former case. In particular from the mathemati-
cal perspective, the Euclidean construction is thus much better under control, in particular the
study of boundary value problems for elliptic operators becomes accessible. The problem is
that Nature seems to be based on a Lorentzian geometry and thus we have to convert the Eu-
clidean results back to spacetimes of pseudo-Riemannian structure. While as mentioned above
this prescription is well understood on flat space, in Quantum Gravity we are dealing with a
generic metric g for which there will be no global time-like Killing vector field. This however,
is crucial for the standard Wick rotation and thus the relation between both description in the
general case is still subject of discussion [26].
3.3 Theories of Quantum Gravity
One remedy to cure the problems intrinsically present in Einstein’s field equation is to reconcile
quantum physics with General Relativity by means of a quantum theory of gravity. There are
standard techniques to quantize a classical theory, which turn out to be inappropriate for gravity.
The obstruction can be traced back to the (re-)normalization requirements one imposes on suit-
able QFT, which on a perturbative level is incompatible with a quantum version of GR. A sim-
ple power-counting argument demonstrates that the theory is perturbative non-(re-)normalizable
due to the negative mass dimension (2−d) of Newton’s coupling GN. In the traditional sense a
QFT should require only a finite number of counterterms that remove divergent contributions in
loop integrals, which corresponds to only a finite number of free parameters that have to be fixed
by experiments. In case of gravity, this is perturbative procedure fails, for in each higher loop
correction supplementary counterterms are needed to compensate divergences. Hence, while
towards higher energy scales the number of free parameters approaches infinity, the predictive
power of the quantum theory tends to zero [70, 87, 88].
After accepting that the standard procedure to link a classical theory with its quantum ana-
log is not applicable for GR, there are several ways to pursue to address the problems describe
in the previous section. First, one may focus on the matter sector instead and retain General Rel-
ativity in its classical shape. Despite of this, one can also think about modifications of GR by
introducing higher order curvature terms, for instance, and then apply the standard quantization
schemes. It turns out that in certain cases indeed perturbative (re-)normalizability is reestab-
132 Chapter 3. Quantum Gravity
lished however with the price of loosing unitarity. The great virtue and also drawback of the
classical theory is the diversity of different mathematical formulations and physical approaches
that are consistent with the experimental results and somehow are closely related to GR. There-
fore, one has various opportunities from which to construct a theory of quantum gravity and if
one direction fails, one may try a different route. In the following we give a incomplete list of
approaches that all share the same objective, namely reconciling the laws of quantum physics
with the gravitational interaction, while the applied methods differ sometimes quite significantly.
Thus, we encounter concepts which stay very close to the standard procedure while modifying
the classical field content severally, or in contrast constructions which question the quantization
scheme while being conservative with the general setting of the physical content. Besides the
ultimate goal to interconnect the light and small scales of Nature described by quantum physics
with its long and heavy scales covered by GR, all these approaches have in common that they
give rise to a new perspective on the whole of physics, new mathematical techniques, and inter-
pretation of the geometric structure of spacetime. In particular objects that combine the heavy
and small features, as for instance black holes, are probably the best candidates to test and ulti-
mately distinguish between the various directions in order to bring scientists back on the right
track.
3.3.1 String theory
The basic idea of string theory is that the elementary particles that appear in the SM can be
divided one step further, revealing a one-dimensional string structure rather than a simple zero-
dimensional point [89–92]. The necessary resolution to its discovery would involve detections
on the level of Planck length scales above which the standard picture of SM is recovered. The
vibration of the strings then correspond to the different particles which also includes the me-
diator of gravity, the graviton. String theory turns out to be a very involved formalism that
needs highly advanced mathematical concepts, which has led to fruitful contributions in both
mathematics and physics in general. In particular the discovery of duality within string theory,
most famous the AdS/CFT correspondence, has brought new aspects and insights by relating
seemingly different theories by means of a duality transformation that ensures certain aspects
being equivalently described in one or the other form. However, very often the mathematical
level is also a drawback since most of the necessary techniques are still under development.
For instance, the scattering of strings is so far based on perturbative methods and the reliability
of those results has to be confirmed by a non-perturbative treatment. Besides the replacement
of point particles, by closed or open strings the theoretical setting is based on supersymmetry,
implementing fermions and bosons in a symmetric way. In addition, in order to have a mass-
less photon, for instance, spacetime has to have at least 11 dimensions, where the additional,
unobserved dimensions are either compactified on very small scales (loosely speaking rolled
up) or thought of as being quite big whereby our familiar four dimensional spacetime is only a
(boundary) slice, the so-called braneworld theory. These somehow exotic supplementary ingre-
dients and the small predictivity power due to the emergence of a huge number of equally likely
universes are the most severe difficulties string theory has to deal with.
3.3.2 Loop Quantum Gravity
We have already mentioned that there is an alternative formulation of GR in terms of non-metric
field content, that even allows for torsion and a more general connection: the Einstein-Cartan
theory. Thereby, the Einstein-Hilbert action gets replaced by the Palatini-Holst functional, i.e.
∫ ∫
1 1
S [e,ω ]≡ ε e j ∧ ekP-H jkℓn ∧ (Ωω)ℓn− e j ∧ ek∧ (Ωω) jk
32πGN 16πγ
3.3 Theories of Quantum Gravity 133
Here, e represents the tetrad, ω the connection, Ωω = dω +ω ∧ω the curvature 2-form and γ
the Immirzi parameter factoring the Holst action part of SP-H[e,ω ]. Based on this description of
the classical theory of gravity, Loop Quantum Gravity (LQG) applies a canonical quantization
using Ashtekar variables which basically puts quantizing matter and spacetime on an equal foot-
ing [72, 93]. Therefore, one uses a Arnowitt-Deser-Misner (ADM) foliation splitting the four
spacetime into three dimensional space slices with induced metric Hab and a time-like direction
including a shift and lapse function ~N and N, respectively. The evolution of the space foliations
is then described by commutation relations between the (densitized) triads of Hab and a SU(2)
related spin connection A. Before the discovery of the SU(2) gauge group within the ADM
formalism by Ashtekar there was little to no progress in this field due to the very difficult nature
of the Hamiltonian constraints. The introduction of the Ashtekar variables lead to a significant
simplification and thus brought LQG back to the game of describing a theory of gravity. The
great advantage of this approach is its Background Independence and its non-perturbative na-
ture. Following LQG space can be understood as some spin network that evolves with time
or in a covariant formalism, as a spin foam that exhibits a porous structure of spacetime. The
challenge of LQG is to recover the low energy (effective) theory from its fundamental prin-
ciples. While in general this is still a very difficult task, for cosmological models involving
homogeneous and isotropic space constraints there are predictions based on LQG calculations
which for instance change the picture of the big bang to a big bounce and thereby removing the
corresponding singularity.
3.3.3 Causal Dynamical Triangulation
A background independent approach denoted Causal Dynamical Triangulations (CDT) attempts
to derive the structure of spacetime from first principle [94–98]. Basically, spacetime is mod-
eled by a set of discrete causal simplices which describe a local patch of flatness while (causally)
gluing them together yields a non-trivial global geometry. One initially replaces the formal path
integral for GR with a lattice regularized version involving the Regge-action and a sum over all
causal triangulations.
∫
CD
d[g] exp(ıSEH[g])−−−−−−T −→ Z[{κ0,κ4,∆}]≡ ∑ exp(−SR[τ ;{κ0,κ4,∆}])
τcausal
Hereby, the causality requirement significantly restricts the number of possible triangulations.
Furthermore the global proper time foliation is used to perform a Wick rotation thus that the
exponent is a real valued action functional, consisting of:
SR[τ ;{κ0,κ4,∆}] =−(κ0+6∆)N0(τ)+κ ( (4,1)4 N4 (τ)+
(3,2)
N4 (τ))+∆(2
(4,1)
N4 (τ)+
(3,2)
N4 (τ))
The coupling constants κ0, κ4, and ∆ correspond to 1/GN, Λ/GN, and an indicator for an asym-
metry of time- and space-like link lengths, respectively. Furthermore, N0(τ) denotes the total
(•)
number of vertices in the triangulation, while N4 represents the number of (•) simplices within
τ . The function Z[{κ0,κ4,∆}] for arbitrary dimensions d is then computed using Monte Carlo
simulations, which reveals a phase diagram with three different phases. Remarkably a classical
four-dimensional de Sitter universe is discovered, which was not implemented by the algorithm
itself but emerged from the causal structure of triangulations. Future Monte-Carlo simulations
can hopefully extend this region to a second order phase transition at which a suitable contin-
uum limit can be performed. Conceptually, CDT has a certain overlap with LQG and even
qualitatively with the Effective Average Action (EAA) approach we study in this thesis, for the
latter see [99, 100].
In the next chapter we present framework to construct a theory of quantum gravity that
is based on the Effective Average Action approach employing the Functional Renormalization
134 Chapter 3. Quantum Gravity
Group Equation to study the Asymptotic Safety conjecture for Quantum Gravity. The following
two parts of this thesis are devoted to an in depth study of a metric based calculation within
the Asymptotic Safety program. At the same time the derivation of the Renormalization Group
evolution demonstrates quite nicely where the technical difficulties reside in studying theories of
quantum gravity and which part a conceptually universal and thus applicable in various branches
of physics.
3.3.4 Experimental verification
Before we conclude our incomplete list by stating the current results of the Asymptotic Safety
program to QG in the next chapter, let us shortly comment on experimental search for physics
beyond classical General Relativity, see ref. [101–104] for further details. The diversity and
difference of the various approaches towards a quantum theory of gravity already indicates that
there is a significant lack of experimental data, which would exclude at least a certain number
of theoretical directions. Opposite to many other inventions in theoretical physics, the concept
of Quantum Gravity is motivated more on a technical and mathematical ground then on a failure
on the phenomenological side. Although there is a sign of incompleteness in Einstein’s field
equations, for instance the prediction of dark matter and -energy, it is so far completely unclear
whether or not the discrepancy is due to the geometrical part of the equation. Nevertheless,
one attempt of every quantum versions of GR is the understanding and possible resolution of
the prescribed classical issues. From the theoretical perspective it seems plausible to go beyond
GR and reconcile quantum physics with the geometry of spacetime. However, in absence of any
experimental justification all description are mere hypothesis or speculations. The only guiding
principle that remains is intrinsic consistency and the emergence of GR as an approximation to
the low energy physics.
While the quantum nature of all other interactions contained in the Standard Model of par-
ticle physics have passed several tests on energy scales now up to the ∼ 10TeV scale, the diffi-
culty with gravity is related to the hierarchy problem, namely the comparatively weak coupling
of gravity. In fact, one assumes that quantum correc√tions for the gravita√tional interaction be-
come relevant only at the Planck scale, i.e. ℓPlanck = GN or mPlanck = 1/ GN. If the classical
value of Newton’s constant is inserted one obtains a length scale of around 10−35m and an en-
ergy scale of about 1016TeV. With a current maximum collision energy of 13TeV there is little
hope that on-Earth experiments may answer the puzzle of a quantum nature of gravity within
the next decades, if at all. Nevertheless, a number of quantum theories of gravity change the
classical value of the Planck scale, which seems to be natural since an observational result is
extrapolated over 15 orders of magnitude, an assumption rather unlikely. Hence, there are even
theories that predict measurable effects on Large Hadron Collider (LHC) energy scales which
should be confirmed or excluded in the near future.
Fortunately, the cosmos provides another playground to test gravity, for it contains many
objects that combine heavy and small properties and thus involve large curvature deformation
of spacetime. Furthermore, on cosmological scales, the gravitational interaction is in a sense
isolated from the other interactions, which have smeared out due to their compensating charges.
However, compared to on-Earth experiments, there is no control over the experimental setting
which has to be modeled on some cosmological assumptions and the reproducibility is usually
not given. For instance, supernovae observations or lensing effects due to massive black holes
are good candidates to study the effect of strong spacetime deformations.
The Cosmic Microwave Background (CMB) is an example where large scale structures
may reveal the nature of the smallest quantum fluctuation of gravity. It is a ‘footprint’ of the
very early form of the Universe before a possible inflationary phase sets in and leads to a rapid
expansion. The remarkable observation in the temperature density pattern of the CMB is its
3.3 Theories of Quantum Gravity 135
uniformity, indicating that even the most departed galaxies were once in equilibrium with each
other and thus causally connected. (Inflation was actually introduced to account for this con-
nection by a phase of rapid expansion.) Still this sea of uniformity contained very tiny seeds of
impurities causing the emergence of galaxies and its clusters, which are assumed to represent
the ‘frozen’ quantum fluctuations at the moment when inflation sets in. Thus, deep inside the
CMB we can search for imprints of quantum fluctuations. In addition, we may find answers
about the field content of the classical (effective) theory, by detection of gravitational waves that
requires tensor modes.
These are the most promising candidates to search for experimental evidence of theories of
quantum gravity: either we look for space-like strong curvatures, or we study the physics close
to a possible Big Bang by considering primordial remnants.
R On the other hand, observations that constraint the classical theory will also affect quan-
tum theories for they have to reproduce this effective theory in a certain limit. Thus,
experiments that underpin or disprove the very fundamental requirements of GR have
also some distinguishing properties for the variety of quantum theories of gravity. For in-
stance a violation of Lorentz invariance of the ground state of gravity would imply quite
different quantum generalizations than those discussed usually.

4.1 Renormalization group
Theory space
General aspects of the Renormalization
Group
4.2 Infrared-cutoff implementation
Eigenmode expansion
Rescaling relation between Φ̄ and τ
Smooth cutoff implementation
4.3 The Functional Renormalization Group
Equation
The derivation
Properties of the FRGE
The normalization factor
4.4 Renormalizability & Asymptotic Safety
4.5 Truncations
4. FUNCTIONAL RENORMALIZATION GROUP
The success of the generating functional approach in studying field theories can be mainly at-
tributed to its universal applicability. It is used to describe the Standard Model of particle
physics and at the same time it is suitable to discuss statistical field theories on very different
scales. The abstract and general structure of this approach is at the same time the source of
its severest problems which bother physicists since its arising. Measure theory in combination
with the infinite dimensional field space demands a thorough regularization procedure and the
practical manipulation of observables is usually beyond the scope of today’s mathematics. Thus,
instead of direct computations addressing the full integration, one systematically reduced the
difficulties of its evaluation by approximating the underlying measures in a suitable way. The
hope is that extrapolating the obtained results may lead to predictions far beyond the involved
scales. With those techniques, known under the name of perturbation theory, we have gained
enormous and very fruitful insights into all interactions up to the TeV scale, except for gravity.
While for the Standard model interactions the described approximation methods seem to be ap-
propriate, at least on certain scales, there is a severe conceptual problem in studying a quantum
theory analog of General Relativity. The prominent role gravity plays in current theoretical
physics is also partly due to the insight one gains in revealing the limitations of the standard
descriptions and in the encouraging work to put the current techniques on a solid ground. From
our point of view it is not that the present results suggest gravity to be not renormalizable, but
rather that we have to go beyond the standard, perturbative, techniques and enter the realm of
non-perturbative effects to judge the ultraviolet behavior of theories.
In this chapter we present a brief introduction into a very promising, intrinsically non-
perturbative method related to the block spin transformations of Kadanoff [105] and its gen-
eralization by Wilson [106–108] and Polchinski [109]. Together they lay the ground for a
very powerful, universal, though highly non-trivial functional differential equation, the FRGE,
which – since almost 20 years – is also available for non-perturbative studies of quantum gravity,
with remarkable success.
We focus here on its implementation based on the Effective Average Action (EAA) [110–
114] and refer the reader to the extensive literature on the full variety of this subject. An excel-
138 Chapter 4. Functional Renormalization Group
lent review on the foundations and different shapes of Exact Renormalization Group Equation
can be found in [115], for instance. At the very beginning we introduce the main idea of the
Renormalization Group (RG) and give a definition of RG steps along the lines of the previous
chapters. In the original form, these RG transformations correspond to a sharp cutoff. In order
to obtain a function differential flow equation it has to be replaced by a smooth implementation
on which we focus next. Subsequently, we come to the main part of this chapter and study the
Functional Renormalization Group Equation, aka flow equation. We first establish a relation
between this highly non-trivial differential equation for the EAA and the path integral proce-
dure based on the generating functional. We briefly comment on its properties and present a
short discussion on the flow on probability theories, which involve an additional normalization
factor. To cope with its universal character, we keep the notation in this section very general
and postpone the particular important case of Quantum Gravity to a later section. In addition
we allow for spacetime manifolds with a non-empty boundary. We then address the notion of
renormalizability in the context of the FRGE and a brief summary on the results for Quantum
Gravity. This chapter and the entire first part of this thesis concludes with some remarks on
truncations which we study in detail for the bi-metric-bulk – pure-background-boundary ansatz
in part II.
4.1 Renormalization group
In studying Nature we see that it reveals itself in very different shapes depending on the scales of
observation. While in our everyday life we are mainly confronted with effective electromagnetic
and a bit of gravitational forces, exploring the world towards larger or smaller scales shows fas-
cinating new phenomena, either dominated by gravity or by the joint product of electroweak and
strong interactions, respectively. This scale dependence of the Universe is expressed in terms
of effective descriptions which model part of a more fundamental underlying theory. Like the
narrow window we cover with our sense of sight, the visible part of the electromagnetic spec-
trum, most of our today’s understanding of Nature relies on relatively small fractions of its
full scale structure. Fortunately, we can observe signs of the underlying fundamental interac-
tions by their influence on macroscopic scales and by testing the limits of our current models.
Furthermore, there are also promising indications from the opposite direction, suggesting that
on cosmological scales, the Universe behaves as a very symmetric isotropic and homogeneous
system. Nevertheless, scientists are always searching for a way to extend the boundaries of
knowledge by digging deeper into the structures and foundations. Depending on how stable the
physical laws behave under a change of scale, we will be relative successful in extrapolating our
macroscopic conception to larger and smaller structures. Usually, the range of validity is quite
restricted and we rapidly touch the edge where conceptual new effects become important.
A very successful guiding principle was discovered by the path integral approach that in
certain cases promotes a classical theory to its quantum counterpart and thus translates physics
on macroscopic scales to a microscopic level. However, the technical difficulties have shown
that in order to take account of the scale dependence of Nature there are different methods
necessary that are particularly designed to give rise to a mathematical realizations of the ‘scale-
evolution’ of physical laws. As an example consider Quantum chromodynamics (QCD), the
theory of strong interactions. On a fundamental level, in the ultraviolet, the associated path
integral ZUV[J;Φ̄] is situated in the vicinity of the free theory which makes perturbative studies
perfectly suitable. Decreasing the resolution we recover effective theories which exhibit very
different structures, Zp2 [J;Φ̄], until in the infrared its elementary constituents given by quarks
and gluons are tightly bound by what is known as confinement, Zp2=0[J;Φ̄]. In this way, we
have a set of theories most suitable in certain branches of energy or momentum scales. Though
seemingly very different, they all descent from the same fundamental interaction ZUV[J;Φ̄] and at
4.1 Renormalization group 139
least in principle can be recovered from this theory. This patchwork of effective descriptions can
be made systematically in that we construct a map p2 7→ Zp2 [J;Φ̄] that interpolates between the
UV- and IR-formulation of the underlying interaction. If all functionals happen to be defined
over the same field space and symmetry class, this amounts to a sequence of flow on theory
space to which we turn next.
4.1.1 Theory space
As we described in the chapter on QFT, 2, the necessary ingredients to specify a gauge theory
are given by a concept of spacetime M, a symmetry principles G, and a set of vector spaces
which can be geometrically promoted to a smooth structure F defining field space. Under these
assumptions any theory which is compatible with this physical content can be associated to a
measure µ , or a generating functional Z[J;Φ̄], on the σ -algebra (e.g. Borel sets) of F. For later
convenience suppose we can transfer all information contained in Z[J;Φ̄] to its effective action
Γ[ϕ ;Φ̄] by means of the Legendre-transform. The respective set of all Γ[ϕ ;Φ̄] that are related
to some suitable Z[J;Φ̄] defines what is known as theory space:
T(M,G,F) ..= {A[ϕ ;Φ̄] | invariant under G, Diff(M), and G} (4.1)
A similar concept, requires in addition that the underlying measure µ is probabilistic, implying
in particular µ(F) = 1. If we impose this further constraint theory space reduces to
TP(M,G,F) ..= {A[ϕ ;Φ̄] ∈ T|dvA[ϕ ;Φ̄]|ϕ0 = 0 ⇒ A[ϕ0;Φ̄] = 0} (4.2)
Both spaces are in general infinite dimensional and inherit, at least on finite subsets, a smooth
vector space structure from its target space R. By specifying a basis on T it is thus possible to
distinguish different theories by their coefficients. For a scalar functional, a basis {P[ϕ ;Φ̄]} is
constructed by means of field monomials which are compatible with the symmetry requirements.
If we have a scalar field content, φ , with a trivial ten∫sor s∫tructu∫re, then th∫e non-lo∫cal part of the-
ory space can be expanded in the basis invariants { φ , φ2, φ3, · · · , φD̄2φ , φ2D̄2φ , · · ·}.
Notice that depending on the symmetry group G and the presence of a non-empty boundary
∂M 6= 0/ , this set reduces or extends. However, for general field content, this procedure is far
less intuitive. Due to the absence of a suitable inner product on T iteratively deriving a linear
independent basis {P[ϕ ;Φ̄]} is more elaborated. There is a different method that makes use
of other well studied vector spaces, for instance Gl(d), to explore T. We have briefly men-
tioned a technique based on Young diagrams [46] to systematically classification a set of linear
independent P[ϕ ;Φ̄]’s that can be used to study metric based theory spaces, see section 1.5.
Each allowed action functional A[ϕ ;Φ̄] ∈T(or ∈TP) defines a point in theory space, which
for a specific set of basis invariants {P(α)[ϕ ;Φ̄]} can be, at least formally, expressed in terms
of their associated coefficients {u(α)}, i.e.
A[ϕ ;Φ̄]≃ ∑u(α)P(α)[ϕ ;Φ̄] (4.3)
α
Different action functionals A[ϕ ;Φ̄] and A′[ϕ ;Φ̄] ∈ T give rise to different coordinates {u(α)}
and {u′(α)}, respectively. If the chosen basis {P(α)[ϕ ;Φ̄]} is implicitly understood, we can thus
use the parameter notation A[ϕ ;Φ̄ |{u(α)}] and A[ϕ ;Φ̄ |{u′(α)}] to denote A[ϕ ;Φ̄] and A′[ϕ ;Φ̄],
respectively. In chapter 2 we made heavily use of this fact without explicitly stating the under-
lying class of basis invariants.
R Notice that {u(α)} are combinations of the coupling constants encountered in QFT and
as such they measure the importance of the related field monomial in the overall theory.
140 Chapter 4. Functional Renormalization Group
This formal description of the class of theories allows to study sequences of measures in
a topological and sometimes smooth way. Perturbative methods provide examples of this pre-
scription where a family of theories Z[J;Φ̄ |{u}] is studied by defining a map for its coordinates
{u}. If this amounts to a continuous, parametrized construction it gives rise to a connected path
on T(M,G,F). In fact, at the end of this chapter, we will focus on smooth trajectories on theory
space which solve a functional differential equation. It provides a way to systematically explore
theory space and its (non-perturbative) renormalizable regions.
R Notice, that in the perturbative case, one concentrates on a small (local) region of the
Gaußian measure and then one tries to extrapolate the fate of the underlying sequence to
global scales. Thus one only studies a fraction of all theories.
Finally, let us mention that with the requirements of symmetry and spacetime invariants, we
have severely restricted the class of possible theories, still there remain in general an infinite
number of linearly independent basis invariants. Each of its elements defines a perfectly suit-
able theory from what we have discussed till now. By means of experiments, we have further
constraints which may help identifying the ‘preferred coordinates’ of the theory realized in Na-
ture. With each measurement we can fix a coefficient for some of the field monomials until (in
the finite case) there is no freedom left. If we can impose additional constraints for which only a
finite number of free coefficients remain the theory can be considered as complete or predictive.
In the sequel we study a method that is particular suitable in exploring the structure of theory
space, the Renormalization Group approach and then classify trajectories (evolution of theories)
according to their limiting behavior. In the end, we describe the notion of renormalizability that
generalizes the perturbative concept in a very natural way which in fact gives rise to strong
evidence for a non-perturbatively renormalizable theory of QG.
4.1.2 General aspects of the Renormalization Group
There are different implementation of the Renormalization Group procedure which have in
common that they describe a path in T connecting the fundamental UV theory with its full
effective description. For the block spin transformation and the Wilsonian RG approach, the
idea is to iteratively evaluate a preexisting UV theory by absorbing the microscopic effects up
to a scale τ in an effective, averaged theory and continue this procedure in a subsequent manner
down to the infrared. We will see that introducing the Effective Average Action and computing
its RG evolution using the FRGE actually leads to a prediction of the fundamental UV theory,
related to the bare action.
However, let us first consider the general receipt to construct suitable sequences Zτ [J;Φ̄]
or Γτ [ϕ̂;Φ̄] in a systematic and consistent way. The corresponding path is parametrized by
the dimensionless renormalization group scale, τ ∈ S ⊆ R, which is related to its dimensionful
counterpart k in a canonical way, i.e. τ = k/k0. If we agree on a particular basis in theory space
Tor in its quantum counterpart, T̂∋ Zτ [J;Φ̄], we may express each of its elements by a full set
of coefficients or couplings, i.e.
∫
Z[J;Φ̄ |{u j}]≡ µ(F |J;Φ̄ |{u j}) = dµG(ϕ̂ ;Φ̄)ρ(ϕ̂ ;Φ̄ |{u })eJ(ϕ̂)j
ϕ̂∈Tϕ̂F
Instead of approximating ρ(ϕ̂;Φ̄ |{u j}) by 1+ · · · as in the perturbative case, which usually
leads to undefined intermediate results, we proceed in a different direction and follow a path
that seems more under our control. The basic idea is to establish a flow on the space of theories
by endowing the σ -algebra of field space with an order structure. That is, we define a filtration
4.1 Renormalization group 141
F τ → 0
τ → ∞
Figure 4.1: The RG procedure visualized as a filtration over field space. With decreasing scale
τ we include more fields in the functional integral.
on ΣF that controls the amount of field modes integrated out:
rg : S⊆ R→ Σ . ′F, τ →7 rg(τ) .=Uτ ∈ ΣF with Uτ ′ ⊆Uτ ∀τ > τ ∧ Uinf{S} = F
Loosely speaking, one chooses a sequence of measurable sets {Uτ} that decrease in their size
when τ is increased and for which every set is contained in its precursor, see fig. 4.1. This pro-
cedure defines a ‘coarse-graining’ of the functional integration, we will consider in a moment.
A Renormalization Group step consists of a coarse-graining and a subsequent field rescaling
part. In technical terms, we define the following τ-dependent generating functional:1
∫
Zτ [J;Φ̄ |{u .j}] .= Nτ [Φ̄ |{u j}] dµG(ϕ̂;Φ̄)ρ(ϕ̂ ;Φ̄ |{u J(ϕ̂)j})e (4.4)
ϕ̂∈Uτ
The differences to the ordinary Fourier transform of µ , i.e. Zτ [J;Φ̄ |{u j}], are the coarse-
graining contribution, resulting in the changed integration range,Uτ ⊆TΦ̄F, and the subsequent
rescaling of the fields which is covered by the pre-factor N [Φ̄ |{u }] =.τ j . Nτ [Φ̄] which satisfies
N0[Φ̄] = 1. Notice that in the limit τ → 0 we recover Z[J;Φ̄ |{u j}] since then both RG effects
vanish.2 Likewise, on the level of the theory, the measure µ , we obtain the following sequence
∫ ∫
τ 7→ µτ with µτ (A) = Nτ [Φ̄] dµ(x)χUτ (x) = dµ(x) ∀A ∈ Σ
x∈A x∈A∩Uτ
The measures, µτ and µ are in general not equivalent. In fact, all µτ are absolutely continuous
with respect to the original (positive) theory µ meaning that for all A with µ(A) = 0 it follows
that µτ (A) = 0. If we omit Nτ [Φ̄], then, due to countable additivity of µ , the RG filtration
generates a monotone sequence on Σ[F]:
µ(A)≡ µ0(A)≥ µτ1 ≥ µτ2 ≥ ·· · ≥ µsup{S} with τ1 ≤ τ2 ≤ ·· · ∈ S
These results also translates to the τ-dependent generating functionals. A very important obser-
vation for theories that are restricted to be probabilistic is that Nτ [Φ̄] is a mandatory τ-dependent
normalization factor which ensures that for each τ we have Zτ [0;Φ̄ |{u j}] = 1. In this way
theories at different RG scales can be directly compared. Furthermore, expectation values of
constant observables are unaffected, for instance 〈Φ̄〉µ ≡ Φ̄, while in case of Nτ [Φ̄] ≡ 1 weτ
would obtain 〈Φ̄〉µ ≡ Φ̄Zτ [0;Φ̄ |{u j}] which now becomes τ-dependent. However, in theoriesτ
for which Background Independence is an issue, Nτ [Φ̄] is an additional source of violation,
though vanishing for τ = 0.
At this point it is instructive to notice the difference between perturbative approximations of
µ , the Wilsonian program and the here presented RG technique. While in the former two cases
1In this discussion we omit a possible necessary UV-cutoff Λ.
2In the following we assume inf(S) = 0 without loss of generality.
142 Chapter 4. Functional Renormalization Group
the density ρ [ϕ̂;Φ̄] is deformed, in the present case, except for a global pre-factor, the actual
measure is kept intact but the range of integration is modified. By changing the scale τ we can
thus study the relative importance of particular subsets on field space to the overall theory. In
other words, we can ‘switch on’ certain fields to contribute to the theory and likewise suppress
others from taking part in the interaction.
In summary, we can say that one introduces an order on field space that is described by the
RG scale τ ∈ S. Decreasing τ amounts to interpolating between the fundamental theory and its
IR effective description.
4.2 Infrared-cutoff implementation
The technique introduced by the RG framework is based on a filtration of ΣF that acts as a
partial evaluation of a theory, more precise of its generating functional. In principle, any such
iterative definition of measurable sets {Uτ}τ∈S is admissible, as long as the integration over the
whole field space is recovered in the limit limτ→0Uτ = F. Thus, it is up to us to decide how to
explicitly implement the RG procedure, though the preferred choice are filtrations with a rich
mathematical structure, as for instance smooth and well understood sequences. In addition, it is
convenient to relate these RG steps to some physical relevant object, even though we will see
that in general only the interpretation of the sequence’s limits might be meaningful.
4.2.1 Eigenmode expansion
The most common and instructive approach is based on a decomposition of field space using
a suitable operator TΦ̄ which besides the background fields might also depend on the gauge
fields of the theory. In this case the discussion follows the same lines, with only changing the
operators target and domain space. Non-negative, self-adjoint operators naturally categorize
field space in terms of the real numbers exploiting their eigenspace decomposition:
TΦ̄(ψλ ) = λψλ
The spectral theorem provides a criterion when {ψλ}λ defines a basis for the space of fluctu-
ation fields at Φ̄ and the index theorem classifies the zero-mode eigenspace related to λ ≥ 0,
which we assume to be trivial for simplification. On the basis of this eigenfunction decompo-
sition, the natural candidate to introduce an order on the space of measurable sets, and thus a
notion of RG steps, is given by
rgT : S⊆ R→ ΣF , τ →7 rgT (τ) = span{ψλ |b ·λ ℓ ≥ τ}Φ̄ Φ̄
This seemingly innocent allocation of the RG scale to some field operator can result in strong
correlations between (background) field rescaling and a corresponding rescaling of τ . Since
Uτ is actually defined via the operator TΦ̄ and hence closely tied to the background fields any
transformation of Φ̄ affectsUτ and all derived concepts.
Therefore, let us assume that we perform a field rescaling Φ̄ ≡ c · Φ̄′ with c> 0 that results
in some rescaled operator TΦ̄ = T
nT ′
c·Φ̄′ = c T ′ . This simple (homogeneous) transformation lawΦ̄
for TΦ̄ (which is for instance satisfied for the Laplacian D̄
2) implies that TΦ̄ and T
′ share the
Φ̄′
same eigenvectors, however for different eigenvalues:
T ′Φ̄′(ψ
′
λ ′) = λ
′ψ ′λ ′ T
′ −nT
Φ̄′ ≡ c TΦ̄ , ψ
′
λ ′ ≡ ψ ′λ , λ ≡ c−nT λ
Depending on which operator the RG filtration is based on, we obtain different measurable sets,
which however are not unrelated. In fact, whenever the operator behaves homogeneous under
4.2 Infrared-cutoff implementation 143
(constant) rescaling of the fields, there is actually a simple identity fulfilled for the RG sets:
( )ℓ
rgT ′ (τ) = span{ψ ′ |b · λ ′′ λ ′ ≥ τ}Φ̄ ( )
= span{ψλ |b · c−nT ·
ℓ
λ ≥ τ}
= span{ψλ |b ·λ ℓ ≥ cℓ·nT · τ} ≡ rgT (cℓ·nT · τ) (4.5)Φ̄
Hence, a rescaling of the background fields can be compensated by a simultaneous rescaling of
the RG scale, keeping the eigenspace decomposition unchanged, i.e.
rgT (τ)≡ rg ℓ·nTT (c · τ) (4.6)cΦ̄ Φ̄
If the spectrum is continuous we can in fact use the same field space filtration {Uτ} for all
rescaling-related operators.
This method of using the eigenspace of an operator to obtain a suitable RG implementation
is a very efficient and well studied technique, among which the Fourier expansion w.r.t. −∂ 2 is
a prominent example. Nevertheless, as usual, simplifications have a price, which in this case is
given by a strong relation between τ and the background fields Φ̄. This renders the RG scale
to be field dependent which will explicitly break the background field independence, at least on
intermediate RG scales.
4.2.2 Rescaling relation between Φ̄ and τ
If we formally solve eq. (4.6) for the background field we find
( )−dΦ̄
Φ̄(τ) = c−1 · ℓ·n · · kΦ̄(c T τ) = Φ̄(k0) , with d ..Φ̄ =−(ℓ · −1nT ) , ∀τ =6 0k0
In the final step we have used τ = k/k0. While mathematics on its own is fully dimensionless,
in physics it is convenient to define relative quantities in terms of which all fields and other
mathematical objects are measured. The canonical factorization of the background field struc-
ture, contained in Φ̄(τ0), and its τ-dependence is usually exploited to define τ-independent –
d
but now dimensionful – fields τ Φ̄0 · Φ̄(τ0) This turns out to be convenient since we can separate
the RG-scale dependence of a function from the geometrical information encoded in the fields,
which is the origin of dimensionful coupling constants.
Using the above operator-based sequence allows to express the field space of fluctuations in
terms of eigenfunctions with respect to the operator T . A very natural, non-negative, self-adjoint
operator associated to a physical theory is the one that defines the underlying free theory and
classifies a Gaußian measure. A further advantage in using this special operator is the feature
that at each RG step we have a natural (well-posed) measure theoretical interpretation, since
the restriction of a Gaußian measure to some subspace is again Gaußian and therefore defines a
free theory.
R For the just described reasons, a very extensively used operator is the Laplacian, ∆ =−D̄2
of the background metric, based on a uniquely related Levi-Civita connection:
D̄2 = ḡ−1ḡ (D̄, D̄) with D̄
2 −1
c·ḡ = c · ḡ−1(D̄, D̄) = c−1D̄2ḡ =⇒ nD̄2 =−1
The implementation of the ordering principle is specified by the parameters b and ℓ, where
the latter is conventionally chosen to be ℓ = 1/2 and b is some positive number usually
set to b = +1. For the Laplacian we find nD̄2 = −1 and hence the field dimension of the
(inverse) background metric can be read off to be dΦ̄ = 2 (and dΦ̄−1 =−2).
144 Chapter 4. Functional Renormalization Group
4.2.3 Smooth cutoff implementation
So far, we have a sharp RG implementation in that all eigenfunctions associated to eigenvalues
below a certain scale τ are completely neglected in the functional integral. It is for several
reasons more convenient to have a smooth RG procedure that allows for differentiation w.r.t. τ ,
in particular. The basic idea is to replace the indicator function χUτ (ϕ̂) in the definition of eq.
(4.4) by a smeared version exp(−∆Sτ [ϕ̂;Φ̄]), i.e.
∫ ∫ ∫
µτ (A) = dµ(ϕ̂ ;Φ̄) = dµ(ϕ̂ ;Φ̄)χUτ (ϕ̂)→ dµ(ϕ̂)exp(−∆Sτ [ϕ̂;Φ̄])
ϕ̂∈A∩Uτ ϕ̂∈A ϕ̂∈A
This additional contribution should be designed such that eigenfunctions of the self-adjoint
operator associated to the Gaußian measure, say TΦ̄, are suppressed if they fall below a certain
threshold. Technically, it should describe the following replacement:3
{
→ 0 for b ·λ
ℓ ≫ τ
T (TΦ̄ +Rτ(TΦ̄)) with Rτ(λ ) = (4.7)
τ1/ℓ for b ·λ ℓ ≪ τ
whereby Rτ(λ ) is (anti-)self-adjoint for (anti-commuting) fluctuation fields, w.r.t. the field
space metric GΦ̄. This is naturally provided, if TΦ̄ fulfills these requirements and Rτ [Φ̄] is a
function of this operator only. Furthermore, let us assume Rτ(λ ) to be a monotone function in
both arguments, ensuring that the original idea of a filtration is still valid.
The full structure of the cutoff action derives from the defining requirement (4.7) yielding a
bi-linear functional of the fluctuation fields:
1 ( )
∆Sτ [ϕ̂ ;Φ̄] = GΦ̄ ϕ̂ ,Rτ [Φ̄]ϕ̂ (4.8)2
In principle, the smearing function related to Rτ [Φ̄] is only subject to some asymptotic condi-
tions, to monotonicity and to be (anti-)self-adjoint for (anti-)commuting fields. Nevertheless, it
is convenient to keep its explicit form close to the sharp cutoff such that its derivative ∂τRτ [Φ̄]
is peaked around τ .
From the scaling properties discussed in the previous subsections we impose the following
additional requirement on the cutoff action:
∆Scτ [c
dϕ ϕ ;cdΦ̄Φ̄] = ∆Sτ [ϕ ;Φ̄] (4.9)
The transformation of the fluctuation fields does not change the range of integration since we
assume a vector space, for which holds:
rgT (τ) = span{ψλ |b · (λ )ℓ ≥ τ}= span{ ℓc ·ψλ |b · (λ ) ≥ τ} ≡ c · rgT (τ)Φ̄ Φ̄
As an example, consider the Laplace-Beltrami operator ∆ ≡ −D̄ associated to some back-
ground metric ḡ. Its tensor structure is trivial and relates to the identity map on the respective
space of fluctuations. Under suitable boundary conditions, e.g. of Dirichlet type on some com-
pact space M, this operator is self-adjoint and strongly elliptic giving rise to a non-negative
spectrum with only a finite number of zero-eigenmodes. In this case, τ ∈ R+ is sufficient to
recover the entire field space integration. Let us choose ℓ= 1/2 and b=+1. Then, the Gaußian
part of the functional integration is replaced by
∆ → (∆+Rτ(∆))
3In this discussion we omit the normalization constant for the sake of brevity.
4.3 The Functional Renormalization Group Equation 145
Expanding ϕ̂ in a complete basis of eigenfunctions ψλ of ∆ and using a relatively sharp cutoff
for simplicity, leads to
(∆+Rτ(∆)) ∑ αλ ψλ = ∑ (λ +Rτ(λ ))αλ ψλ
λ∈R+ λ∈R+ ( )
≈ ∑ λ αλ ψλ + ∑ λ + τ2 αλ ψλ
λ>τ λ<τ
= ∆ϕ̂ 2λ>τ + τ ϕ̂λ<τ
This illustrates the low-eigenmode suppression of the smooth cutoff procedure.
Since the tensor structure, here 1, and pre-factors, u, are usually inherited by the associated
operator TΦ̄, it is convenient to introduce the cutoff shape function which contains the essence
of the cutoff-implementation:
Rτ [Φ̄] ..= R
(0)(T ) ·u = τ1/ℓτ Φ̄ 1 R(0)(T /τ1/ℓΦ̄ ) ·u1 (4.10)
Notice that R(0)(T /τ1/ℓΦ̄ ) is invariant under a simultaneous rescaling of τ and Φ̄ in the preferred
ratio.
4.3 The Functional Renormalization Group Equation
In this section we perform a coordinate free derivation of the Functional Renormalization Group
Equation [112, 116–118] from the generating functional under very general conditions. We
extend the derivation by an additional normalization factor, NΛτ [Φ̄], which is τ- and ḡ-dependent.
Due to its dependence of ḡ which, in addition to ∆Sτ [ϕ̂ ;Φ̄] and the gauge fixing sector breaks
the split-symmetry, it is actually a source of possible issues. Nevertheless, since this factor is
quite interesting from a mathematical point of view, we keep it in the derivation. Anyhow, the
traditional result is recovered by NΛτ [Φ̄] = 1. We will discuss its influence at the end of this
section. Furthermore, we will also briefly comment on the general properties of the FRGE,
irrespectively on the choice of NΛτ [Φ̄].
4.3.1 The derivation
In this subsection we establish the link between the generating functional Z[J;Φ̄] and the Func-
tional Renormalization Group Equation, in a coordinate-free version. We initially start by in-
troducing the τ-dependent construction Z[J;Φ̄] by means of a smooth cutoff action ∆Sτ [ϕ̂ ;Φ̄]
which is assume (to be a bilin)ear functional in its first argument. Explicitly it is given by
∆Sτ [ϕ̂;Φ̄] =
1
2GΦ̄ ϕ̂ ,Rτ [Φ̄]ϕ̂ . Along the lines of the previous discussions transfer Z[J;Φ̄]
to a sequence of functionals ZΛτ [J;Φ̄] with a ultraviolet cutoff Λ, yielding
∫ ( )
ZΛ[J;Φ̄] = NΛτ τ [Φ̄] dµ(ϕ̂ ;Φ̄) exp −∆Sτ [ϕ̂ ;Φ̄]+ J(ϕ̂) (4.11)
ϕ̂∈UΛ[Φ̄]
Here, we retained the τ-dependent normalization factor NΛτ [Φ̄] for later convenience. However,
since NΛτ [Φ̄] is independent on J it does not influence the main part of the derivation and the
usual result is recovered by setting Nτ [Φ̄] = const. So, let us write ZΛτ [J;Φ̄] ≡ NΛτ [Φ̄]zΛτ [J;Φ̄]
with NΛ[Φ̄]≡ (zΛ[0;Φ̄])−1τ τ such that in each RG step the generating functional corresponds to
a probability measure, ZΛτ [0;Φ̄] = 1.
The sequence of Schwinger functionalsWΛτ [J;Φ̄] is given by the logarithm of Z
Λ
τ [J;Φ̄] ∈R.
In the above notation,WΛτ [J;Φ̄] assumes the following form:
WΛτ [J;Φ̄]≡ lnZΛτ [J;Φ̄] = lnzΛτ [J;Φ̄]+ lnNΛτ [Φ̄] (4.12)
146 Chapter 4. Functional Renormalization Group
Notice that NΛτ [Φ̄] does not affect the correlation functions, but ensures that for J = 0 the
Schwinger functional vanishes, i.e. WΛτ [0;Φ̄] = 0. Assuming that the underlying theory gives
rise to a continuous differential and convex WΛτ [J;Φ̄], we can transfer all its information into
the Legendre-transform, which in the limit τ → 0 corresponds to the effective action. For its
τ-dependence we obtain
{ }
Γ̃Λ[ϕ ;Φ̄]≡ LFT(WΛ Λτ τ [•;Φ̄])[ϕ ]≡ sup J(ϕ)−Wτ [J;Φ̄] (4.13)
J∈U⋆Λ[Φ̄]
In terms of the normalization factor, this expression can be recast into the following form, mak-
ing again use of NΛτ [Φ̄] being independent on J:
{ }
Γ̃Λτ [ϕ ;Φ̄] = sup J(ϕ)− lnzΛτ [J;Φ̄] − lnNΛτ [Φ̄] (4.14)
J∈U⋆Λ[Φ̄]
Under the above assumptions, the supremum can be evaluated at the global maximum of the
bracket, Jmaxτ ;Λ [ϕ ;Φ̄], which solves
v(ϕ) = dvW
Λ
τ [J;Φ̄] = dv lnz
Λ
τ [J;Φ̄] v ∈ TU⋆Λ[Φ̄] (4.15)
At this point we suppose that we can invert this relation to obtain Jmaxτ ;Λ [ϕ ;Φ̄] and insert this
solution into eq. (4.14) that results in:
Γ̃Λτ [ϕ ;Φ̄] = J
max Λ max
τ ;Λ [ϕ ;Φ̄](ϕ)−Wτ [Jτ ;Λ [ϕ ;Φ̄];Φ̄]
= Jmax Λ maxτ ;Λ [ϕ ;Φ̄](ϕ)− lnzτ [Jτ ;Λ [ϕ ;Φ̄];Φ̄]− lnNΛτ [Φ̄] (4.16)
In the later derivation, it will prove very useful that Jmaxτ ;Λ [ϕ ;Φ̄] is a solution of eq. (4.15). For
simplicity, let us denote Jw ≡ Jmaxτ ;Λ [ϕ ;Φ̄], whereby w absorbs the full set of dependencies. Then
the application of the chain rule w.r.t. the derivative of w results in4
( )
∂ J (ϕ)−WΛ[J ;Φ̄] = ∂ J (ϕ)−∂ WΛw w τ w w w w τ [Jw;Φ̄][ ]
= (∂wJw)(ϕ)−d∂ J WΛτ [J ;Φ̄] + J (∂ ϕ)−∂ WΛw w w w τ [J;Φ̄]|w w J=Jw
= Jw(∂wϕ)−∂wWΛτ [J;Φ̄]|J=Jw
In the last step we made use of eq. (4.15) for v ≡ ∂wJw to cancel the bracket. Notice that the
LHS of eq. (4.17a) corresponds to the derivative of the Legendre transform ∂ ΛwΓ̃τ [ϕ ;Φ̄]. For
our purpose the differentiation of Γ̃Λτ [ϕ ;Φ̄] w.r.t. ϕ and τ are most important. For these special
cases, eq. (4.17a) reads
d Γ̃Λδϕ τ [ϕ ;Φ̄] = J
max
τ ;Λ [ϕ ;Φ̄](δϕ) (4.17a)
τ∂ Λ Λτ Γ̃τ [ϕ ;Φ̄] =−τ∂τWτ [J;Φ̄]|J=Jmaxτ;Λ [ϕ ;Φ̄] (4.17b)
The first expression can be thought of as a τ-dependent effective field equation.
In the sequel, we focus on the second equality, eq. (4.17b), which represents the starting
point for the derivation of the FRGE. If we insert the relation betweenWΛτ [J;Φ̄] and the gener-
ating functional, we can extend this chain of dependencies up to the definition of the underlying
theory, i.e.
τ∂ Γ̃Λ[ϕ ;Φ̄] =−τ∂ WΛτ τ τ τ [J;Φ̄]| ΛJ=Jmax[ϕ ;Φ̄] =−τ∂τ lnzτ [J;Φ̄]| Λτ;Λ J=Jmaxτ;Λ [ϕ ;Φ̄]− τ∂τ lnNτ [Φ̄]
(4.18)
4For simplicity, we use the shorthand notation ∂w to denote both, functional variation (in case of Φ̄ or ϕ) and
ordinary differentiation (if τ or Λ are involved).
4.3 The Functional Renormalization Group Equation 147
Thus, once we know the RG behavior of zΛτ [J;Φ̄] we can directly compute the τ-dependence
of Γ̃Λ[ϕ ;Φ̄].5τ In the remainder of this section, we cast the RHS of eq. (4.18) into a more
convenient, and above all, closed form that gives rise to the definition of the Effective Average
Action (EAA), ΓΛτ [ϕ ;Φ̄].
Therefore, let us consider zΛτ [J;Φ̄] and notice that its entire τ-dependence is due to the cutoff
action ∆Sτ [ϕ̂;Φ̄], a bilinear functional of ϕ̂:
∫ ( )
zΛτ [J;Φ̄] = dµ(ϕ̂ ;Φ̄) exp −∆Sτ [ϕ̂;Φ̄]+ J(ϕ̂) (4.19)
ϕ̂∈UΛ[Φ̄]
Hence, the application of τ∂τ on zΛτ [J;Φ̄] affects only the exponential and corresponds to the
evaluation of a ‘propagator-like’ expectation value −〈τ∂τ∆Sτ [ϕ̂;Φ̄]〉zΛτ [J;Φ̄].
∫ ( )
τ∂ Λτzτ [J;Φ̄] =− dµ(ϕ̂ ;Φ̄) exp −∆Sτ [ϕ̂;Φ̄]+ J(ϕ̂) τ∂τ∆Sτ [ϕ̂;Φ̄]
ϕ̂∫∈UΛ[Φ̄]
1 ( ) ( )
=− dµ(ϕ̂ ;Φ̄) exp −∆Sτ [ϕ̂ ;Φ̄]+ J(ϕ̂) GΦ̄ ϕ̂ , τ∂τRτ [Φ̄]ϕ̂2 ϕ̂∈UΛ[Φ̄]
(4.20)
In the second step, we introduced the defining expres(sion for the cut)off oper(ator. Notice tha)t
Rτ [Φ̄] satisfies a generalized symmetry property GΦ̄ v , τ∂τRτ [Φ̄]w = GΦ̄ w , τ∂τRτ [Φ̄]v ,
which is equivalent toRτ [Φ̄] being (anti-)self-adjoint in case of (anti-)commuting fields.
In order to obtain(a more conveni)ent form of eq. (4.20) it will be useful to rewrite the
expectation value 〈GΦ̄ ϕ̂ , τ∂τRτ [Φ̄]ϕ̂ 〉 Λ in terms of derivatives of J. Therefore, considerzτ [J;Φ̄]
the second variation, associated to the Hessian, of zΛτ [J;Φ̄] w.r.t. the sources v and w.
( )
dvdwz
Λ
τ [J;Φ̄] = 〈v(ϕ̂)w(ϕ̂)〉zΛ[J;Φ̄] = 〈G v∗Φ̄ , ϕ̂ ·w(ϕ̂)〉zΛ( τ ) τ [J;Φ̄]
= 〈G v∗Φ̄ w(ϕ̂) , ϕ̂ 〉zΛτ [J;Φ̄]
Here, we have introduc(ed the )dual vector v
∗ in the canonical way, see Lax-Milgram theorem
in section 1.4, v = G v∗Φ̄ , • , and made use of the bilinearity of GΦ̄ noticing that w(ϕ̂) ∈
C. Recapitulating the definition of the trace we notice that v∗w(ϕ̂) can be interpreted as an
endomorphism (v∗⊗w)ϕ̂ , which due to the properties of w and v is also linear. Hence, if we
choose w= u◦ τ∂τRτ [Φ̄], we obtain an expression which is quite similar to eq. (4.20):
( ∗ )d Λvdu◦τ∂ R zτ [J;Φ̄] = 〈GΦ̄ (v ⊗u◦ τ∂τRτ [Φ̄])ϕ̂ , ϕ̂ 〉τ τ Λ( ∗ )
zτ [J;Φ̄]
= 〈GΦ̄ (v ⊗u)τ∂τRτ [Φ̄](ϕ̂) , ϕ̂ 〉zΛ (4.21)τ [J;Φ̄]
If we symmetrize this relation, we obtain the Hessian of zΛτ [J;Φ̄] that is associated to the Hessian
operator in the following way
[ ] ( [ ] )
Hess zΛJ τ [J;Φ̄] (v,w)≡ 12 (dvdw+d Λvdw)zτ [J;Φ̄]≡GΦ̄ w , ĤessJ zΛτ [J;Φ̄] v (4.22)
In a final step utilize the definition of the trace given in subsection 1.4.2 to get rid of the extra
tensor structure v∗⊗u and make use of the (anti-)self-adjointness of Rτ [Φ̄]:
1 [ [ ] ]
τ∂τz
Λ
τ [J;Φ̄] =− STr Ĥess ΛJ zτ [J;Φ̄] ◦ τ∂τRτ [Φ̄] (4.23)2
5If the τ-dependence of NΛτ [Φ̄] is retained, hence Z
Λ
τ [J;Φ̄] is normalized, notice that N
Λ
τ [Φ̄] = z
Λ
τ [0;Φ̄] provides
the missing information to solve eq. (4.18).
148 Chapter 4. Functional Renormalization Group
Here, for bre(vity we omitted) the necessary field space metric insertions, writing τ∂τRτ [Φ̄]
instead of G • , Rτ [Φ̄]G−1 . Equation (4.23) is a central result in our derivation and we
should pause for a moment and subsume the necessary assumptions we had to impose. First of
all, we required Rτ [Φ̄] to be (anti-)self-adjoint, which compensates the extra minus sign of the
supertrace in case of anti-commuting fields. Considered as a L2-operator it yields to restrictions
on the set of allowed boundary conditions if M 6= 0/ . Similarly, in case of non-empty boundary
the construction of the Hessian operator is usually more involved. Either field space is such
that all surface contribution cancel or one has to rely on more elaborated techniques in which
bulk-boundary terms appear, see ref. e.g. [119, 120].
In eq. (4.23) we have found a closed RG flow equation for zΛτ [J;Φ̄] that in principle can be
inserted on the RHS of (4.18). However, we are more interested in deriving a closed differential
equation for the effective (average) action and therefore going to rewrite the trace in (4.23) in
terms of the Hessian of Γ̃Λτ [ϕ ;Φ̄]. To this end, we first have to relate the Hessian of z
Λ
τ [J;Φ̄]
with the Schwinger functional counterpart. From the chain rule and definition (4.12) we obtain
d d WΛv w τ [J;Φ̄] = d
Λ
vdw lnzτ [J;Φ̄]+d
Λ
vdw lnNτ [Φ̄]
= (zΛτ [J;Φ̄])
−1dvdwz
Λ
τ [J;Φ̄]− (zΛ[J;Φ̄])−2τ d Λwzτ [J;Φ̄]dvzΛτ [J;Φ̄] (4.24)
Notice that NΛτ [Φ̄] does not affect this relation for it is independent of J. For the Hessian, we
have to symmetrize this relation. The first term in eq. (4.24) gives rise to the Hessian of zΛτ [J;Φ̄]
while the second contribution has first to be converted into a more convenient form:
dwW
Λ
τ [J;Φ̄]dvW
Λ
τ [J;Φ̄] = (z
Λ
τ [J;Φ̄])
−2d Λ Λwzτ [J;Φ̄]dvzτ [J;Φ̄]
= 〈w(ϕ̂)〉WΛ[J;Φ̄] 〈v(ϕ̂)〉WΛτ τ [J;Φ̄]
≡ (w⊗ v)(〈ϕ̂〉WΛτ [J;Φ̄] ,〈ϕ̂〉WΛτ [J;Φ̄])
Evaluated under the trace of (4.23), we find
[
− [ ] ] [ [ ] ](zΛ[J;Φ̄]) 1STr Ĥess zΛτ J τ [J;Φ̄] ◦ τ∂τRτ [Φ̄] = STr Ĥess ΛJ Wτ [J;Φ̄] ◦ τ∂τRτ [Φ̄]
( )
+GΦ̄ 〈ϕ̂〉WΛ , Rτ [Φ̄]〈ϕ̂〉τ WΛ (4.25)τ
The second contribution results from converting v to v∗ as above and the self-adjointness of
Rτ [Φ̄]. Notice that this term represents the τ∂τ derivative of the cutoff action evaluated for the
connected expectation values of ϕ̂ . Hence, the flow equation forWΛτ [J;Φ̄] reads
Λ 1
[ [ ] ] ( )
τ∂τWτ [J;Φ̄] =− STr Ĥess ΛJ Wτ [J;Φ̄] ◦ τ∂τRτ [Φ̄] − τ∂τ ∆Sτ [〈ϕ̂〉 ΛΛ ;Φ̄]−N [Φ̄]2 Wτ τ
(4.26)
Here, we used τ∂ WΛ[J;Φ̄] = (zΛ[J;Φ̄])−1τ∂ zΛ[J;Φ̄] to cancel a global factor of (zΛ[J;Φ̄])−1τ τ τ τ τ τ
on both sides and consider the normalization factor.
For the final step, we need to translate the Hessian of the Schwinger functional to its Legen-
dre transform. The derivation of this relation is found in section 2.2, while we here only state
its result
[ ]
Λ ∣∣ [ ]∣ −1ĤessJ Wτ [J;Φ̄] = Ĥessϕ Γ̃Λτ [ϕ ;Φ̄] (4.27)
J=Jmaxτ;Λ [ϕ ;Φ̄]
Here we omit insertions of the(field space me)tric. Keeping in mind that under all traces
τ∂τRτ [Φ̄] actually stands for G • , Rτ [Φ̄]G−1 we see that the final result is free of addi-
tional contributions of GΦ̄. Furthermore notice that 〈ϕ̂〉WΛ[J;Φ̄] |J=Jmax[ϕ ;Φ̄] ≡ ϕ reduces to theτ τ;Λ
expectation value of the fluctuation field, thus the argument of Γ̃Λτ [ϕ ;Φ̄].
4.3 The Functional Renormalization Group Equation 149
From eq. (4.18) and relation (4.27) we obtain a closed flow equation for the Legendre
transform ofWΛτ [J;Φ̄], which reads
[ ]
Λ 1
( [ ])−1 ( )
τ∂τ Γ̃τ [ϕ ;Φ̄] = STr Ĥessϕ Γ̃
Λ Λ
2 τ
[ϕ ;Φ̄] ◦ τ∂τRτ [Φ̄] + τ∂τ ∆Sτ [ϕ ;Φ̄]− lnNτ [Φ̄]
(4.28)
A more convenient form of this expression can be obtained by absorbing the additional τ∂τ
derivative in eq. (4.28) into a new functional which is denoted the Effective Average Action
(EAA) [110–114]:
ΓΛτ [ϕ ;Φ̄]
..= Γ̃Λτ [ϕ ;Φ̄]−∆Sτ [ϕ ;Φ̄] (4.29)
Since the Hessian oper[ator is line]ar in its functional argument and furthermore that for the cutoff
action we have Ĥessϕ ∆Sτ [ϕ ;Φ̄] ≡Rτ [Φ̄] we succeeded in a coordinate-free derivation of the
Functional Renormalization Group Equation (FRGE) for the EAA:
[
1 ( [ ] )− ]1
τ∂τΓτ [ϕ ;Φ̄] = STr Ĥess
Λ
ϕ Γτ [ϕ ;Φ̄] +Rτ [Φ̄] ◦ τ∂τRτ [Φ̄] − lnNτ [Φ̄] (4.30)2
Compared to the standard form found in the literature, we have an additional term lnNΛτ [Φ̄]
that originates from the normalization of ZΛτ [J;Φ̄]. If this factor is set to N
Λ
τ [Φ̄] = 1, as in the
standard derivation, we recover the familiar definition of the FRGE:
[( [ ] )− ]1 1
τ∂τΓτ [ϕ ;Φ̄] = STr Ĥessϕ Γτ [ϕ ;Φ̄] +Rτ [Φ̄] ◦ τ∂τRτ [Φ̄] (4.31)
2
Equation (4.31) was first established by Wetterich in [112] and has found many applications
in various fields. It is quite remarkable that the Renormalization Group procedure gives rise to
a closed and differential equation that covers in a very universal way the RG behavior of field
theories. Notice, in particular, that as it stands, it is an exact reformulation of the underlying
theory described by the generating functional and as such is non-perturbative. Furthermore,
the structure of this equation and the required properties of Rτ [Φ̄] justify the removal of the
ultraviolet cutoff Λ by taking the limit Λ → ∞ on both sides. We will briefly comment on this
aspect in the next subsection and conclude with some remarks on the normalization factor.
4.3.2 Properties of the FRGE
The FRGE is an exact equation and a particular realization of the generating functional for QFT
or statistical systems. As such, it is non-perturbative in nature and the question of renormaliz-
ability can be studied on the basis of different approximation schemes than usually employed
within the perturbative methods. From the mathematical point of view, we are confronted with a
highly complicated expression, a functional non-linear differential equation. In a suitable basis
of field monomials, this corresponds to an infinite number of coupled ODE. In its final form, eq.
(4.31), it describes the RG evolution of the EAA, a functional related to the familiar effective
action and as such it is defined on the field space of expectation values.
Now, let us comment on the limit Λ → ∞. Since we are dealing with a differential equation,
both sides of eq. (4.31) describe the change w.r.t. the RG scale τ . So far we have not yet
stated a suitable criterion that distinguishes the predictivity and regularity of solutions to eq.
(4.31) we will discuss this important aspect in the next section. Here, our focus is on whether
or not we can safely remove the ultraviolet cutoff from the differential equation. It turns out
that the way we implemented the cutoff operator Rτ amounts to a sharp localization of eq.
(4.31) at τ . Therefore notice that the inverse operator that appears under the trace of the FRGE
150 Chapter 4. Functional Renormalization Group
contains a the cutoff operator Rτ [Φ̄] which acts as an infrared regulator for small eigenvalues.
On the other hand, Rτ [Φ̄] is chosen such that it only changes significantly within in a small
region of τ . For instance, if Rτ [Φ̄] is associated to Laplacian-type operator, we may write
τ∂τRτ [Φ̄]≈ δ (τ1/ℓ−∆) and thus eq. (4.31) is also UV-finite.
Next, let us consider the limit τ → ∞. Therefore, we make use of an equivalent description
of the FRGE (4.30) in terms of a functional integro-differential equation:
∫ ( ∫ )
e−Γ [ϕ ;Φ̄]
δΓ
τ τ= [dϕ̂ ] exp −S[ϕ̂;Φ̄]−∆Sτ [ϕ̂ ;Φ̄]+ ddx ϕ̂(x) [ϕ ;Φ̄] (4.32)
δϕ(x)
The saddle point approximation for fluctuations around ϕ yields, under the assumption that
δΓτ
δϕ( ) does not diverge,x
lim Γτ [ϕ ;Φ̄] = S[ϕ ;Φ̄]+ · · · (4.33)
τ→∞
Here, the dots represent higher order loop corrections [121].
4.3.3 The normalization factor
Notice that NΛτ [Φ̄] does not contribute to the RHS of the FRGE for it is independent on the
fluctuation fields. It only appears as an extra contribution in the definition of the EAA. In
later considerations we always set NΛτ [Φ̄] = 1, resulting in the standard formulation of the flow
equation. The τ-dependence of NΛ Λ −1τ [Φ̄]≡ (zτ [0;Φ̄]) is obtained from eq. (4.23)
[ [ ] ]
τ∂ NΛ
1
τ τ [Φ̄] =− STr ĤessJ zΛτ [J;Φ̄] ◦ τ∂τRτ [Φ̄] (4.34)2 J=0
If included in the definition of Zτ [J;Φ̄] it ensures that
It is instructive to present a specific example that demonstrates the effect of NΛτ [Φ̄]. There-
fore, consider the free Gaußian theory of a scalar field based on the Laplacian operator ∆=−D̄2.
The corresponding generating functional is given by
∫ ( ∫ √ )
Z[J;Φ̄]≡ dµG(ϕ̂;Φ̄)eJ(ϕ̂) = exp 1 ddx ḡJ(x)∆−12 J(x)
ϕ̂∈TΦ̄ϕ̂
The implementation of the RG scale amounts to replace the Laplacian with ∆ +Rτ [ḡ] and
further add a global factor Nτ [ḡ] to the measure. The Fourier transform of µ∆G with the additional
∆+Rτ [ḡ] Hence, we obtain
∫
Zτ [J;Φ̄]≡ dµ (ϕ̂ ;Φ̄)eJ(ϕ̂)e−∆Sτ [ϕ̂ ;Φ̄]G
ϕ̂∈TΦ̄ϕ̂√ ( ∫
det(∆) 1 √
)
= N d −1τ [ḡ] det(∆+ [ ]) exp 2 d x ḡJ(x)(∆+Rτ [ḡ]) J(x)Rτ ḡ
√
In the following we consider the two cases of Nτ [ḡ] = 1 and Nτ [ ] =
det(∆+Rτ [ḡ])ḡ det( ) , the lat-∆
ter yielding a suitable probability theory. It is straightforward, see section 2.2, to derive the
associated effective action which then gives rise to Γτ [ϕ ;Φ̄]:
∫ √ ( √ )
Γ [ϕ ;Φ̄] = 1 ddτ 2 x ḡϕ(x)∆ϕ(x)+ ln Nτ [ ]
det(∆+Rτ [ḡ])ḡ det(∆)
Notice that if we retain the probabilistic interpretation the second term vanishes and what re-
mains is the bilinear τ-independent term. In this case the RHS of eq. (4.30) vanishes, that is
τ∂τΓτ [ϕ ;Φ̄] = 0 and the RG effects of the trace can be entirely attributed to the τ-dependence
of the normalization factor Nτ [ḡ]. On the other hand, if we set Nτ [ḡ] = 1 than the free, scale
invariant theory shows an τ evolution.
4.4 Renormalizability & Asymptotic Safety 151
4.4 Renormalizability & Asymptotic Safety
Renormalizability is a condition on a mathematical theory which requires that by removing the
ultraviolet cutoff the description remains regular with only a finite number of free parameters
which have to be fixed by experiments. In this section we are going to classify solutions of
the FRGE according to their UV behavior. We will make use of the renormalization condition
in the form stated above and show its implementation into the RG language that significantly
reduce the number of ‘proper’ theories in T. We will see that the notion of perturbative renor-
malizability is only a special case of a more general class of renormalizable theories.
Special, renormalizable points in theory space are those which are fixed points of the RG
procedure and therefore scale independent. For the EAA we have to perform a joint transfor-
mation of the RG scale τ and a field rescaling whereby the associated dimensionalities of the
fields are obtained from the scaling property of the cutoff action:
∆S dϕcτ [c ϕ ;c
dΦ̄Φ̄] = ∆Sτ [ϕ ;Φ̄] (4.35)
Formally expanding the EAA in terms of basis invariants we can read off the canonical scaling
behavior of the coefficients {uα(τ)}. At scale invariant theories the following requirement has
to hold:
u(α)
!
(τ [Φ̄])P [ϕ ;Φ̄] = u(α)(τ [cdΦ̄ Φ̄])P [cdϕ(α) (α) ϕ ;c
dΦ̄ Φ̄] ∀α (4.36)
A fixed point under the RG evolution is thus given by the following condition:
Γτ [ϕ ;Φ̄]≡ Γ[τdϕ ϕ ;τdΦ̄Φ̄] (4.37)
Notice that in particular ∂τ Γτ [ϕ ;Φ̄] =6 0 at fixed points! If we rewrite the flow equation in terms
of dimensionless couplings we recover the familiar picture that critical points of the dimension-
less beta-functions signal the occurrence of fixed points of the RG flow.
The important observation is that fixed points are natural candidates to describe a renor-
malizable field theory. Due to their scale independence, we can safely remove the UV cutoff Λ.
Furthermore, notice that the coordinates of the fixed point are determined by the zero conditions
and thus there are no further experiments needed to specify the underlying theory.
In the space of dimensionless couplings, fixed points correspond to zeros of the RG vector
field and as such determine the topological landscape of the space of solutions. The fate of RG
trajectories is very sensitive to the way they approach or depart from these critical points. The
linearized regime of a fixed point already furnishes very valuable particulars about its structure
and the behavior of bypassing trajectories. In fact the critical exponents, see chapter 9, already
contain all essential information that characterize the RG flow in the vicinity of the fixed point.
The observation that conceptual very different models of Nature describing very distinct phys-
ical systems, may approach a fixed point of the same structure is encoded in the concept of
universality classes. That is, while on general scales the RG flow for these two systems exhibits
very contrary properties and in fact may be defined on completely different theory spaces, it may
happen that in the vicinity of a fixed point both are described by the same linearized differential
equations and thus are closely related on these scales.
Besides revealing connections of in general unrelated systems at criticality, the linearized
regime of a fixed point contains essential information what happens to RG trajectory that enter
its realm. In any case the RG solution feels the presence of a fixed point and is either attracted
or repulsed from its position. Associated to each fixed point there are (ir-)relevant directions in
which it acts as an (repeller) attractor when increasing the scale τ . While for relevant directions
the trajectory is always pulled towards the fixed point, for irrelevant directions only those which
admit the fixed point value are not repelled, but stay in the close vicinity of this critical point.
152 Chapter 4. Functional Renormalization Group
If this happens to be the case for all irrelevant directions the UV behavior of the corresponding
RG trajectory is fully controlled by the fixed point which it approaches. Thus, the underlying
theory shows more and more signs of scale-invariance and converges to a point in theory space,
which is nothing else than the condition of renormalizability given above. Hence, from this per-
spective a theory has a well behaved UV limit if it is pulled towards a fixed point for increasing
τ . For a specific fixed point the collection of all these RG-trajectories defines its UV-critical
hypersurface, SUV. Its linearization closed to the fixed point coincides with the hypersurface
spanned by the relevant field monomials. All trajectories not contained in SUV will depart from
the corresponding fixed point and their UV fate is either untamed or under the control of a dif-
ferent critical point. From this perspective it is clear that the dimensionality of SUV is related
to the predictivity of the RG flow in that for each relevant direction (equivalent for each dimen-
sion of SUV) there is one free parameter that has to fixed by additional constraints. Besides
experimental results this includes other fundamental requirements of the underlying setting, for
instance Background Independence in case of Quantum Gravity. Hence, the fewer the number
of fixed points that are compatible with the classical limit and the lower the dimensionality of
the associated UV-critical hypersurfaces, the higher the predictivity of the RG flow. The other
way around, if dimSUV is infinite and no further mathematical constraint can be imposed that
reduce the physical subspace to finite dimension, we cannot obtain full predictivity from the
results since an infinite number of experiments would be needed in order to uniquely identify
the RG trajectory realized in Nature.
In conclusion, we are prepared to present the concept of renormalizability in the context of
the Functional Renormalization Group:
Definition 4.4.1 — Renormalizability of RG trajectories. Let Γτ [ϕ ;Φ̄] describe the RG
solution of the FRGE subject to a certain theory space T(M,G,F). The RG trajectory
corresponds to a (non-perturbative) renormalizable QFT, if and only if it is contained within
a finite-dimensional SUV-critical hypersurface of a (non-)Gaußian, aka (non-)trivial, fixed
point of the RG flow.
We distinguish the type of renormalizability by the fixed point properties which controls the
UV of a RG-trajectory. The first classification refers to the underlying theory of the critical
point itself, i.e. we denote a fixed point (non-)trivial or being (non-)Gaußian if it corresponds
to a (interacting) free theory. In this case, the critical exponents reflect the canonical scaling
of the dimensionful couplings. Based on this distinction we have the following notation for
renormalizable theories, see fig. 4.2:
Definition 4.4.2 — Classification of Renormalizability. Let Γτ [ϕ ;Φ̄] be renormalizable in
the above sense, thus the RG-trajectory lies within a UV-critical hypersurface of a fixed
point FP. We say that Γτ [ϕ ;Φ̄] is . . .
◮ . . . trivial :⇔
FP is Gaußian and or all A[ϕ̂;Φ̄] ∈ SUV A[ϕ̂;Φ̄] are Gaußian (triviality problem)
◮ . . . asymptotically free :⇔
FP is Gaußian and there exists an interacting relevant direction
◮ . . . asymptotically safe :⇔ FP is non-Gaußian
Notice the subtle difference between triviality and asymptotic freedom. In the former case, all
trajectories that are attracted towards the Gaußian fixed point correspond to free theories. On
the contrary, asymptotic free theories become only Gaußian in the UV limit but are interacting
on lower scales. A prominent example of asymptotic freedom is given by QCD. Notice that
perturbation theory describes the local vicinity of Gaußian fixed points and perturbative renor-
malizable theories lay within their UV-critical hypersurface. Thus, if we perturbative results
4.5 Truncations 153
S NGFPUV
GFP NGFP GFP
NGFP
GFP
SUV SUV SUV
(a) Asymptotic freedom or trivi- (b) Asymptotic Safety (c) Perspective of perturbation theory
ality
Figure 4.2: These plots describe the types of renormalizable RG-trajectories. If the UV-critical
hypersurface, SUV, is spanned by only non-interacting relevant directions SUV describes a
regime of triviality. Otherwise, a theory is asymptotically (safe) free if it is contained in the
critical manifold of a (non-)Gaußian fixed point. On the right, we have sketch a RG trajectory
that is perturbatively non-renormalizable but turns out to be non-perturbatively renormalizable
for it approaches a non-Gaußian fixed point.
suggest that a theory is perturbative non-renormalizable it simply states the corresponding RG-
trajectory does not emanate from the trivial fixed points. Whether or not its UV behavior is
controlled by a different (non-trivial) critical point depends on the structure of the RG flow.
The last case of Asymptotic Safety is particular important for studying Quantum Gravity.
There, exactly the above described scenario arises: from perturbative analysis we observe that
associated RG-trajectories pass by the Gaußian fixed point and are repelled towards the UV.
The crucial question that people try to answer is whether or not there is another, non-trivial
fixed point that takes over this role and renders at least a single RG-trajectory asymptotically
safe.
Up to now there has been strong evidence for the existence of such a non-trivial fixed point,
first obtained in the Einstein-Hilbert truncation almost twenty years ago [122], for the concept
of truncations see the next subsection. This picture of an asymptotically safe theory of Quan-
tum Gravity has prevailed in a number of different extensions of the groundbreaking work. Till
now, there are confirmation of the existence of a suitable fixed points from various directions,
including f (R)-truncations [123–129], the square of the Weyl tensor [128] and also by coupling
matter to gravity [130, 131]. In addition the influence of the ghost and gauge sector has been
analyzed [132–134] and [135]. While these confirmations are based on single-metric computa-
tions, first studies in taken the full bi-metric character of the EAA into account became available
recently, [136, 137], again confirming the general results. Studies on the difference of the usual
Euclidean truncations and the Lorentzian case revealed re-ensuring confidence in the previous
results [138, 139] While one focuses mainly on metric field content there is also evidence based
on different field spaces, as for instance the Einstein-Cartan formalism [140–143] tetrad formu-
lation [144]. In conclusion one can say that taken together the Asymptotic Safety conjecture for
Quantum Gravity has a very solid foundation.
4.5 Truncations
The Functional Renormalization Group Equation is an exact representation of the path inte-
gral and thus intrinsically non-perturbative. From the technical point of view it combines the
154 Chapter 4. Functional Renormalization Group
difficulties of functional differential equations with the problem of non-linearity:
[
1 ( [ ] )− ]1
∂tΓτ [ϕ ;Φ̄] = STr Ĥessϕ Γτ [ϕ ;Φ̄] +Rτ [Φ̄] ◦∂tRτ [Φ̄] (4.38)
2
Notice that we have introduced the RG-time t ≡ lnτ = lnk/k0. In order to evaluate this
differential equations one usual employs its coordinatized version, expanding Γτ [ϕ ;Φ̄] in terms
of a full linearly independent basis of field monomials {Pa[ϕ ;Φ̄]}:
T∋ A[ϕ ;Φ̄]≡ ∑ ūa ·Pa[ϕ ;Φ̄]
a
We can employ this expansion formally into an infinite number of ordinary differential equa-
tions for the coefficients using the coordinate expansion. The LHS of the flow equation (4.38)
assumes the form
( )
LHS : ∂tΓτ [ϕ ;Φ̄] = ∂t ∑ ūaτ ·Pa[ϕ ;Φ̄] = ∑(∂t ūaτ) ·Pa[ϕ ;Φ̄]
a a
Since we assume that {Pa[ϕ̂ ;Φ̄]} forms a complete, linearly independent basis, the RHS of eq.
(4.38) can also be expanded onto the basis invariants {Pa[ϕ̂;Φ̄]} in a unique way. This defines
the dimensionful beta-functions β̄ :
[ ]
1 ( [ ] )−1
RHS : STr Ĥessϕ Γτ [ϕ ;Φ̄] +Rτ [Φ̄] ◦∂ R [Φ̄] =.. ∑ β̄ at τ ({ūbτ};τ) ·Pa[ϕ ;Φ̄]2 a
Since the P’s are linearly independent we can compare coefficients and find
∂ ūa = β̄ at τ ({ūbτ};τ) ∀a
This defines an infinite set of coupled differential equations which is no improvement compared
to a single functional differential equation. As long as there are no systematic techniques to, for
instance iteratively, attack this differential system, we have to rely on approximations to study
the flow equations. The idea is to project this equation onto a (possibly still infinite) subset
of field monomials, which is denoted a truncation. However, such a truncated theory space is
in general not invariant under the RG flow, since the non-linearity together with the second
variation of the basis invariants will lead to a generation of terms on the RHS which are not
present in the original ansatz for Γτ .
In general, we introduce a projection operator that reduces an action functional to a specific
direction of theory space:
( )
ΠP (A)≡ Π a . bb P ū ·P .= ū ·Pb ∑ b a
a
Assume that we can find a suitable linearly independent basis of field monomials on T, in the
general case this space will not be endowed with an inner product and projection operators are
difficult to obtain. Still, let us proceed on a formal level and apply ΠP to the FRGE:b
( ) ( [ ])1 ( [ ] )−1
ΠP ∂tΓτ [ϕ ;Φ̄] = ΠP STr Ĥessϕ Γτ [ϕ ;Φ̄] +Rb b τ [Φ̄] ◦∂tRτ [Φ̄]2
Since for dimensionful coefficients the fields and thus the basis invariants are τ-independent.
Hence, ∂t and ΠP commute and we can change the order on the LHS. In the coordinatizedb
version we find
( [( )− ])1 [ ] 1
(∂ ūbt τ) ·Pb[ϕ ;Φ̄] = ΠP STr Ĥess ab ϕ ∑ ūτ ·Pa[ϕ ;Φ̄] +Rτ [Φ̄] ◦∂tRτ [Φ̄]2 a
4.5 Truncations 155
The supertrace gives rise to the dimensionful beta-functions and thus the application of ΠPb
reproduces the differential equation for the coefficient ūbτ :
( )
(∂ ūb) ·P [ϕ ;Φ̄] = Π ∑ β̄ a({ūb};τ) ·P [ϕ ;Φ̄] = β b bt τ b P τ a ({ūτ};τ) ·Pb[ϕ ;Φ̄]b
a
Thus, even if we are only interested in the RG evolution of a single basis invariant, we still have
to compute the Hessian of a generic action functional in full theory space, for the projection is
applied only after evaluating the trace. The true approximation, or truncation, is employed if
we project the EAA on the RHS before computing the Hessian operator, i.e.
ΠSTrunc(Γk) = Π
S a
Trunc(∑ ūτ ·Pa[ϕ ;Φ̄]) ..= ∑ ūaτ ·Pa[ϕ ;Φ̄]
a a∈S
Here S is usually a finite subset of basis invariants which span the truncated theory space. The
evaluation of the flow equation is then given by
1 [( [ ] ) ]
∂ ΠS (Γ [ϕ ;Φ̄]) = ΠS S
−1
t Trunc τ Trunc(STr Ĥessϕ ΠTrunc(Γτ [ϕ ;Φ̄]) +Rτ [Φ̄] ◦∂tRτ [Φ̄] )2
+ corrections
Notice, that the corrections describe the contribution from the infinite number of neglected
directions and indicate that the RG flow is not closed. Improving truncations and test their
stability for different cutoff operators Rτ [Φ̄], for instance, provides a way to test the reliability
of the results.

Calculations
II
5 The truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.1 Theory space
5.2 Preliminary truncation
5.3 The final truncation
5.4 Cutoff-operator
6 The Hessian operator . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1 Hessian of the cutoff action
6.2 Hessian of the gauge fixing functional
6.3 Hessian of the ghost functional
6.4 Hessian of the gravitational functional
6.5 Hessian of the full truncation ansatz
6.6 First variation formula
6.7 Implementation of the gauge fixing condition
6.8 The level expansion
7 Trace evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
7.1 Inversion of the Hessian operator
7.2 Decomposition of symmetric 2-tensor fields
7.3 Projection techniques for truncations
7.4 The ‘Ω deformed’ α = 1 harmonic gauge
7.5 Heat kernel
8 The beta-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.1 Threshold functions
8.2 Beta-functions for the coefficients
8.3 Beta-functions for dimensionful couplings
8.4 Beta-functions for dimensionless couplings
8.5 The semiclassical approximation
8.6 Beta-functions in d = 4
8.A Expanded beta-functions in d = 2+ ε
8.B Beta-functions in d = 3
The first part of this thesis focused on the conceptual aspects of the Functional Renormalization
Group Equation (FRGE) along with the general framework it is embedded in. We have seen that
it provides a very universal and – for the present case more important – non-perturbative method
based on the Effective Average Action (EAA) approach to Quantum Field theory (QFT).
In the Functional Renormalization Group formulation, a QFT is renormalizable in a more
general, non-perturbative sense, namely if the underlying probability theory assumes a fixed
point under the Renormalization Group (RG) evolution. The usual notion of perturbative renor-
malizability is then recovered in the vicinity of a Gaußian fixed point, however in the general
case a non-trivial fixed point with a finite number of relevant operators suffices to render a
theory asymptotically safe and thus (non-perturbative) renormalizable.
In the FRGE setting, the difficulty in proving that a QFT is asymptotically safe, resides
in the nonlinear, functional nature of the underlying differential equation. For a specific field
content and symmetry constraint, this amounts to an infinite dimensional theory space Twith
the compatible field monomials as basis and the couplings related to the coefficients. In absence
of any systematic treatment of the resulting infinite dimensional ordinary but coupled differen-
tial system, one has to rely on truncations that investigate only a (probably finite dimensional)
subset of T. Notice that truncations are performed on the level of the EAA and thus are in-
trinsically non-perturbative and in fact correspond to a functional integration over the full field
space, in particular over all fluctuations. Rather one should visualize this approximation to the
complete RG flow as restricting the analysis of allowed theories (measures) to only a subset, by
projecting the FRGE to the corresponding basis invariants. By construction projections only
reveal part of the information and in fact can lead to ‘optical illusions’ that can destroy or simu-
late certain features which are or are not present in the RG evolution on full theory space. Thus,
as for any other approximation technique, the significance of truncations has to be analyzed
and confirmed in order to define the range of validity of the employed truncation. Basically,
one either tests the truncations to possess certain fundamental properties of the full solution
that can be read off by the structure of the FRGE, or by stability analysis whereby the trunca-
tion is deformed in one or the other direction. Concerning the first technique that investigates
the consistency of truncations, in part III of this thesis we present and study a monotonicity
property in the sense of Zamolodchikov’s C-function to test and design truncations, which is
non-perturbative and universal in nature. Furthermore, the Ward Identities for split-symmetry
(WISS) and other exact equations can in principle, though usually very difficult, be utilized to
set the range of reliability. The second way to judge the validity of truncations involves the
stability under changes of field space, symmetries, and general topological modifications, as
well as increasing the number of basis invariants.
While theories of Quantum Gravity (QG), in particular those based on General Relativ-
ity (GR), are known to either spoil unitarity or renormalizability on the perturbative level, the
hope is to cure those fundamental issues when leaving the realm of perturbation theory and
employ non-perturbative techniques, as for instance the FRGE. In this setting, the Asymp-
totic Safety conjecture of Quantum Gravity states that even though QG fails to be perturbative
renormalizable, there exists a non-trivial fixed point of the RG evolution with a finite dimen-
sional ultraviolet (UV)-critical hypersurface [122, 145]. As an additional initial condition one
requires a classical regime in infrared (IR) where GR or a consistent classical modification is
recovered. In the pioneering work [122] a first (single-metric) truncation of Quantum Einstein
Gravity (QEG) based on a metric field content and diffeomorphism symmetry was studied for
the Einstein-Hilbert functional. The promising result of the existence of a non-Gaußian fixed
point (NGFP) – hence the first evidence of the Asymptotic Safety conjecture – has been con-
solidated since then in various different truncations and modifications [99, 100, 121, 123, 126,
159
130–135, 138, 139, 141, 142, 146–171]. Besides Einstein-Cartan formulations, cutoff-scheme
dependency checks, and even infinite dimensional f (R)-truncations based on single-metric tech-
niques, there is also evidence from bi-metric calculations [136, 137, 172, 173]. As Background
Independence (BI) is very central to Quantum Gravity bi-metric studies are in fact mandatory,
however the difficulties that arise once the EAA depends on two metrics, a dynamical g and a
background field ḡ, has led almost exclusively to developments in the single-metric approxima-
tion scheme.
This part of the thesis is therefore devoted to an in depth bi-metric calculation of the FRGE
of Quantum Einstein Gravity (QEG) that can be used to address the question of coexistence of
Background Independence and Asymptotic Safety. We further allow for a topological modifi-
cation that is quite important, for instance in cosmology, namely we allow spacetime to have a
non-vanishing boundary. Both, performing a bi-metric calculation and a changing the topology
to spacetime with non-vanishing boundary, are non-trivial extensions to the well-studied cases,
and thus we expect subtle issues to arise due to the non-linearity and functional nature of the
FRGE and in constructing the Hessian operator.
We start this part of the thesis by introducing theory space and then present the steps leading
from the truncation ansatz for the Effective Average Action to the beta-functions for the running
couplings it contains. In order to carefully disentangle fluctuation and background fields, study
a truncated theory space spanned by two separate Einstein-Hilbert actions for the dynamical
and the background metric, respectively, with a background Gibbons-Hawking-York functional.
A new powerful method is used to derive the corresponding RG equations for the Newton- and
cosmological constant, both in the dynamical and the background boundary and bulk sector.
In part III we classify and analyze the solutions of the derived beta-functions of this part,
determine their fixed point structure, and identify an attractor mechanism which turns out in-
strumental in the split-symmetry restoration. This then allows to study global properties of
the RG flow in particular the coexistence of Background Independence and Asymptotic Safety.
We then propose a C-function like property of the FRGE and analyze the beta-functions of
the present (bulk part of the) truncation, as well as for a different bi-metric calculation based
on a full transverse-traceless (TT)-decomposition, and the single-metric approximation. The
same beta-functions are used for study applications to propagating gravitons and to Black Hole
thermodynamics. For the latter case the running of Gibbons-Hawking-York boundary term is
essential.

5.1 Theory space
Spacetime
Field space
Space of functionals
Notation and convention
5.2 Preliminary truncation
Gravitational functional
Gauge fixing functional
Ghost functional
5.3 The final truncation
Split-symmetry
Coefficients and couplings
5.4 Cutoff-operator
5. THE TRUNCATION
In this chapter we present the details of the considered bi-metric Einstein-Hilbert–Gibbons-
Hawking-York truncation. We start by specifying theory space and then described the prelimi-
nary truncation ansatz containing a total of eight coupling constants in the gravitational sector
plus gauge fixing and ghost parameters. During the process of evaluation, we slightly mod-
ify this initial ansatz in order to simplify the nonlinear functional trace on the right-hand-side
(RHS) of the flow equation. The actual analyzed truncated theory space then consists of six
running couplings all appearing as coefficients of invariants in the gravitational functional. We
describe its explicit form in the background-dynamical as well as the level-language.
5.1 Theory space
In the FRG approach there are three essential ingredients that specify the entire RG evolution.
The first concerns spacetime, the second field space, and the third the operators that are about to
appear in the theory. In this section we present these general constituents of theory space Tstep
by step. Our objection is a theory space build on a manifold with non-vanishing boundary, met-
ric field content, a Levi-Civita field space connection, and with diffeomorphisms as symmetry
group.
5.1.1 Spacetime
In our setting, spacetime does not contain an a priori geometry, in a sense that it is naturally
equipped with a Riemannian metric. Rather, it is a special class of topological manifolds with
boundary endowed with a smooth structure, for details we refer to section 1.1. Only when the
question about the nature of spacetime is settled, we can move on an define fields and operators
on top. It is thus a very central ingredient on which all the following concepts sensitively depend
on.
In what follows, we actually consider an entire collection of spacetimes that share certain
important features, while others are kept arbitrary for a moment and are represented by a param-
eter that is fixed only for practical applications, for instance the dimension of spacetime.
162 Chapter 5. The truncation
Since we assume spacetime to be continuous rather than lattice-like having discrete build-
ing blocks, the underlying point set XM has uncountably infinite cardinality. Furthermore, we
only consider inequivalent topologies τM which are locally Euclidean with boundary, Hausdorff
and second countable, hence which define a topological manifold with boundary. In addition,
we attach a maximal smooth atlas to the topological space (XM,τX) that results in a smooth
manifold with boundary. These requirements still allow for a large class of different spacetimes,
in particular the constraints do not fix the spacetime dimension d. Though for a full specifica-
tion of theory space this is a necessary ingredient, for truncated subspaces a general dimension
d suffices in deducing the RG evolution for several choices of d at once. However, for later
convenience we restrict d being finite and larger equal 2.
Since a manifold with boundary is more general than one without, the presence of a non-
vanishing boundary in the following calculation generalizes the usual considered truncations
where ∂M is empty. These special cases are recovered in setting ∂M along with its boundary
invariants and coefficients to zero. The price we have to pay in order to extend spacetime to
account for non-vanishing boundaries appear on the level of field space and operators. Most
of the very technical details in constructing a suitable smooth structure on the set of fields is
only proven for ∂M = 0/ . However, in particular assuming spacetime to be compact is usu-
ally sufficient to recover most of those features even in the presence of boundaries. Replacing
compactness with a weaker condition might also be suitable in certain cases. In particular, in
frameworks for which the rigorous mathematical treatment is yet to be established one consid-
ers the results to be valid even without compactness. Then future work especially theorems in
mathematics have to either verify or reduce the validity of these results. In this work, at least
formally we assume a compact manifold whenever boundary effects are studied.
S Thus, the class of spacetimes, {M}, we consider in this thesis are d-dimensional smooth
manifolds with non-vanishing boundary, whereby 2≤ d < ∞ holds true andM is assumed
to be compact whenever ∂M 6= 0/ .
5.1.2 Field space
Now that the arena is set by a class of smooth manifolds, we can consider field space, F, defined
over someM. Since we have only put a lower bound on the dimensionality of spacetime, F will
also depends on d.
Even though the original theory of GR is based on a metric field content, this could in fact
be an artifact of an effective description and various generalizations are possible. For instance,
the Einstein-Cartan formalism is a simple extension of GR whereby the field content is not
necessarily related to a Levi-Civita connection but based on a tetrad formulation of gravity.
Studies of the Asymptotic Safety conjecture for QG pursuing the Einstein-Cartan formalism
were performed in [141–144, 162, 163, 174].
In this thesis, we consider a field content that is given by the set of Riemannian metrics,
Riem(•), defined over M. Since spacetime is assumed finite, it is always paracompact and thus
admits a Riemannian structure. Every Riemannian metric represents a section of an associ-
ated bundle of a principal bundle with structure group related to Gl(d;R). The group of gauge
transformations G is thus generated by infinitesimal diffeomorphisms defining the set of invari-
ant action functionals. Hence, the physical field space [F] consists of equivalence classes on
Riem(•) identified by the action of the diffeomorphism group. As already mentioned Riem(•)
can be endowed with a principal bundle structure where the base space constitutes the physical
sector [Riem(•)] and the structure group is given by G. In order to avoid the use of equivalence
5.1 Theory space 163
classes, we introduce (anti-)ghost fields η and η̄ and define F to be
F= Riem(M)×θ ·Γ (TM)×θ ′ ·Γ (T∗M) (5.1)
The (anti-)ghost field represents an anti-commuting (dual) vector field, whereby θ (or θ ′) indi-
cates that the algebraic field of these vector spaces is replaced by anti-commuting numbers.
R Notice that we define the space of expectation fields rather than the fundamental field
content. Depending on the actual measure (theory) both coincide or differ, for instance
when bosonization of the field content take places in the transition from µ to Γ. All
assumptions we impose on 〈[F]〉 are confined to the arguments of the effective action
and the very difficult task is then to extract the bare action functional along with the
fundamental field content from its precise structure.
Whereas in the ghost sector, the Maurer-Cartan form acts as a (trivial) connection, the space
of Riemannian metrics is non-linearly constrained due to the non-degenerate property of its
elements and turns out to have a non-vanishing curvature in general. This complicates matters
a lot and we have to rely on a geometrical definition of functional variation, involving the non-
trivial connection. Though we are not going to make explicit use of this connection, one can
in principal define the Levi-Civita connection of associated to some field space metric GΦ̄ :
TΦ̄F×TΦ̄F→ R. Here, we choose the following class of field space metrics [39]:   
( ) ∫ (v )
µνρσ µν µν ρ
√ h µνg g ρ g  
(wh)ρσ
G v , w = ddx ḡ  (v )µ  g ρσ g g ρ  ρ Φ̄ ξ µ µ ρ µ  (wξ ) 
M
(vξ̄ )
µ ρσ
µ g g
µ gµ ρρ (wξ̄ )ρ
( )T
Hereby, Φ̄ = ḡ,Ξ, Ξ̄ represents the set of background fields.
The field space metric assumes the following form in the metric-metric component evalu-
ated at ḡ, see DeWitt [39]:
gµνρσ = ḡµρ ḡνσ −ϖ ḡµν ḡρσ (5.2)
The second term affects only the trace part of the metric fluctuations and contains a free param-
eter ϖ , which in GR assumes the value ϖ = 1/2. Notice that the inverse of gµνρσ is given
by
g−1
ϖ
µνρσ = ḡµρ ḡνσ + − · ḡµν ḡρσ ∀ϖ 6= 1/d(1 d ϖ)
Except for the metric-ghost entries, which are irrelevant in the present context, the remain-
ing structure of G is described by the ghost components. Since the ghost fields are Grassmann
valued and thus anti-commute, in particular ξ̄ ξ = −ξ ξ̄ holds true, G will be degenerate and
anti-symmetric in the ghost sector, while its associated matrix representation is symmetric:
( ) ( )
gµ ρ g
ρ
µ ḡµρ δ
ρ
µ
= (5.3)
gµ gµ ρ δ µ µρρ ρ ḡ
This subtlety will be relevant when evaluating the traces of the Hessian operators and in the
definition of the cutoff-operator.
Due to the non-vanishing boundary of M we have to impose supplementary constraints on
field space in order to obtain self-adjoint differential operators. For a moment we assume that a
particular choice is made, however only after computing the Hessian, we explicitly fix those to
be of Dirichlet type. Up to this point the considerations involve a completely general boundary
operator B∂M whose kernel defines field space.
164 Chapter 5. The truncation
R Notice that the fluctuation fields ϕ ∈ TΦ̄F are constraint by the tangent map of this opera-
tor, in particular for Dirichlet conditions this results in
B∂M( );Φ̄(Φ)
..= (Φ− Φ̄)(•)δ∂M(•) with ker(B∂M( );Φ̄)≡ {Φ |Φ|∂M = Φ̄|∂M}D D
Obviously, while F does not define a vector space since the sum of two fields fulfills
different boundary conditions, its tangent space TΦ̄F is indeed restricted by the linear∗ ∗
operator B∂M with ker(B∂M( ); ) = {ϕ ∈ TΦ̄F |ϕ |(D);Φ̄ D ḡ ∂M = 0}.
5.1.3 Space of functionals
Finally, theory space can be constructed on the basis of certain (differential) operators D̄ and D,
the field content F, and spacetime M. In order to give rise to functionals that are independent
on the gauge fixing condition, we require theory space to consist of diffeomorphism invariant
functionals only.
T(M,F,{D̄, D}) = {A[g, ḡ,η , η̄] | diffeomorphism invariant }
On the level of the Effective Average Action, which is an element of T, the cutoff implementa-
tion and the gauge fixing contribution introduce a true ‘bi-metric’ dependence of theory space
that in general spoils split-symmetry, the manifestation of Background Independence in the
present context. Thus, in generic elements of theory space are bi-metric functionals with two in-
dependent metric arguments: the dynamical- g and the background metric ḡ. Likewise, utilizing
the background field method, we can express T in terms of fluctuation fields, which reads
T(M,F,{D̄, D}) = {A[h,ξ , ξ̄ ; ḡ] |(h,ξ , ξ̄ ) ∈ TΦ̄F ∧ diffeomorphism invariant }
In addition to the diffeomorphism constraint we will later impose boundary conditions on
the fields, that leads to a vanishing of certain field monomials.
The local part of theory space can be systematically expanded in terms of basis invariants
with increasing number of covariant derivatives. For a single metric ansatz, the field monomials
up to fourth order in D are given by
∫
d √
∫ √ ∫ √
d x g ddx gR ddx gD̄2R̄
∫M ∫M ∫M
dd
√ √ √
x gR̄2 ddx gR̄µν R̄µν d
dx gR̄µνρσ R̄µνρσ
M M M
In case of non-vanishing boundaries, we have supplementary monomials on ∂M. The bound-
ary conditions for the fields will decide whether or not they appear as basis invariants in the
corresponding theory space.
∫ √ ∫ √ ∫
dd−1x H dd−1x HK dd−
√
1x HK2
∫∂M ∫∂M√ √ ∫
∂M
− √dd 1x HKµνKµν dd−1x H R dd−1x H nµnνRµν
∂M ∂M ∂M
Here, nµ∂µ denotes the outward-pointing normal vector, Kµν = Dµnν the extrinsic curvature
tensor, and Hµν = gµν −nµnν the induced metric on ∂M.
Once a second metric enters the game, as the background field ḡ in the present case, an
infinite number of mixed invariants arise in each order of D and D̄, for instance
∫ √ (√ ) j ∫g √ ∫d √d x ḡ √ ddx ḡR(g) ddx gR̄(ḡ)
M ḡ M M
Thus, Background Independence is a severe restriction on the set of functionals that constitute
theory space.
5.2 Preliminary truncation 165
5.1.4 Notation and convention
In this part of the thesis, we use the following list of conventions and notations.
◮ Einstein sum convention & indices: If not stated otherwise, an implicit summation over
(double) repeated indices is assumed. Small Greek letters denote spacetime indices with
values µ ∈ [1,d]. In case of symmetric 2-tensor fields we sometimes use capital Latin
symbols A≡ µν to represent its pair of indices.
◮ Curvature tensor fields: In our convention, the Riemann-, Ricci-, and scalar curvature
tensor are locally defined as follows:
Rσ ρµν = ∂µΓ
σ
νρ −∂ Γσ λ σν µρ +ΓρνΓλ µ −Γλ σρµΓλν
Rµν = R
σ
ρσν
R= gµνRµν
◮ Boundary geometry: In our setting, the normal vector field n of ∂M is pointing outwards.
If the normal vector is normalized using the dynamical g or the background metric ḡ,
we write n or n̄, respectively. However, in case of Dirichlet boundary conditions n = n̄
holds true. The projection of the metric tensor field to the boundary is defined by Hµν ≡
gµν −nµnν . Furthermore, the extrinsic curvature tensor assumes the form Kµν ≡ Dµnν .
◮ Field space: We assume a single background field Φ̄ = (ḡ,Ξ, Ξ̄) is sufficient to cover the
entire field space. Thus, the field space metric is evaluated at Φ̄ and we usually write
G≡GΦ̄. Furthermore, the dynamical fields are denoted Φ = (g,η , η̄) and its fluctuation
fields ϕ = (h,ξ , ξ̄ ) are given by Φ = exp (ϕ).
Φ̄
5.2 Preliminary truncation
To understand the actual process of evaluating truncations it is very illuminating to start with a
preliminary ansatz for the class of action functionals that constitute the truncated theory space
and then present the final ansatz that is most suitable for practical reasons, we encounter later
on. The purpose of this section is to motivate the initial truncation ansatz and define the general
theme that should be preserved in the finial form of truncated theory space. It is therefore
instructive to start once more with the properties of field space in order to decompose theory
space into conceptual different portions.
In the previous section we have introduced the frame in which our investigation takes place
by specifying theory space, T. Therefore, we defined spacetime, the space of physical fields,
and the invariance group of compatible action functionals. Usually, instead of working with
the physical relevant fields and operators, one considers an enlarged field space while reducing
the set of operators by certain symmetry constraints, as e.g. Becchi, Rouet, Stora and Tyutin
(BRST) invariance. To account for the additional (infinite1) contribution to the path integral
measure, one typically implements the Faddeev-Popov insertion by restricting the path integral
to a hypersurface that intersects equivalence classes of physical fields exactly once.2 This is
realized by adding a gauge fixing and a corresponding ghost term to the action, see subsection
2.1.5.
However, there is a subtlety that occurs when variations on the field space appear in the
formalism. One source, a covariant way to define the above mentioned gauge fixing condition,
is the background field method, see subsection 2.1.5. Thereby, the dynamical field Φ̂ (dynamical
metric ĝ) is expressed in terms of a fixed, but arbitrary background field Φ̄ (background metric
1The factor is related to the size of an orbit and thus it depends on the symmetry group whether it is finite or
infinite.
2The existence of such global hypersurfaces is in general not given [36, 54].
166 Chapter 5. The truncation
ḡ) and a fluctuation field ϕ̂ (or ĥ) being an element of the tangential space at Φ̄ (or ḡ), where
the identification is given by the exponential map, i.e. for metric field content ĝ = expḡ(ĥ) or
ĝ = ḡ+ ĥ[ĝ, ḡ]. In general, this relation is valid only in a local region of Φ̄ (or ḡ) and thus we
need several patches to cover the entire field space. (On the level of the path integral this is
reflected in a summation over different background fields which carries over to the definition of
the effective action, depending now on the expectation value of the dynamical metric g = 〈ĝ〉
and ḡ = 〈ḡ〉 or likewise h = 〈ĥ〉.) However, in certain cases the field space topology allows to
identify each dynamical field with a tangential vector globally; then the path integral can be
either written in terms of ĝ or h and the derived effective action depends on a single field only.
In Quantum Einstein Gravity the physical field content is given by the equivalence classes
of Riemannian metrics under the action of the diffeomorphism group. Unfortunately, this space
turns out to be curved and we have to related the dynamical and the background fields in a
non-trivial way. This results in a theory space spanned by all diffeomorphism invariant field
monomials P(α) depending on a dynamical metric g, a background field ḡ, and ghost fields ξ ,
ξ̄ which are in total ghost number neutral.3 Each point in theory space can thus be described
by an effective action of the form Γ[g,ξ , ξ̄ , ḡ] = ∑ I II ū P [g,ξ , ξ̄ , ḡ]. Since we consider a pure
gravity truncation, there are no additional matter fields present, though there is no obstruction
for including them, see ref. [175]. Hence, the present theory space can be decomposed into
pure gravitational contributions, gauge fixing terms and ghost functionals:
Γ [ϕ ;Φ̄] = Γgrav[ϕ ;Φ̄]+Γgf[ϕ ;Φ̄]+Γghk k k k [ϕ ;Φ̄] (5.4)
Here Γghk [ϕ ;Φ̄] ≡ Γ
gh
k [h,ξ , ξ̄ ; ḡ] is the only contribution that contains basis invariants of ghost
fields.
Notice that from the perspective of functionals Γgrav[ϕ ;Φ̄] and ΓgfR k k [ϕ ;Φ̄] are indistinguish-
able, for they describe actions with the same field arguments. However, the relation be-
tween ghost- and gauge fixing functional allow to separate the latter from the gravitational
contribution, at least formally.
The general way to design a truncation is to put a special emphasize on one of the three
constituents of eq. (5.4) and modify the remaining parts as needed. Surely, the most interesting
part is the RG evolution of the gravitational functional whereby the gauge fixing- and ghost
terms are kept as simple as possible.4 The word ‘simple’, however, is very sensitive to the
actual truncation ansatz considered and thus it will only become clear during the evaluation
process which modifications turn out to be extremely helpful.
In the remainder of this section we specify the initial truncation ansatz in this threefold
manner, starting with the gravitational contribution and then define a gauge fixing condition,
from which we deduce the ghost functional.
5.2.1 Gravitational functional
In this thesis the emphasize lies on the gravitational part of theory space, Γgravk [ϕ ;Φ̄]≡Γ
grav
k [h; ḡ].
We want to consolidate a bi-metric calculation on the bulk with boundary terms that seem to be
interesting from the cosmological point of view. In its full beauty the (untruncated) Effective
Average Action contains an infinite number of different basis invariants of any order in the ghost
fields, curvatures and fluctuation- and background fields.
3Here, we make use of the identification of the ghost background fields and their tangent vectors.
4Different directions in which the running of the ghost or gauge fixing sector is studied can be found in ref.
[132–134] and ref. [135], respectively.
5.2 Preliminary truncation 167
The Einstein-Hilbert truncation, a central part of the following derivation, is for at least two
reasons, a suitable choice or testing ground. First of all, in an successive derivative expansion it
constitutes all invariants up to second order on manifolds without boundary. Secondly, General
Relativity is found to be a very good description of IR physics of gravitational interaction. Thus,
its invariants should be included in any truncation that wants to reproduce GR as an effective
field theory at small scales k.
In the bi-metric setting and for manifolds with non-vanishing boundaries, theory spaces
gets significantly extended, for there are now two metrics and two covariant derivatives which
allow for an infinite number of field monomials at each le√vel of√the de√rivativ√e expansion. For
i√nstance, the Einstein-Hilbert term is now represented by ḡR̄, gR, ḡR, gR̄ or in generaln+1 √
ḡ / gnR. Furthermore, there are now additional basis invariants defined on the boundary
of spacetime, as for instance the Gibbons-Hawking-York term, particular important to obtain
Einstein’s field equations by a variational principle in the presence of ∂M 6= 0/ :
∫
1 − √S d 1GHY[g] =− d x H K(H)
8πGN ∂M
Notice that compared to the Einstein-Hilbert action SEH[g] = − 1
∫ √
16π Md
dx g R(g) there is a
GN
relative prefactor of 2 multiplying the extrinsic curvature K.
In the sequel, we will focus on fully disentangled dynamical and background terms and ne-
glect the basis directions of the ‘mixed’ interactions. The thus considered subspace is described
by EAA’s of the form:
Γgrav[Φ,Φ̄]≡ Γgrav;Dk k [g]+Γ
grav;B
k [ḡ] (5.5)
The single-field functionals Γgrav;D grav;Bk [g] and Γk [ḡ] only differ by their coefficients. Our trun-
cation ansatz includes all non-mixed basis invariants up to second order in the bulk and corre-
spondingly first order on the boundary, thus a bulk and boundary volume term as well as the
Einstein-Hilbert- and the Gibbons-Hawking-York functional, yielding four field monomials for
each metric, hence a total of eight basis invariants:
Γgrav[Φ,Φ̄] = ( D λ̄D) ·Γgrav D gravū ∂D ∂D grav ∂D gravk k k (A) [g]+ (ūk ) · Γ( ) [g]+ (ūk λ̄k ) ·Γ(∂ )[g]+ (ūk ) · Γ(∂ )[g]B A B
+( B λ̄ B) ·Γgrav[ ]+ ( B) · Γgravū ḡ ū [ḡ]+ (ū∂B λ̄ ∂Bk k ( ) k ( ) k k ) ·Γ
grav
(∂ )[ḡ]+ (
grav
ū∂B
A B A k
) · Γ(∂ )[ḡ]B
(5.6)
Hereby, ū•k and λ̄
•
k are dimensionful coefficients which are subject to the RG flow and related
to Newton’s coupling and the cosmological constant, respectively. In increasing order in the
covariant derivatives D or D̄ we have the following list of basis invariants:
∫
Γgrav( ) [g]≡+2 · d
d √x g (5.7a)
A
∫M
Γgrav [g]≡+2 · dd−
√
1
(∂ ) x H (5.7b)A
∫∂M √
Γgrav d−1(∂ )[g]≡−2 · d x H K(H) (5.7c)B
∫ ∂M
Γgrav
√
( ) [g]≡− d
dx g R(g) (5.7d)
B
M
Notice that we have defined the coefficients of the Einstein-Hilbert- (ūDk ) and the Gibbons-
Hawking-York term (ū∂Dk ) such that if both agree we recover the classical matching condition
that results in the standard field equations.
168 Chapter 5. The truncation
5.2.2 Gauge fixing functional
To specify the second term, the gauge fixing contribution, we have to define the operator F[g, ḡ]
acting on the fluctuation field h. While the entire derivation of the Hessian operator is valid
for an arbitrary linear F[g, ḡ] that ‘commutes’ with the background metric ḡ, there are general
arguments in favor of a certain class of gauge fixing conditions where F[ḡ] is only background
dependent. Therefore, consider the following.
Whenever the background field method gives rise to a bi-field formulation of the path in-
tegral, the gauge fixing hyperplane can be a functional of ĝ as well as ḡ. In the bi-field con-
struction, infinitesimal gauge transformation of the dynamical field, δvĝ ≡ Lvĝ, assumes the
form
δvĝ= δvḡ+δvĥ[ĝ, ḡ] for all v ∈ Γ (TM) (5.8)
In decomposing the full dynamical field into a background and a fluctuation field one has to
decide in which way the gauge transformation affect ḡ. A natural choice are the quantum gauge
transformations where the background field is unaffected by δv:
( )
δvḡ≡ 0 δvĥ[ĝ, ḡ]≡ Lvĝ= Lv ḡ+ ĥ[ĝ, ḡ] for all v ∈ Γ (TM) (5.9)
However, for generic linear F[g, ḡ] based on the quantum gauge transformation the EAA is not
manifestly diffeomorphism invariant off-shell, though on-shell the Ward identities ensure coor-
dinate invariance. Thus, instead of absorbing the full gauge transformation into the fluctuation
fields, one may equally distribute its action on both, the fluctuation and background field:
δvḡ≡ Lvḡ δvĥ[ĝ, ḡ]≡ Lvĥ[ĝ, ḡ] for all v ∈ Γ (TM) (5.10)
This split is usually denoted background gauge transformation that keeps eq. (5.8) intact. In
sharp contrast to its ‘quantum’ counterpart gauge fixing condition which are invariant under
background gauge transformations result in a ḡ-covariant Effective Average Action suitable to
study QEG. The corresponding gauge fixing functional is given by
( )
Γgf[ϕ ;Φ̄]≡ 1 gfk 2 ūk G F[ḡ](h) · êξ̄ , F[ḡ](h) · ê∫ ξ̄
1 √= gf2 ūk d
dx ḡ F[ḡ]µ(h)ḡ
µνF[ḡ]ν(h) (5.11)
M
It is bilinear in the fluctuation fields and thus contributes to the propagator and hence the Hessian
operator. Here gfūk = Z
D
k /αk contains a running α-parameter, which will be constraint in the end.
A family of appropriate gauge fixing conditions, invariant under background gauge trans-
formations and linear, is the family of generalized harmonic gauge conditions that we will use
later on
( )
ρσ
F[ḡ] = ḡαρ ḡβσ −ϖ ḡαβ ρσµ ḡ ḡαµD̄β (5.12)
The kernel of F[ḡ] yields the gauge fixing hyperplane which includes the zero-fluctuation cor-
responding to gµν = ḡµν . Notice that this general ansatz, with an up to now free parameter ϖ ,
covers the harmonic gauge (ϖ = 1/2) as well as the ‘anharmonic gauge’, used in [172], for
instance, which has ϖ = 1/d, respectively.
5.2.3 Ghost functional
The third term in eq. (5.4) is correspondingly the ghost functional containing in principle arbi-
trary powers of the ghost fields. However, we retain only a bilinear term which is associated to
5.3 The final truncation 169
the Faddeev-Popov trick w.r.t. the gauge condition αβFµ [ḡ]h
µν .
√ ( )
Γghk [ϕ ;Φ̄] =− 2
gh
ūk G ϕξ̄ · êξ̄ , M[g, ḡ](ϕ∫ ξ
)
√ √
=− 2 ghū ddk x ḡ ξ̄µM[g, ḡ](ϕξ ) (5.13)
M
Here, M[g, ḡ] denotes the Faddeev-Popov operator that acts on contravariant vectors according
to
( )
M[g, ḡ](ξ ) = F[ḡ] Lξg · êξ̄ (5.14)
Thus, in our setting the ghost functional is fully determined once we specify the gauge fixing
condition.
5.3 The final truncation
The final form of the truncation ansatz is mainly oriented by the structure of the Hessian operator
that has to be simplified in order to apply the standard heat kernel techniques for evaluating
Laplacian like operators. Since we are most interested in the purely gravitational part of theory
space most of the modifications affect the gauge and ghost sector, while Γgravk [g, ḡ] is kept as
close as possible to the initial ansatz (5.6).
The RHS of the flow equation involves the second variation of Γk w.r.t. the field content
and thus it is convenient to expand the action functionals in terms of the fluctuation fields. For
the gravitational part, we obtain
∣ ∣
Γgrav[ , ]≡ Γgrav[ , ]+ ( Γgrav ∣ 1 gravk g ḡ k ḡ ḡ dh k [g, ḡ]) + 2(dhdhΓk [g, ḡ])∣ +O(h3)g=ḡ g=ḡ
This expansion defines a sum of functionals with inc∣reasing degree of homogeneity in h, de-
noted the level. Hence, Γgravk [ḡ, ḡ] or ( Γ
grav
dh k [g, ḡ])
∣ correspond to level-(0) or level-(1),
g=ḡ
respectively, and the associated k-dependent coefficients play different roles in physics; the
level-(1) couplings appear in the field equations for self-consistent backgrounds, while level-
(2) coefficients affect the propagator for instance. Employing the linear field parametrization
dvg = v and imposing Dirichlet boundary conditions h|∂M = 0, eq. (5.6) is almost identical to
the following level-expansion
∫ ( ) ∫ √ ( )
Γgrav[ ; ] =− ( √0) · dd (0) ∂ (0) ∂ (0)k h ḡ ūk x ḡ R̄−2λ̄ − ū · dd−1k k x H̄ 2K̄−2λ̄k
∫M ∂M√ { }
+ (1)ū · ddx ḡ Ḡµν + λ̄ (1) ḡµνk k hµν
∫M { { }
+ ( ) · √1ū ddx ḡ h 1 µνρσ µνρσk µν 4 −K̄ + Ū
M ( )}
− 1 λ̄ (1) · ḡρµ ḡσν − 1 ḡρσ ḡµν2 k 2 hρσ (5.15)
Here, Ḡµν ≡ R̄µν − 1 ḡµν2 R̄ denotes the Einstein tensor. The two operators that appear in the bi-
linear part of Γgrav[h; ḡ] are a kinetic-like and a potential-like contribution, K̄µνρσ ≡Kµνρσk (ḡ)
and Ūµνρσ ≡ Uµνρσ(ḡ), respectively. They are defined as
Kµνρσ(g)≡ (gµνDρDσ +gρσDµDν −2gνσDρDµ)− (gµνgρσ −gµρgνσ )D2 (5.16a)
Uµνρσ(g)≡ gρσGµν +gµνRρσ +Rgνρgµσ −2gµρRνσ −2gνρRµσ (5.16b)
The relation between the level-(p) and the ‘D’/‘B’ coefficients used in eq. (5.15) and (5.6),
respectively, will be spelled out in a moment. First of all, notice that the difference between the
170 Chapter 5. The truncation
final (5.6) and preliminary gravitational part of the truncation (5.15) consists of a surface term,
which is omit in the final result omitted in order to apply the conformal projection technique and
evaluate the functional trace. We have the following correspondence between the preliminary
and the final gravitational action functional for the linear field parametrization:
∫ √
Γgrav;
( )
init[ ; ] = Γgrav;final[ ; ]+ ( ∂ (1)k h ḡ k h ḡ ūk −
(1)
ūk ) · dd−1x H̄ n̄µD̄ν − ḡµν n̄ρ D̄ρ hµν
∂M
The additional boundary term is linear in h and thus does not affect the Hessian operator. It is
proportional to the discrepancy of the classical matching condition for the Einstein-Hilbert and
Gibbons-Hawking-York action: ∂ (1) = (1)ūk ūk . Thus, if the classical fine tuning survives the RG
evolution this term will absent for all trajectories that preserve this fixed ratio of bound∫ary√and
bulk∫term√. While the heat kernel expansion comes with the right relative factors for M ḡR̄
and ∂M H̄K̄, one would expect that the imbalance of bulk and boundary contributions in the
Hessian operator will spoil the matching in general.
Notice that For the choice of Dirichlet conditions, i.e. h|∂M ≡ 0, the set of boundary in-
variants reduces significantly, in particular the boundary volume element is identical to the
background counterpart:
∫ ∫
dd−
√ ∣ √
1x H∣ ≡ dd−1x H̄ since g
(D) µν
|∂M = ḡµν |∂M
∂M ∂M
Hence, once we impose Dirichlet boundary conditions for the metric fluctuations there is liter-
ally no corresponding volume term in level-(1) or higher levels. This has to be clearly distin-
guished from the omitted level-(1) Gibbons-Hawking-York contribution, for which the invariant
itself does not vanish.
In this final setting we have translated the original coefficients, related to the background or
dynamical functionals in eq. (5.6), into the level-language. The new coefficients are combina-
tions of the original couplings and in each step one can easily switch between both descriptions.
For the level-(0) monomials the correspondence is given by:
(0)λ̄ (0)ū ..k k =
(0)
ūD D B B .. Dk λ̄k + ūk λ̄k ūk = ūk + ū
B
k (5.17a)
∂ (0) ∂ (0) ∂ (0)
ū λ̄ .. ∂D ∂D ∂B ∂B .. ∂D ∂Bk k = ūk λ̄k + ūk λ̄k ūk = ūk + ūk (5.17b)
For the invariants of higher orders in h, there is no contribution from the background sector and
only the D-coefficients appear.√This will be different if we include mixed field monomials in
the truncation ansatz, as e.g. gR̄. Nevertheless, in the present case we have the following
correspondence between the D- and the higher level-coefficients:
(p)λ̄ (p)ū ..k k =
(p)
ūD D ..k λ̄k ūk = ū
D
k ∀ p≥ 1 (5.18a)
∂ (p)λ̄ ∂ (p)ū ..= ū∂D λ̄ ∂D ∂ (p)ū ..k k k k k = ū
∂D
k ∀ p≥ 1 (5.18b)
Notice, that again due to Dirichlet conditions for the fluctuation fields, certain boundary mono-
mials of non-zeroth order in h vanish in Γk[ϕ ;Φ̄] and thus do not define independent basis
directions in the field space of Dirichlet constraints.
Next, let us consider the gauge fixing functional, which is now evaluated for the harmonic
gauge fixing condition with the k-dependence of gfūk fixed to the gravitationa(l propagator coeffi- )
cient (2) ≡ (1)ūk ūk , i.e.
gf
ū =α−1 (1)k ūk . For the harmonic gauge condition,
ρσ
F = ḡαρ ḡβσ −ϖ ḡαβ ḡρσµ ḡαµD̄β ,
we obtain the following type of functionals
∫ √ {
Γgf[ ; ] = α−1 (1)h ḡ ū ddx ḡ h 1K̄µνρσ + ḡνσ R̄µρ − R̄ρµνσk µν 2
M ( ) }
− 1 ḡµρ ḡνσ − 1 ḡµν ḡρσ D̄22 2 hρσ (5.19)
5.3 The final truncation 171
Notice that the RG flow is ‘artificially’ constrained along the corresponding basis invariants
by allocating a specific k-dependence for the coefficient. The resolution of this field monomial
would require a k-dependent αk, while in the present case it could only be used as a consistency
check for the identification gfū = α−1 (1)k ūk .
The associated ghost functional is projected on a k-independent direction of the correspond-
ing basis invariant, i.e. we choose ghū = ūghk constant. This results in the following truncation
of the ghost sector
√ ∫ √ { ( )}
Γgh[ϕ ; ḡ] =− 1 gh d µν ρσ2 2ū d x ḡ ξ̄µ ḡ Fν D̄λgρσ +gλρ D̄σ +gλσ D̄ρ ξ
λ (5.20)
M
As already mentioned, our emphasis is laid on the gravitational sector and we use the gauge
fixing and ghost functional to simplify the trace evaluation of the RHS of the flow equation:
[
1 ( [ ] )− ]1
∂tΓk[ϕ ;Φ̄] = STr Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄]
2
As we see, we make heavy use of modifying the gauge sector, which will become clear in the
process of evaluation.
This concludes the final truncation ansatz for which we later compute the beta-functions.
Next, we discuss what split-symmetry means for this truncated theory space and then present
the conversion of the coefficients into more familiar quantities, as the Newton and cosmological
couplings.
5.3.1 Split-symmetry
One of our objectives to study bi-metric truncations is the question of whether or not Asymp-
totic Safety is compatible with the fundamental requirement of Background Independence in
theories of Quantum Gravity. In the EAA approach based on the FRGE this translates to
the investigating the status of split-symmetry of the BRST invariant part of theory space,
here Γgravk [g, ḡ]. Technically speaking, split-symmetry is the condition that Γ
grav
k [g, ḡ] reduces
to a functional of the dynamical metric gravg, i.e. Γk [g] or in terms of the fluctuation fields
Γgravk [h; ḡ] ≡ Γ
grav
k [exp
grav
ḡ(h)] ≡ Γk [ḡ+ h], whereby the last equality holds true for the linear
field parametrization.
For the truncation ansatz (5.15) notice that Γgravk [h; ḡ] corresponds to the Volterra expansion
of a single-field functional Γgravk [g] if all coefficients of different levels agree. For the bulk
invariants these conditions read
(0) = (1) = (2)ūk ūk ūk = · · ·=
(p)
ūk for all p≥ 0 (5.21a)
λ̄ (0) = λ̄ (1)k k = λ̄
(2)
k = · · ·= λ̄
(p)
k for all p≥ 0 (5.21b)
Part of these requirements are already implemented into the truncation ansatz by identifying
all level-(p) coefficients for p ≥ 1. While in general one expects that this connection breaks
the considered truncation is anyway insensitive to a separation of the corresponding basis in-
variants. Compared to the usually employed single-metric calculations the disentanglement of
level-(0) and level-(1) coefficients is an important progress in understanding the infinite number
of Newton-type couplings isolate their respective RG evolution.
For the boundary coefficients, there is a∫sub√tlety involved, since for Dirichlet boundary
conditions all field monomials associated to ∂M H vanish above level-(0). Hence,
∂ (0)λ̄ ∂ (0)ūk k
can assume any value without spoiling split-symm∫etry.√For the curvature boundary term re-
sulting from the Gibbons-Hawking-York invariant, ∂M HK, Dirichlet conditions give rise to
two independent field monomials, one of level-(0) and one of level-(1). However, the latter is
172 Chapter 5. The truncation
not resolved within the present truncation and thus we have to focus the study of Background
Independence on the bulk part of this truncation only.
A different way to present split-symmetry is given by the D/B description, where ūD (p)k ≡ ūk
equals the level-coefficients for all p≥ 1, for instance. The case of Γgravk [g, ḡ] loosing its ‘extra’
ḡ-dependence, [122], is equal to the vanishing of all background coefficients, i.e. the following
conditions have to hold true:
ūB = 0 ∧ ūBλ̄ B = 0 ∧ ū∂B = 0 ∧ ū∂B ∂Bk k k k k λ̄k = 0 (split sym.) (5.22)
In general, due to the cutoff-action and the gauge fixing contribution that explicitly introduce
extra factors of ḡ that do not combine to purely dynamical invariants, split-symmetry will be
broken along RG trajectories. In fact, even for k= 0 where the cutoff-dependence vanishes, the
gauge fixing functional is still a potential source of split-symmetry violation.
R For single-metric calculations, split-symmetry is assumed to be preserved by the full
RG evolution. This is a very severe constraint, a priori not justified by any algebraic or
analytic argument. Indeed, we will later demonstrate that there bi-metric calculation seem
to support the statement that there exists not a single trajectory for which split-symmetry
is intact for all k. Consequences of these violation of the single-metric assumption are
regions in theory space where the single-metric results are unreliable and in fact exhibit
properties incompatible with exact solutions.
5.3.2 Coefficients and couplings
In order to make contact to the couplings encountered in classical General Relativity let us
reformulate the coefficients ūk in terms of the dimensionful Newton-type couplings Gk:
I ≡ 1ūk for all I ∈ {(p), ∂ (p), B, ∂B, D, ∂D} (5.23)16πGIk
Within these more familiar couplings, our truncation ansatz (5.15) reads
∫
1 √ ( ) ∫1 − √ ( )Γgrav[ (0)k h; ḡ] =− · ddx ḡ R̄−2λ̄k − · dd 1x H̄ 2K̄−2λ̄
∂ (0)
16π (0) ∂ (0) kGk M 16πGk ∂M∫
1 √ { }
+ · dd (0)x ḡ Ḡµν + λ̄ µν
16π (1) k
ḡ hµν
Gk M∫
1 √ { { }
+ · ddx ḡ h 1 µνρσ µνρσ
(1) µν −K̄ + Ū (5.24)16πG 4k M ( )}
− 1 λ̄ (2) · ḡρµ ḡσν − 1 ḡρσ µν2 k 2 ḡ hρσ +O(h3)
The relation between the level- and D/B-formulation is given by the following conversion
rules:
1 1 1 ∧ λ̄
(0)
λ̄ Bk k λ̄
D
= + = + k ∧ (1) (2)Gk = Gk = · · ·= GD (5.25)(0) GB GD (0) GB GD kGk k k Gk k k
And similarly for the boundary couplings. Hence, a truncation ansatz with the six independent
couplings GB, λ̄ B, GD, λ̄Dk k k k ,G
∂D
k , λ̄
∂D
k is equivalent to an ansatz of the type (5.24) whose a priori
infinitely many couplings (p), λ̄ (p)Gk k , · · · are not independent but satisfy
GD B(0)
G = k
Gk , (1) = (2)k Gk Gk = · · ·= GDGD+GB k (5.26)k k
5.4 Cutoff-operator 173
and likewise for the cosmological constant type couplings:
(0) λ̄
DGBk + λ̄
BGD
λ̄ = k , λ̄ (1) = λ̄ (2)k = · · ·= λ̄D (5.27)(GD+GBk k)
It will be instructive to switch back and forth between the (g, ḡ)-language employing the ‘B’
and ‘D’ parameters, and the (h; ḡ)-language with (p)Gk , λ̄
(p)
k , p= 0,1,2, · · · .
In case of intact split-symmetry where the different levels stem from a Volterra expansion
we have the following set of relations among the various Newton- and cosmological constant
type couplings:
(0) = (1)Gk Gk =
(2)
Gk = · · · , and λ̄
(0) (1) (2)
k = λ̄k = λ̄k = · · · (split sym.) (5.28)
In the (g, ḡ)-language these relations are satisfied if the ‘B’ part of the EAA completely vanishes
1 λ̄ B
= 0 , and = 0 (split sym.) (5.29)
GBk G
B
k
or, stated differently when Γgravk [g, ḡ] is actually independent of its second argument.
Again, we should mention that there is no guarantee for the existence of trajectories sat-
isfying condition (5.28) or likewise (5.29). However, there are points in theory space where
these conditions are fulfilled or are closed to be. In these regions a single metric description
is well applicable. From eq. (5.29) we see that if GBk ≫ GDk and λ̄ B ≈ λ̄D split-symmetry is
approximately restored.
5.4 Cutoff-operator
The FRGE defines a vector field on theory space which in turn describes the RG evolution of
certain action functionals. The RG flow that is described by a sequence of functionals Γk[Φ,Φ̄]
is triggered by the cutoff operator Rk[Φ], which loosely speaking gives a k-dependent mass to
all fluctuation fields. Its implementation is based on a bilinear functional, the so-called cutoff
action:
1 ( )
∆Sk[ϕ ;Φ̄] = G ϕ ,Rk[Φ̄]ϕ (5.30)
2
This action, essentially the cutoff-operator Rk, divides field space into suppressed and unsup-
pressed field modes, depending on k. Lowering k corresponds to gradually increasing the
amount of fields one integrates over and in the limit k → 0 one recovers the usual effective
action limk→0 Γk = Γ.
In the sequel, we use the Laplacian operator of the background metric ∆ ≡−D̄2 to express
field space in terms of its eigenfunctions. For Dirichlet boundary conditions, ∆ is strongly
elliptic, which implies that all its eigenvalues are non-negative, and there is only a finite number
of zero eigenmodes. Hence, when relating the cutoff procedure to the eigenfunction expansion
of ∆, the scale k assumes values from 0 to ∞ only.
The Gaußian part associated to ∆ in the present truncation reads
∫ √
Γfreek [ϕ ;Φ̄] = +
1 ūD dd2 k x ḡ hµν (g )
µν ρσ
hh (−D̄2)hρσ
√ M ∫ √ ( )− 1 2 ūgh dd ξ̄ ν 2 µ µ νx ḡ 22 ν (gξ̄ ξ ) (−D̄ )ξ −ξ (gµ ξ ξ̄ )µ (−D̄ )ξ̄νM
174 Chapter 5. The truncation
whereby g ≡ g[ḡ] denotes the local part of the field space metric, i.e. (g )µν ρσ ≡ ḡµρhh ḡνσ −
1 ḡµν ḡρσ2 for the gravitational sector (for the harmonic choice ϖ = 1/2) and (g )
ν ≡ δ νξ̄ξ µ orµ
(gξ ξ̄ )
ν
µ ≡ δ νµ for the ghost parts.
The cutoff procedure is now understood as replacing ∆ with ∆+ (0) (0)Rk (∆), whereby Rk (∆)≡
k2R(0)(∆/k2) denotes the shape-function. The matrix form related to Γfreek [ϕ ;Φ̄] is thus given by
 
grav
Rk [ḡ] 0 0 
Rk[ḡ] = 0 0 R [ḡ]ghk  (5.31)
0 R [ḡ]ghk 0
A comparison with eq. (5.30) reveals the explicit structure of the L2-operators in terms of the
shape-function:
grav
Rk [ḡ] = +
(0)
ūDk Rk (−D̄2) ·1[h] (5.32a)√
gh[ ] =− 2 gh (0)Rk ḡ ū Rk (−D̄2) ·1[ξ̄ ] (5.32b)√
gh[ (0)Rk ḡ] = + 2 ū
ghRk (−D̄2) ·1[ξ ] (5.32c)
For this choices, the required ‘symmetry’ property of the cutoff-operator is indeed satisfied:
( ) ( )
G v , Rk[Φ̄]w =G w , Rk[Φ̄]v ∀v,w ∈ TΦ̄F,
R Notice that the cutoff operator does not depend on the background fields of the ghost
sector. Thus, there is no violation of ‘ghost split-symmetry’ between the background
and dynamical ghost fields. The situation is quite different for the metric field content,
explaining one source of split-symmetry violation which however only appears for k 6= 0.
6.1 Hessian of the cutoff action
6.2 Hessian of the gauge fixing functional
6.3 Hessian of the ghost functional
6.4 Hessian of the gravitational functional
6.5 Hessian of the full truncation ansatz
6.6 First variation formula
6.7 Implementation of the gauge fixing con-
dition
6.8 The level expansion
6. THE HESSIAN OPERATOR
The vital piece in the evaluation of the RHS of the Functional Renormalization Group Equation
is the Hessian operator of the Effective Average Action. Besides the non-linearity of the FRGE
that emanates from the inversion of this operator, determining the trace of the Hessian operator
represents the major difficulty in the derivation of beta-functions especially in the present of
boundaries for spacetime. In this chapter we compute the second variations of the preliminary
truncation ansatz under very general conditions, in particular for generic boundary constraints,
any linear gauge fixing condition, and arbitrary field parametrization. This allows future work
to modify the settings in the range of validity and deduce new results with less effort. Towards
the end of the chapter, we then evaluate the Hessian operators for Dirichlet boundary conditions
and for the family of generalized harmonic gauge conditions that will be used in the sequel of
the derivation.
6.1 Hessian of the cutoff action
The cutoff action, as defined in eq. (5.30), is a bilinear functional of the fluctuation field ϕ ≡
[F]
ϕ(Φ̂,Φ̄) that defines a geodesic ∇ϕ ϕ = 0 on field space F interconnecting a base point Φ̄ with
the dynamical field Φ̂:
1 ( ) ( ) ( )
∆Sk[ϕ ;Φ̄] = G ϕ ,Rk[Φ̄]ϕ with G v ,Rk[Φ̄]w =G w ,Rk[Φ̄]v ∀v,w ∈ TΦ̄F2
The latter condition is equivalent to Rk[Φ̄] being (anti-)self-adjoint for (Grassmannian) fields.
Since Rk[Φ̄] and the field space metric G are linear the functional variation of ∆Sk[ϕ ;Φ̄]
with respect to the fluctuation field ϕ is a straightforward computation. Since the tangent space
is a u(sua)l vector space we have the trivial identification TΦ̄F ≃ TϕTΦ̄F with exponential map
expϕ t v ≡ ϕ +TF ι(v), whereby ι : TϕTΦ̄F → TΦ̄F denotes the inclusion map. Hence, any
functional variation on TΦ̄F reduces to a simple Gateaux derivative. We are going to demon-
strate the general procedure of functional variation for this particular elementary example mak-
ing each step explicit. In the later derivation of the Hessian operators we present a reduced form
in which some of the rather technical details are omitted.
176 Chapter 6. The Hessian operator
For now, let v ∈ TϕTΦ̄F be any tangent vector of ϕ with ṽ ≡ ι(v) its natural identification
in TΦ̄F. The first variation of the cutoff action yields
1 ( )
dv∆Sk[ϕ ;Φ̄] = lim ∆Sk[ϕ +TF (ε · ṽ);Φ̄]−∆Sk[ϕ ;Φ̄]
ε→0 ε
1 ( ( ) ( ))
= lim G ϕ +TF (ε · ṽ) ,Rk[Φ̄]ϕ +TF (ε · ṽ) −G ϕ , Rk[Φ̄]ϕ
ε→0 2ε
1 ( ( ) ( ) ( ))
= lim ε ·G ϕ , Rk[Φ̄] ṽ + ε ·G ṽ ,Rk[Φ̄]ϕ + ε2 ·G ṽ , R→ k[Φ̄] ṽε 0 2ε
1 ( ) 1 ( )
= G ϕ ,Rk[Φ̄] ṽ + G ṽ ,Rk[Φ̄]ϕ
2 2
As expected, the resulting functional dv∆Sk[ϕ ;Φ̄] is linear in ϕ and ṽ. Thus the second field
derivative w.r.t. w ∈ TϕTΦ̄F, whereby w̃≡ ι(w), assumes the form:
( ) 1 (( )∣ )
dw dv∆Sk[ϕ ;Φ̄] = lim dv∆Sk[ϕ ;Φ̄] ∣ · −dv∆Sk[ϕ ;Φ̄]ε→0 ε ϕ=ϕ+TF(ε w̃)
1 ( ( ) ( ))
= lim ε ·G w̃ ,Rk[Φ̄] ṽ + ε ·G ṽ , Rk[Φ̄] w̃
ε→0 2ε
1 ( ) 1 ( )
= G w̃ , Rk[Φ̄] ṽ + G ṽ , Rk[Φ̄] w̃
2 2
The second functional variation of the cutoff action is already symmetric and hence agrees with
the Hessian of ∆Sk[ϕ ;Φ̄]. Using the (anti-)self-adjoint property of the cutoff operator Rk[Φ̄] the
previous result further simplifies to
[ ] 1 ( ( ) ( ))
Hessϕ ∆Sk[ϕ ;Φ̄] (v,w) ≡ dw dv∆Sk[ϕ ;Φ̄] +dv dw∆Sk[ϕ ;Φ̄]
2
1 ( ) 1 ( )
= G w̃ , Rk[Φ̄] ṽ + G ṽ ,Rk[Φ̄] w̃
2 ( ) 2( )
=G w̃ ,Rk[Φ̄] ṽ ≡G ṽ , Rk[Φ̄] w̃ (6.1)
The second equality of eq. (6.1) is the very definition of the associated Hessian operator which
in this case coincides with the cutoff operator:
[ ]
Ĥessϕ ∆Sk[ϕ ;Φ̄] =Rk[Φ̄] (6.2)
Crucial in the derivation of this Hessian operator is the (anti-)self-adjoint property ofRk[Φ̄] that
has to be satisfied for any explicit choice.
6.2 Hessian of the gauge fixing functional
Working with an enlarged field space F instead of the set of physical indistinguishable fields [F]
we have to introduce a gauge fixing condition that selects a single representative Φ for each Φ.
On the level of the EAA a very general form of the gauge fixing contribution is given by
( )
Γgfk [ϕ ;Φ̄]≡ 1
gf
2 ūk G F[ḡ](h) · êξ̄ , F[ḡ](h) · êξ̄
While the full transformation F[ḡ](•) · ê ⊗ ê∗ξ̄ h is indeed an operator on TΦ̄[F] in the usual sense,
its field coordinate form F≡ F[ḡ] is not, for it lacks the endomorphism property mapping
metric fluctuation into dual vectors: T F→Γ (T∗ḡ M). Hence, the adjoint mapF∗[ḡ] :Γ (TM)→
TḡF has to be lifted to a transformation on field space – rather than one on its components – to
give rise to the adjoint operator F∗[ḡ](•) · ê ⊗ ê∗h .ξ̄
6.3 Hessian of the ghost functional 177
For the sake of generality, we will keep the explicit form of F[ḡ] arbitrary for a moment,
however besides locality we assume it satisfies the following requirements:
F[ḡ](c ·h1+h2)≡ c ·F[ḡ](h1)+F[ḡ](h2) (linear)
[F[ḡ](•) , ḡ] = 0 (commutes with ḡ)
This still give rise to a very large class of gauge fixing functionals built from the covariant deriva-
tive D̄ and the background field ḡ. In this general setting, consider the L2 operator associated to
the gauge fixing condition and determine its adjoint form, which is given by
(F∗ µ
( )
[ḡ])A =−√
1
g−1
√
B µν −1 B µν
ḡ AB
Fν ḡ ḡ • ≡ −gAB Fν (g •)
Now, the linearity of F[ḡ] results in a bilinearity of Γgfk [ϕ ;Φ̄] which in turn makes the evaluation
of the Hessian very simple:
[ ] ( ( ) ( ))
Γgf
1
Hessϕ k [ϕ ;Φ̄] ( , )≡
gf gf
v w dw dvΓk [h; ḡ] +dv dwΓk [h; ḡ]2 ( )
= gfūk G w , ê
∗
h · (F [ḡ]◦F[ḡ])(êh · v) (6.3)∫ √ ( )
+ gfū ddx ḡ FAk µ ḡ
µν(w B gfh)AFν (vh)B ‖ =̂ I [v,w]
M
The second term in this final result stems from the chain rule and can be usually transformed
into a boundary contribution, depending on the explicit form of FBν . This additional boundary
piece in the Hessian is an obstruction to deduce a Hessian operator. Hence, assuming that in
following we find a way to absorb or eliminate this second contribution, we formally define the
Hessian operator of Γgfk [ϕ ;Φ̄] in the usual way, i.e.
[ ] ( [ ] )
Hess Γgf[ϕ ;Φ̄] (v,w) =G w , Ĥess Γgf[h; ḡ] (v) +Igfϕ k ϕ k [v,w] (6.4)
By a comparison of eq. (6.3) and eq. (6.4) we obtain the matrix form in field space coordinates:
 
gh
[ ] Ĥesshh 0 0
Ĥessϕ Γ
gf
k [ϕ ;Φ̄] ≡
gf
ū ·k  0 0 0
0 0 0
Obviously, this operator on its own is not invertible for its vanishing in the ghost sector. The
only non-vanishing contribution is a pure background field dependent L2-operator given by
( )
gf A ( )A
Ĥess ∗ −1 C µν Ahh ≡ (F [ḡ]◦F[ḡ])B ≡−gBC Fµ g Fν •
B
Once we have collected all the contributions to the Hessian, we are going to consider a very
gh
important class of gauge fixing conditions. Then, the explicit form of Ĥess gfhh and I [v,w] for
these choices will be inserted.
6.3 Hessian of the ghost functional
In order to extract the superfluous information of the gravitational functional that is due to
an over-counting of gauge equivalent fields we had to add the ghost action which couples the
gravitational fluctuation field with the corresponding fluctuations of the ghost sector. Using
178 Chapter 6. The Hessian operator
the metric on field space this final contribution of Faddeev-Popov method can be cast into the
following form:
√ ( )
Γghk [Φ,Φ̄] =− 2
gh
ūk G ϕξ̄ · êξ̄ , M[g, ḡ](ϕξ ) (6.5)
The appearing Faddeev-Popov operator, M[g, ḡ], considered as a function on full field space, is
based on the not yet specified gauge fixing condition and the infinitesimal gauge transforma-
tions:
( )
M[g, ḡ](ξ ) = F[ḡ] Lξg · êξ̄ (6.6)
Notice however that we assume a linear, local gauge fixing condition. This term is the only
source of graviton-ghost interactions and as such will generate the ghost sector on the RHS of
the flow equation.
For the derivation of the Hessian of Γghk [Φ,Φ̄] we can use its linearity w.r.t. every field,
yielding three contributions which we consider separately in what follows:
[ ] √ ( )
ΓghHessϕ k [Φ,Φ̄] (
gh
v,w) =− 2ūk G dvϕξ̄ · êξ̄ , dwM[g, ḡ](ξ ) ‖ =̂ NI(v,w)√ ( )
− 2 ghūk G dwϕξ̄ · êξ̄ , dvM[g, ḡ](ξ ) ‖ =̂ NI(w,v)√ ( )
− 2 ghūk G ϕξ̄ · ê 1ξ̄ , 2 (dvdw+dwdv)M[g, ḡ](ξ ) ‖ =̂ NII(v,w)
Due to the linear structure, the Hessian will contain only off-diagonal terms and furthermore
both NI and NII involve derivatives of the Faddeev-Popov operator. Since F[ḡ] is independent
on the fluctuation fields and linear, the functional variation of M[g, ḡ](ξ ) affects only the Lie
derivative:
( )
dvM[g, ḡ](ξ ) = F[ḡ] dvLξg · êξ̄
At this point the ghost field ξ and the dynamical metric g appear as different constituents in
the Lie derivative. Again linearity in both arguments allows the following expression of the
functional variation of Lξg:
( )
dv Lξg = Lξd( ) v
g+Lv hξ
dwdv Lξh = Lξdwdvg+Lw d g+L d gξ v vξ w
For a symmetric two-tensor, as encountered for g, the Lie derivative can be written using the
covariant derivative of the background metric ḡ:
( )
µ µ µ
Lξg = ξ ∂µgρσ +gµρ∂σ ξ +gµσ ∂ρξρσ
= ξ µD̄ µ µµgρσ +gµρD̄σ ξ +gµσ D̄ρξ
Hereby, we substituted D̄µgρσ = ∂ λ λµgρσ − Γ̄µρ gλσ − Γ̄µσ gλρ in the last step and notice that
all connection coefficients Γ̄ cancel each other. It is now convenient to express the functional
variation of the Lie derivative using its explicit action on the fluctuation field leading to1
( )
dv Lξg = ξ
µD̄µ (dvg)ρσ +(d
µ
vg)µρD̄σ ξ +(dvg)
µ
ρσ µσ
D̄ρξ
µ µ µ
+ v D̄µgρσ +gµρD̄σv +gµσ D̄ρv (6.7a)
( ) ξ ξ ξµ µ µ
dwdv Lξg = w D̄µ (dvg)ρσ +(d g) D̄ w +(d g)ρσ ξ v µρ σ ξ v µσ D̄ρwξ
µ µ µ
+ vξ D̄µ(dwg)ρσ +(dwg)µρD̄σvξ +(dwg)µσ D̄ρvξ (6.7b)
1In the following we omit contributions of dwdvg. While in the linear parametrization employed later on this
terms identically vanish, in a geometric parametrization those terms contribute, however only to the ghost sector.
6.3 Hessian of the ghost functional 179
This information is sufficient to deduce the Hessian of the ghost functional and as mentioned
above we will proceed in three steps. The main effort originates from rewriting the second
va(riation of Γ
gh
[k [Φ,Φ̄] in]to a)symmetric expression suitable to read off the Hessian operator, i.e.
gh
G w , Ĥessϕ Γk [Φ,Φ̄] (v) . Due to the non-empty boundary of the manifold there will be ob-
structions in form of boundary contributions that we have to get rid off by imposing appropriate
constraints on field space.
6.3.1 Hessian contribution of NI(v,w)
The first two termsNI(v,w) andNI(w,v) involve field variations for the anti-ghost field ξ̄ . In the
former case, the one we consider first, we have to revert the order of v and w using the chain
rule and the assumption of FAν commuting with ḡ:
√ ∫ √ ( )
(− 2 ghū )−1k NI(v,w) ≡ ddx ḡ (vξ̄ ) ḡµνµ Fν dwLξg
∫M √ ( ( ) )
= ddx ḡ FAν (vξ̄ )µ ḡ
µν dwLξg ‖ =̂NIII(v,w)A
M∫ √ ( ) ( )
+ ddx ḡ FAν ḡ
µν(vξ̄ )µ dw Lξg A
M
In most cases NIII(v,w) can be transferred into a boundary term and thus introduces some ad-
ditional constraints on field space. Once we specify the gauge fixing condition we are going
to evaluate this piece of contribution. So far, let us focus on the second part and insert the
functional variation of the Lie derivative from eq. (6.7a) that yields
√
(− 2 ghūk )−1NI(v,w)−NIII(v,w)∫ √ ( ) [ ]ρσ
= ddx ḡ Fν ḡ
µν(v ) λξ̄ µ ξ D̄λ (dwg)ρσ +(d
λ
wg)λρ D̄σ ξ +(dwg)λσ D̄
λ
ρξ
∫ M √ ( ) [ ]ρσ
+ ddx ḡ F ḡµνν (vξ̄ ) w
λ λ λ
µ ξ D̄λgρσ +gλρD̄σwξ +gλσ D̄ρwξ
M
To establish the required order of the variation fields, we have to integrate by parts contributions
containing covariant derivatives of w. Basically, these are of two types: the first affects the
gravitational sector and assumes the form:
∫ √ ( ) ∫ ( )
dd ρσx ḡ F ḡµν(v ) ξ λ D̄ (d g) = dd−
√
1 ρσ µν λ
ν ξ̄ µ λ w ρσ x H nλ Fν ḡ (vξ̄ )µ ξ (dwg)ρσ
M ∫∂M √ ( ) ( )− dd ρσx ḡ(d λ µνwg)ρσ D̄λ ξ Fν ḡ (vξ̄ )µ
∫M √ ( )− dd ρσx ḡ (dwg)ρσ ξ λ Fν ḡµνD̄λ (vξ̄ )µ
M
The second type consists of covariant derivatives acting on the ghost component ξ which has to
be converted using integration by parts:
∫ √ ( ) ∫ ( )
dd ρσ
√
x ḡ F ḡµν(v ) g D̄ wλ
ρσ
= dd−1x H n F µν λν ξ̄ µ λρ σ ξ σ ν ḡ (vξ̄ )µ gλρwξ
M ∂∫M √ ( ) ( )
− dd ρσx ḡ wλ D̄ g µνξ σ λρ Fν ḡ (vξ̄ )µ
∫M √ ( )− dd ρσx ḡ wλ µνξ gλρFν ḡ D̄σ (vξ̄ )µ
M
Notice that in both cases we interchanged anti-commuting fields, vξ̄ with ξ or wξ , producing
the minus sign in the bulk contributions. In addition, due to integration by parts we encounter
further boundary terms, which in general obstruct the construction of a Hessian operator.
180 Chapter 6. The Hessian operator
Combining the previous results we carefully reorder all integrals such that we obtain the
desired form by introducing additional signs whenever anti-commuting fields are swapped. The
final result forNI(v,w) including boundary contributions affects two different entries in the field
space matrix of a possible Hessian operator, namely the off-diagonal ξ -ξ̄ and h-ξ̄ entries:
√
(− 2 ghū )−1k NI(v,w)−N∫ { III
(v,w)
√ ( ) }
=− dd ρσx ḡ wλξ D̄λg + D̄ g µνρσ σ λρ + D̄ρgλσ +gλρ D̄σ +gλσ D̄ρ Fν ḡ (vξ̄ )
∫ M µ
− √
{( ( )) }
dd β β β ρσx ḡ (d g) D̄ ξ αδ + D̄ ξ αw αβ σ ρ ρ δσ +δ
α λ λ µν
ρ δσ D̄λ ξ +ξ D̄λ Fν ḡ (vξ̄ )
∫M √ ( )( )
µ
+ dd−1 ρσx H F ḡµνν (v )
λ
ξ̄ µ nσgλρ wξ +nρg
λ λ
λσ wξ +nλ ξ (dwg)ρσ
∂M
While the first contribution, the one appearing in the ξ -ξ̄ component, does not contain any
Ghost- or Anti-ghost fields, the second contribution is linear in ξ . Hence, it is a possible source
of ghost-functionals on the RHS of the FRGE.
6.3.2 Hessian contribution of NI(w,v)
The next piece in the derivation of the Hessian operator is actually already cast in the right form
and does not involve any further manipulation. It suffices to insert the explicit form of eq. (6.7a)
into NI(w,v) which leads to:
√ ∫ √ ( )
(− 2 ghū −1k ) NI(w,v) ≡ ddx ḡ (wξ̄ ) ḡµνµ Fν dvLξg
∫ M√ { ( )}
dd ρσ β β β= x ḡ (w ) ḡµνF λ α α αξ̄ µ ν ξ D̄λ δρ δσ + D̄σ ξ δρ + D̄ρξ δσ (dvg)αβ
∫M √ { ρσ ( )}
+ ddx ḡ (wξ̄ )
µν
µ ḡ Fν D̄λg
λ
ρσ +gλρ D̄σ +gλσ D̄ρ vξ
M
Notice that there is no boundary term appearing. Furthermore, the above contributions populate
the reverted entries in the field space matrix of the Hessian operator, i.e. the components ξ̄ -h
and ξ̄ -ξ , respectively. While the latter is independent on the ghost fields, the former explicitly
depends linearly on ξ and affects the ghost sector in the RG flow.
6.3.3 Hessian contribution of NII(v,w)
Finally, we turn our attention to the last ingredient NII(v,w) which is obviously a linear func-
tional in ξ̄ . Since in our final truncation we ignore the running of the ghost sector NII(v,w) is
actually projected to zero in the final result. Nevertheless it is worth going through the details
of its derivation to pave the ground for later enlarged truncations.
From the second variation of the Lie derivative of eq. (6.7b), which is central in the present
contribution, we observe that the result is already symmetric in v and w. Thus, it suffices to
consider only dwdv-terms instead of their symmetrized versions. Applying the chain rule and
again using the commutativity of Fν with ḡ we obtain:
√ ∫
(− 2 ghū )−
√ ( )1N d µνk II(v,w) ≡ d x ḡ ξ̄µ ḡ Fν dwdvLξg
∫M √ ( ( ) )
= ddx ḡ FAν ξ̄µ ḡ
µνdwdv Lξg ‖ =̂NIV(v,w)A
M∫ √ ( ) ( )
+ ddx ḡ FAν ḡ
µν ξ̄µ dwdv Lξg A
M
6.3 Hessian of the ghost functional 181
Usually, a choice for the gauge fixing condition transforms( the fi)rst summand NIV(v,w) into a
boundary contribution. Inserting the explicit form for dwdv Lξg of eq. (6.7b) we find
√
(− 2 ghūk )−1N∫ II(v,w)−NIV(v,w)√ ( )[ ]
dd ρσ= x ḡ F ḡµνν ξ̄µ w
λ
ξ D̄λ (dvg)ρσ +(dwg)λρD̄σv
λ
ξ +(dwg)λσ D̄
λ
ρvξ
∫ M √ ρσ ( )[ ]
+ ddx ḡ F µν λν ḡ ξ̄µ vξ D̄λ (dwg)ρσ +(dvg) D̄ w
λ
λρ σ ξ +(d
λ
vg)λσ D̄ρwξ
M
In the second integral all terms appear in the non-desired order, thus we have to revert it using in-
tegration by parts. Concerning the components in the metric sector we can derive the following
identity
∫ √ ∫
dd ρσ
( ) √ ρσ ( )
x ḡ Fν ḡ
µν ξ̄ vλ d−1 µν λµ ξ D̄λ (dwg)ρσ = d x H nλ Fν ḡ ξ̄µ vξ (dwg)ρσ
M ∂∫M √ ( )
+ dd ρσx ḡ (d g) µν λw ρσ Fν ḡ D̄λ ξ̄µ vξ
∫M √ ( )
+ dd ρσx ḡ (d g) F ḡµνw ρσ ν ξ̄ D̄ v
λ
µ λ ξ
M
Besides a boundary contribution we have two bulk terms whereby no need of interchanging
anti-commuting fields was necessary. On the other hand, for the remaining terms a reordering
of the ghost variations will generate additional signs yielding
∫ √ ρσ ( ) ∫ √ ( )ddx ḡ F ḡµν ξ̄ (d g) D̄ wλ d−1 ρσ µν λν µ v λρ σ ξ = d x H nσ Fν ḡ ξ̄µ (dvg)λρ wξ
M ∂∫M
− √ ρσ
( )
ddx ḡ wλξ F ḡ
µν
ν D̄σ ξ̄µ (dvg)λρ
∫M √ ( )
− ddx ḡ wλ ρσξ Fν ḡµν ξ̄µ D̄σ (dvg)λρ
M
This identity has to be substituted twice into the previous result giving rise to two additional
boundary contributions. Thus, finally we arrive at the overall result for the last ingredient of the
Hessian:
√
(− 2 ghū −1k ) N∫ II(v,w)−N (v,w)√ {(
IV ) ( )
=− ddx ḡ wλ β β ρσξ δ δ σ σ µνρ γ +δρ δγ Fν ḡ D̄γ ξ̄ δ αµ λ
M ( ) }
ρσ ( )
+F ḡµν ξ̄ D̄ δ α
β
δ + D̄ δ α
β β
ν µ λ ρ σ σ ρ δλ + D̄ρδ
α
λ δσ (dvg)αβ
∫ √ { αβ ( )
+ ddx ḡ (dwg) F ḡ
µν
αβ ν D̄λ ξ̄µ
M ( )}
ρσ ( )
+F ḡµν ξ̄ δ α
β α β α βδ λν µ λ ρ D̄σ +δλ δσ D̄ρ +δσ δρ D̄λ vξ
∫ ( )
+ dd−
√ ρσ ( )1x H F ḡµνν ξ̄ λ λ λµ nσ (dvg)λρ wξ +nρ(dvg)λσ wξ +nλvξ (dwg)ρσ
∂M
It fills the up to now empty off-diagonal entries in the field space matrix of the Hessian operator
and introduces some more boundary terms that have to vanish in its combined form.
6.3.4 Hessian for the ghost functional
The ghost functional is a vital piece of the effective action. Together with the gauge fixing
condition, it ensures that the redundancies in the description are canceled and condense the
182 Chapter 6. The Hessian operator
theory to the physical content. Hence, it is also an important ingredient in the effective action
though its general form may vary significantly from its ‘bare’ counterpart. Quite general, the
Hessian of the ghost functional Γghk [Φ,Φ̄] is given by
[ ] ( [ ] )
gh gh
Hessϕ Γk [Φ,Φ̄] (v,w) =G w , Ĥessϕ Γk [Φ,Φ̄] (v) +I
gh[v,w]
Besides the formal Hessian operator there are usually boundary contributions, here subsumed
into Igh[v,w]. Only for suitable boundary conditions for the fluctuation fields we can actually
arrange for a well-behaved Hessian operator. For linear gauge fixing conditions, the associated
field space matrix contains only off-diagonal entries denoted by
 
gh gh
[ ] √  0 Ĥesshξ Ĥesshξ̄
gh gh gh gh

Ĥess  ϕ Γk [Φ,Φ̄] ≡− 2ūk ·Ĥessξh 0 Ĥessξ ξ̄
gh gh
Ĥessξ̄h Ĥessξ̄ ξ 0
For our later purpose only the ghost block turns out be relevant for the respective operators
are ghost field independent and thus generate terms on the RHS of the FRGE that affect the
gravitational or gauge fixing terms only. These two relevant contributions can be read off to be
gh ( )
(Ĥess µξ ξ̄ ) =−
λ ρσ
(g−1) D̄ µν
ξ̄ξ • λ
gρσ + D̄σgλρ + D̄ρgλσ +gλρD̄σ +gλσ D̄ρ Fν ḡ (6.8a)
{
gh ( )}
(Ĥess ) = +(g−1
•
) ḡµν
ρσ
ξ̄ξ λ Fν D̄λgρσ +gλρ D̄σ +gλσ D̄ρ (6.8b)ξ ξ̄ µ
The remaining terms which interrelate metric and ghost variations are linear in either ξ or ξ̄ and
hence are ultimately projected to zero in the final truncation. Nevertheless, we list them here
for the linear parametrization dvg= v:
{( )
gh λ β β ρσ ( )
(Ĥess αβ −1 σ σ µν αξh) =−(g ) δ δ +δhξ • ρ γ ρ δγ Fν ḡ D̄γ ξ̄µ δλ (6.9a)
( )( )}ρσ
+F ḡµν α
β β β
ν ξ̄µ D̄λ δρ δσ + D̄σ δ
α
ρ δλ + D̄ρδ
α
λ δσ
gh • { αβ ( )
(Ĥesshξ )λ =+(g
−1
ξ ) F ḡ
µν
ν D̄λ ξ̄µ (6.9b)h αβ
ρσ ( )( )}β β β
+Fν ḡ
µν ξ̄µ δ
α
λ δρ D̄σ +δ
αδ αλ σ D̄ρ +δσ δρ D̄λ
{(
gh µ β β(Ĥesshξ̄ ) =−(g−1) D̄σ ξ αδ αρ + D̄ρξ δσ (6.9c)ξ̄h •αβ ( )) }
β
+δ αδ D̄ ξ λ λ
ρσ
+ξ D̄ F ḡµνρ σ λ λ ν
{ ( )}
gh ρσ β β β
(Ĥessξ̄h)
αβ =+(g−1) ḡµνF ξ λν D̄λ δ
α
ρ δσ + D̄ ξ
αδ + D̄ ξ αδ (6.9d)
hξ̄ • σ ρ ρ σµ
The next step would be to get rid of the boundary contributions by somehow constraining
the fluctuation fields. The total obstruction to a Hessian operator of Γghk [Φ,Φ̄] is absorbed into
the following functional:
√
(− 2 ghūk )−1Igh[v,w]∫ √ ( ( ) ( ) )≡ ddx ḡ FA ξ̄ ḡµν µνν µ dwdv Lξg +(v ) ḡ d L gA ξ̄ µ w ξ A
∫ M √ ( )( )
+ dd−1 ρσx H F ḡµνν (vξ̄ )µ nσg
λ
λρ wξ +nρgλσ w
λ
ξ +nλ ξ
λ (dwg)ρσ
∫∂M √ ( )( )
dd−1 ρσ+ x H F ḡµνν ξ̄µ nσ (dvg)λρ w
λ
ξ +nρ(dvg) w
λ +n vλλσ ξ λ ξ (dwg)ρσ
∂M
6.4 Hessian of the gravitational functional 183
For the last two integrals Dirichlet boundary conditions are sufficient for their vanishing, how-
ever for the first integral the explicit form of the gauge fixing condition is crucial. We will come
back to this issue when having the full Hessian at our disposal.
6.4 Hessian of the gravitational functional
Except for the ghost functional, in all previous derivations of Hessian operators, the underlying
functionals were defined in terms of the fluctuation fields, i.e. elements of TF. For the gravi-
tational contribution the full dynamical fields Φ are generally used to define the corresponding
functional Γgravk [Φ,ϕ ]. This approach is very suitable to interpret the chosen truncation for
the appearing invariants a well studied geometrical quantities in classical gravitational theory.
However, in the light of deducing the Hessian it has the main disadvantage that the functional
variation is based on geodesics on field space. Due to the chain rule, the occurring complica-
tions of this geodesic approach can be absorbed into the variation of the dynamical fields only.
In the present context, the ghost part of field space is trivial and only the metric field content
requires this involved technique, hence
dvΦ = dv (g,η , η̄)
T = (dv g,v
T
ξ ,vξ̄ ) ∈ TΦFh
Notice that v ∈ TΦF is a tangent vector at Φ contrary to the usual case of being in the tangent
space of Φ̄. While the geometric construction of dv g based on geodesics through g retainsh
the meaning or interpretation of Γgravk [Φ,ϕ ] different prescription for dv g give rise to a differ-h
ent truncation ansatz. Hence, the interpretation of Γgravk [Φ,ϕ ] in terms of the Einstein-Hilbert
functional is only approximately true, while the considered truncation is a valid independent on
this choice. Nevertheless, once we leave the realm of truncations or aim for a more systematic
treatment of the latter, it is crucial to rely on the geometrical formalism.
In this section we start very generally and derive the Hessian for arbitrary boundary con-
ditions and choices of d 2v g. This will allow future work to draw on at this point by slightlyh
changing one or the other assumption and test the sensitivity of the derived results on these
choices. However, the present context, we are ultimately going to simplify the structure of the
Hessian employing a linear geodesic flow, i.e. we use dv g≡ vh h.
In the present truncation we consider only invariants that are either constructed by the back-
ground or the dynamical metric. In the general case, on the level of the effective action, merged
monomials may appear of which there is an infinite number. Due to the complexity of the
FRGE we focus here on only two Einstein-Hilbert like monomials with respective Gibbons-
Hawking-York like terms and cosmological volume elements:
Γgravk [Φ,Φ̄]≡ Γ
grav grav
k [g]+Γk [ḡ] (6.10)
= (ūDk λ̄
D) ·Γgrav Dk ( ) [g]+ (ūk ) · Γ
grav[g]+ (ū∂D λ̄ ∂D) ·Γgrav [g]+ (ū∂D( ) k k (∂ ) k ) · Γ
grav
(∂ )[g]A B A B
+( B λ̄ B) ·Γgrav[ ]+ ( B) · Γgrav[ ]+ ( ∂B λ̄ ∂B) ·Γgrav [ ]+ ( ∂B) · Γgravūk k (A) ḡ ūk ( ḡ ū ḡ ū [ḡ]B) k k (∂A) k (∂B)
Notice, that this truncation consists of two equivalent contributions of background and dynam-
ical metric, however each factored with a different coefficient. Hereby, the following abbrevia-
2Except for omitting terms dw dv g in the ghost functional, which anyway are quadratic in the ghost fields.h h
184 Chapter 6. The Hessian operator
tions are used
∫
Γgrav d
√
( ) [g]≡+2 · d x g (6.11a)A
∫M
− √Γgrav(∂ )[g]≡+2 · d
d 1x H (6.11b)
A
∫ ∂M
Γgrav d
√
( ) [g]≡− d x g R(g) (6.11c)B
M∫ √
Γgrav(∂ )[g]≡−2 · d
d−1x H K(H) (6.11d)
B
∂M
Another very important remark concerns the bulk-boundary matching which is usually intro-
duced on the classical level to assure a well-behaved variational principle. In the present ansatz
we explicitly distinguish the coefficients of Einstein-Hilbert and Gibbons-Hawking-York term
in order to obtain their RG evolution and study whether or not the classical correspondence can
be recovered on the effective quantum level. Therefore, we have to deduce the Hessian operator
of Γgravk [Φ,Φ̄] which by construction contributes only in the metric sector of field space. The
background terms do not affect the RHS of the FRGE for it vanishes under functional variation
w.r.t. the dynamical fields. We thus proceed by evaluating the Hessians for each term of eq.
(6.11) separately. After combining these intermediate results to form the complete Hessian of
the gravitational functional, we also state the principal of variation for the present truncation
ansatz.
6.4.1 Hessian for the volume term
The cosmological constant in the classical theory factors the volume element of the dynamical
metric. In our notation it corresponds to Γgrav( ) [g] where the entire field dependence is containedA
in the square root of the determinant of g. Its functional variation can be translated into a
variation of (dv g) that we are going to replace only in the very end:h
√ ∫
d g= 1
√ µν √
v 2 gg (dv g)µν =⇒ dv (Γ
grav[g]) = + ddx ggµν(d g)
h h h (A) vh µν
M
The second variation involves the functional derivative of the inverse metric which follows form
the identity µδ = gµρν gρν :
( )µν
dv g
−1 ≡ dv gµν =−gρµgσν (dv g)h h h ρσ
√
The Leibniz-rule applied to dv g results inh
√ { √ ( )}
d d g= (d g) − 1 g gρµgσν − 1gρσ µν √w v w ρσ 2 2 g (dv g) 1µν − 2 gg
µν(d
h h h h w d g)h vh µν
The term in brackets (· · · ) is the standard form of the DeWitt field space metric with ϖ =−1/2.
In the generic – geometrical – way the field variation (dv g) is still dependent on g and henceh
(dw dv g) =6 0. However, later we will project the preliminary ansatz using (dv g) = vh andh h h
(dw dv g) = 0 onto the final truncation which simplifies the preceding calculations.h h
Combining the above results gives rise to the Hessian of the volume contribution in the
gravitational sector:
[ ] ∫ √ {( )}
Hess Γgravϕ ( ) [g] (v,w) =− d
dx g(d g) gρµw ρσ g
σν − 12gρσgµν (d g)A h vh µν
∫ M ( )
− dd √x g gµν 12 (dw dv +dv dw )gh h h h µν
M
6.4 Hessian of the gravitational functional 185
As long as (dv g) does not involve any derivative operators there are in fact no obstructions inh
reading off the associated Hessian operator:
[ ] ( [ ] )
Γgrav[ ] ( , ) = ΓgravHessϕ ( ) g v w G w , Ĥessϕ ( ) [g] (v)A A
Notice that for ϖ = −1/2 the first term in the Hessian operator is proportional to the identity
in the metric sector and vanishes elsewhere. In the current state of the analysis we still have to
specific (dv g) by means of a field space connection which we however postpone to the veryh
end of the gravitational derivation part.
6.4.2 Hessian for the boundary volume term
Due to the smoothness of the Riemannian metric and the manifold character of the boundary
∂M, one can equip ∂M with a Riemannian metric that is induced by g (or ḡ) and defines a
symmetric tensor field in d−1 dimensions. The boundary of a manifold brings in another piece
of information into the spacetime formalism usually presented in form of a hyperplane equation
in some embedding space. Here, we skip all intermediate steps and only work with an outward
pointing normal vector field n (n̄) which should be normalized w.r.t. the dynamical (background)
loc.
metric g (or ḡ), i.e. 1 = g(n,n) = nµg νµνn . Rather than working with the induced metric
directly, we introduce a projection operator for all p ∈ ∂M that projects onto the boundary’s
tangent hyperplane:
loc. µ ( µ )
Π∂M : TpM → Tp∂M, v 7→ Π ν µ ν∂M(v) = (Π∂M)ν v ∂µ ≡ δν −n nν v ∂µ
Hereby, we lower indices with the respective metric field, in the present case the dynamical one:
nν = g
µ
ν µn . The induced metric can thus be related to a projected version of g by means of the
pullback, which locally agrees with subtracting the normal directions of vector fields:
H ..= Π ∗(g) locally: H ≡ g −n n ∧ Hµν ≡ gµν −nµnν∂M µν µν µ ν
Notice that this indeed defines a suitable metric only on the boundary, for it is degenerate in the
loc.
perpendicular direction in the sense that H(n,•) = 0 = H nµµν vanishes.
The boundary volume term, abbreviated Γgrav(∂ )[g], is the analog of the usual bulk volumeA
contribution now projected onto ∂M. It involves the determinant of the projected metric H
and hence its field derivative requires the variation of the normal vector field. To this end, we
exploit the properties of n that should be kept intact after the variation procedure, in particular
its normalization:
0= (d 1) = (d nµn ) = n d nµ µv v µ µ v +n dh h h v nh µ
This provides a mean to interchange the variation with respect to n and its dual object g(n,•),
but still does not associate one or the other with the variation of the metric field. Therefore,
we start with the first summand nµd µv n and artificially convert it to depend only on the dualh
g(n,•):
( )
nµdv n
µ = nµ(d (g
µν
v nν)) = nµ −nνgρµgσν(dv g) +gµνρσ dh h h v nh ν
=−nσnρ(d g) +nµv ρσ dv nh h µ
=−nσnρ(d µv g)h ρσ −nµdv nh
In the final step we inserted the above derived relation n µ µµdv n = −n dv nµ . Hence, rearrang-h h
ing the last equality yields the field variation of the normal vector field n and thus its dual
186 Chapter 6. The Hessian operator
correspondence g(n,•):
(d n)µ ≡ d nµ =− 1nµv v nσnρ(d g)h h 2 vh σρ
(dv g(n,•))µ ≡ dv nµ = 12nµn
σnρ(d
h h v g)h σρ
For the projected boundary metric H we only have to substitute the results for the variation
loc.
of g and g(n,•) successively and likewise for its dual version H∗ = Hµν:3
(dv H)µν ≡ dv Hµν = (dvg)µν − (dvnµ)nν −nµ(dvnh h ν)
= (dv g)
σ ρ
h µν −nνnµn n (dv g)h σρ
(d H∗)µνv ≡ dv Hµν = d gµν − (d nµ)nν −nµ νh h v v (dvn )
=−gµρgνσ (dv g)σρ +nνnµnσnρ(dh v g)h σρ
The central geometrical object of interest in the currently considered invariant is determinant
of H . Taking the functional derivative amounts to a variation of the inverse boundary metric H
and thus yields
√ √ √ ( )
d H =− 1 HH d Hµν =− 1v 2 µν v 2 HH −g
µρgνσµν (dv g)σρ +n
νnµnσnρ(d g)
√ √ h vh σρ
=+ 12 HH g
µρgνσµν (dv g)σρ ≡+ 1 µν2 HH (dv g)h h µν
In the last step we used the transverse property of H , i.e. H νµνn = 0, hence only the first
summand remains. The volume boundary invariant adds the following contribution to the first
variation of the preliminary truncation ansatz:
∫ √
( gravdv Γ(∂ )[g]) = + d
d−1x H Hµν(dv g)h A h µν
∂M
In combination with all other gravitational terms it has to cancel whenever a well-behaved
variational principle is required to hold true.
√ { √ ( )}
d d H = (d 1 1 ρσ µν µρ νσ ν µ σ ρw v w g)σρ 2 H 2H H −g g +n n n n (dh v g)h µν√
+ 12 HH
µν(dw dv g)h h µν
Again, in the case of a true geometrical variation the last term in general produces a non-
vanishing contribution, except for special choices of boundary conditions. It is very instructive
to rewrite the first expression of this result using Hµν = gµν −nµnν :
√ { √ ( ( ) )}
d d H = (d g) 1 H − Hµρ νσ 1 ρσ µν µρ ν σ νσ ρw v w σρ 2 H − 2H H −H n n −H n n
µ (dv g)h h µν
√
+ 12 HH
µν(dw dv g)h h µν
What appears is the gravitational part of the field space metric for ϖ = −1/2 projected onto
the boundary with two additional contributions in the orthogonal direction. In conclusion, the
Hessian of the boundary volume invariant assumes the form:
[ ] ∫ √ {( )
Hess grav d−1 µρ νσ 1 ρσ µνϕ Γ(∂ [g] (v,w) =− d x H (dA) w g)σρ H H −h 2H H
∂M }
+Hµρnνnσ +Hνσnρnµ (dv g)h µν
∫ √ ( )
− dd−1x H Hµν 12 (dw d +d d )gh vh vh wh µν
∂M
3 Notice thatH∗ is not the inverse ofH in the general sense. In fact H as such has no inverse due to its degeneracy,
however restricted to the boundary (thus considered as the induced metric) the induced form of H∗ coincides with
the inverse of the induced form of H.
6.4 Hessian of the gravitational functional 187
In contrast to the bulk volume term the second variation is a purely boundary contribution
as expected. While there are techniques to study generalized Hessian operators that contain
bulk-boundary terms [119, 120] it is not clear how these methods translate into the FRG set-
ting. Rather, we require that all boundary terms have to vanish, either by suitable boundary
conditions or by matching certain coefficients, as in the classical Gibbons-Hawking-York case.
However, the latter method may be incompatible with the RG evolution and thus should be
omitted whenever possible.
6.4.3 Hessian for the curvature term
When curvature enters the game, derivatives appear which are a potential source of boundary
terms in the Hessian of bulk invariants. Besides its appearance as a∫central ingredient of General
Relativity in the shape of the Einstein-Hilbert action, Γgrav
√
[g] =− dM d x g R(g) is also special(B)
for it is the only bulk invariant containing two derivatives D, assuming d≥ 2. In our conventions
the scalar curvature is the full contraction of the Riemannian tensor in local coordinates given
by
R σ =−∂ Γσ +∂ Γσ +Γκ Γσ −Γκ Γσµνλ µ νλ ν µλ µλ νκ νλ µκ ,
Rµν = δ
ν
σR
σ
µνλ and R= g
µνRµν
Comparing different literature on General Relativity or on its quantum versions exhibits various
sign conventions that may alter certain results, so one has to be careful in this process.
Based on the above form of the scalar curvature, the Ricci tensor, and the Riemann tensor
we now deduce the Hessian of the Einstein-Hilbert functional in the presence of boundaries.
The crucial constituent is the covariant derivative Dwhich builds up these second derivative ten-
sors. Under field variation the partial derivative is unaffected, while the connection coefficients4
transform as
( )
d Γλ
ρ
v µν =
1 gλρ δ σD λρ σ λκ σ
h 2 ν µ +g δµ Dν −g δµ δν Dκ (dv g)h σρ
This result suffices to arrive at the functional variation of the Riemann tensor which can be
directly contracted to the respective form for the Ricci tensor. Finally, the behavior of the scalar
curvature under infinitesimal changes of the metric is deduced from the Leibniz-rule acting on
gµνRµν ≡ R, i.e.
( ( ) )
d R = 1 Dµ δ νD +δ νD −gµν µ νv ρσ 2 σ ρ ρ σ DρDσ −δρ δσ D
2 (d
h v g)h µν
( )
d R= −Rµν +DµDν −gµνD2v (dv g)h h µν
√
In combination with the outcome of the metric variation of g we derived previously the first
order field derivative of the Einstein-Hilbert functional is given by
∫ ( )
( Γgrav
√
dv ( ) [g]) =− d
dx g 1 µν µν2g R−R +D
µDν −gµνD2 (d
h B v
g)
h µν
∫M √ ( )
=+ ddx g Rµν − 1 µν2g R (dv g)h µν
∫M √ ( )
− dd−1x H nµDν −gµνnρDρ (dv g)h µν
∂M
In the last step we converted the total derivatives into some boundary contributions. The re-
maining bulk term constitutes the Einstein tensor Gµν ≡ Rµν − 1 µν2g R and is one of the central
4In the present case we have a torsion free connection, for which the connection coefficients coincides with the
Christoffel symbols. The entire derivation is based on this assumption.
188 Chapter 6. The Hessian operator
objects in the field equations for General Relativity. The obstruction is exactly the boundary
term that in a classical theory is usually canceled by introducing the Gibbons-Hawking-York
functional with an appropriate coefficient. We come back to this point in the end of this section
after deriving the full Hessian of Γgravk [Φ,Φ̄].
The second variation of Γgrav( ) [g] that contributes to the RHS of the FRGE is treated in twoB
steps, the first focusing on the bulk, the second on the boundary term:
∫ √ ( )
( gravd d µν 1w dv Γ( ) [g]) = +dw d x g R − 2g
µνR (dv g)µν ‖ =̂ NV(v,w)h h B h h
∫M √ ( )
−d d−1w d x H nµDν −gµνnρDρ (dv g)µν ‖ =̂ Nh h VI(v,w)
∂M
In fact, we have all the variations of the individual constituents at our disposal which – together
with the Leibniz-rule – yields the final result for the Hessian of the Einstein-Hilbert functional.
Though tedious it is straightforward to determine its two contributions from the previous func-
tional variations. An important intermediate result is the derivative of the Einstein tensor:
(d Gµν) = 1 {−Kµνρσ +(gµνRρσ +Rgνρgµσw 2 −2g
µρRνσ −2gνρRµσ)} (d
h w g)h ρσ
with Kµνρσ ≡ (gµνDρDσ +gρσDµDν −2gνσDρDµ)− (gµνgρσ −gµρgνσ )D2
Here we introduced the operatorK B≡ g−1KCBA AC , the propagator of the field modes. It is the only
source of not ultra-local contributions to the gravitational Hessian except for possible boundary
terms that we assume to vanish in the very end. Now, the full bulk variation splits into three
components, one which disappears whenever we rely on a flat field space connection:
∫ √ (√ √ )
NV(v,w) = + d
d −1x g g (dw g)G
µν +(d µνw G ) (dv g)h h h µν
∫M
+ dd
√
x g Gµν (dw dv g)h h µν
M
Substituting the variation of the determinant of g the first integral reduces to
∫ √ { }
NV(v,w) = + d
dx g (dv g)
1
µν 2 (−K
µνρσ + Uµνρσ) (dw g)h h ρσ
∫M
+ dd
√
x g Gµν (dw dv g)h h µν
M
whereby we condensed the notation using the following abbreviation for the completely ultra-
local (potential like) tensor
Uµνρσ(g)≡ gρσGµν +gµνRρσ +Rgνρgµσ −2gµρRνσ −2gνρRµσ (6.12)
In the Hessian the symmetric version of dw dv is relevant which requires both K and U beingh h
self-adjoint. W(hile for the u)ltra-local part this simply translates in considering the symmetric
combination 1 UAB2 + U
BA instead, for the ‘kinetic’ term additional boundary conditions are
in need. In detail we are looking for a way to express the symmetric sum NV(v,w)+NV(w,v)
as an operator relation giving rise to a suitable Hessian operator for the gravitational curvature
term:
∫ √ { }
(NV(v,w)+NV(w,v)) = + d
dx g (dv g)
1 (−Kµνρσ + Uµνρσµν 2 ) (dw g)h h ρσ
∫M
d √ { }+ d x g (d 1w g)ρσ 2 (−Kρσ µν + Uρσ µν) (dv g)h h µν
∫M ( )
+ dd
√
x g Gµν 12(dw dv +dh h v d )gh wh µν
M
6.4 Hessian of the gravitational functional 189
For a natural symmetric operator, we reorder the first term such that all the operators act on
(dv g). As mentioned above, for the ultra-local part this is trivial in that we only have to relabelh
the indices. Concerning Kµνρσ a stepwise derivation reveals the appearing of boundary terms.
First of all, consider the adjoint of the following integral using integration by parts twice
∫ √ ( ) ∫ √ ( )
ddx g (d αβ κ λv g)µν g D D (dw g)ρσ =+ d
dx g (d αβ λw g)ρσ g D D
κ (d
h h h v g)h µν
M ∫M
+ dd−
√
1x H gαβ nκ(dv g)µνD
λ (d
h w g)h ρσ
∫∂M
− √− dd 1x H gαβ nλ (d κw g)h ρσD (dv g)h µν
∂M
Notice that in the absence of spacetime boundaryKµνρσ is symmetric and under suitable bound-
ary conditions, as for instance Dirichlet constraints, it even holds if ∂M 6= 0/ .
∫ ∫
dd
√
x g (d g) KAB
√
v A (dw g)B =+ d
dx g (d BA
h h w g) K (dh B v g)h A + NVII(v,w)
M M
This last contribution, NVII(v,w), represents the boundary terms arising from integration by
parts, explicitly given by
∫ √
N (v,w) = + dd−1VII x H (d
µν ρ σ
v g)µν (g n D −2gνσnνDρ +gρσnνDµ)(dw g)h h ρσ
∫ ∂M
− √− dd 1x H (d µν σ ρ νσ ρ µw g)ρσ (g n D −2g n D +gρσnµDν)(d g)h vh µν
∫∂M (
+ dd−
√
1x H (gµνgρσ −gµρgνσ ) · (d λv g)µνn Dλ (dw g)h h ρσ
∂M )
− (dw g)ρσnλDh λ (dv g)h µν
We thus arrive at the final symmetric version of the gravitational bulk contribution to the Hes-
sian, which reads
( )
1
2NV(v,w)+
1
2NV(w,v)∫ √
=− 12 d
dx g (d g) {Kµνρσv µν }(dw g)h h ρσ
∫M
+ 1 dd
√
2 x g (dw g)
µνρσ
h ρσ {U }(dv g)h µν
∫M √ ( )
+ 1 ddx g Gµν 12 2(dw dv +dv dw )g +
1N
h h h h µν 2 VII
(v,w)
M
The missing part constitutes a pure boundary term containing the normal vector field n and
covariant derivatives acting on the metric fluctuation. For general Dirichlet boundary conditions
this term spoils the variational principle; it has to be canceled by either Neumann conditions or
by introducing the matched version of the Gibbons-Hawking-York term. Nevertheless, as we
will see in a moment, its second variation vanishe√s under either Dirichlet or Neumann bound-
ary constraints. Besides the field derivative of H, we have to determine the variation of
nσgαβDρ(dv g)µν , which is given byh
d (nσgαβw Dρ(dv g)
σ αβ
µν) = (dw n )g Dρ(dv g)
σ αβ
µν +n (dw g )Dρ(dv g)h h h h h h µν
+nσgαβ (dw Dρ(dv g)h ( h µν
)
)
=−nσ (d g) 1 λ κ αβ κα λβwh κλ 2n n g Dρ(dv g)µν +g g Dρ(dh v g)h µν
+nσgαβ (dw Dρ(dh v g) )h µν
190 Chapter 6. The Hessian operator
Hence it necessary to compute the variation of the covariant derivative acting on (dv g):h
( )
dw (Dρ(dv g)µν) = dw ∂
λ λ
h h h ρ(dv g)h µν −Γρµ(dv g) −Γh λν ρν(dv g)h µλ
= Dρ(dw dv g)
λ
µν − (dv g)λν(dw Γρµ)− (dv g)µλ (d Γλh h h h h wh ρν)
For brevity we are not going to further expand (d λw Γρµ) but notice that each of its terms is pro-h
portional to some Dκ(dw g), hence the corresponding boundary terms vanish whenever Dirich-h
let or Neumann conditions are imposed. The complete variation of the boundary contribution is
thus given by
∫ √ { ( )}
NVI(v,w) = + d
d−1x H (d g) 1Hρσ −nµDν +gµνnλw ρσ 2 Dλ (dv g)h h µν
∫ ∂M { ( )
+ dd−
√
1x H (dw g)
1 σ
ρσ 2n n
ρ nµDν −gµνnλD
h λ
∂M ( )}
+gρν nµDσ −gσ µnλDλ (dv g)h µν
∫ √ ( )( )
+ dd−1x H nλgµν −nµgνλ Dλ (dw dv g)µν −2(dv g)αν(dw Γαλ µ)h h h h
∂M
This concludes the derivation of the Hessian associated to the bulk curvature invariant. The
symmetric second variation of Γgrav( ) [g] readsB
[ ]
Hessϕ Γ
grav
( ) [g] (v,w)∫B
− 1 d √= 2 d x g (dv g)µν {K
µνρσ}(d
h w g)h ρσ
∫M
1 d √+ 2 d x g (dw g)ρσ {Uµνρσ}(dh v g)h µν
∫M √ ( )
+ 1 dd2 x g G
µν 1
2(dw dv +dv dw )g + NVI(v,w)+
1N
h h h h µν 2 VII
(v,w)
M
Hereby, a symmetrization of the boundary terms NVI(v,w) and NVII(v,w) is implicitly under-
stood but omitted for the sake of brevity. We are now going to derive the final piece in the
gravitational functional that on a classical level was introduced to get rid of the boundary terms
appearing in the construction of field equations for GR: the Gibbons-Hawking-York term.
6.4.4 Hessian for the boundary curvature term
∫ √
Finally, we arrive at the boundary curvature term Γgrav(∂ )[g] = −2 dd−1∂M x HK(H) of theB
present truncation, a Gibbons-Hawking-York like invariant including the trace of the extrin-
sic curvature K = D nµµ . There are various ways to derive the variation of this field monomial
all related by some algebraic transformations involving identities of the form nµnµ = 1 and
H νµνn = 0, for instance. For our purpose we will follow a path that directly yields the standard
shape of the functional derivative for K such that the resulting boundary contributions are com-
parable with those occurring in the derivation of the Einstein-Hilbert functional. To this end,
we present two of the previously obtained results in a new fashion that is more suitable for the
current analysis. First of all, we replace µnσnµ = δσ −Hσ µ in the variation of g(n,•) which
leads to
µ
(d 1 ν 1 λ µ νv nh σ ) = 2δσ n (dv g)µν +bσ with bσ ≡− 2gσλH n (dv g)h h µν
Here, we introduced a co-vector field b for reasons that will be obvious in the final result.
Moreover, we consider the contracted form of the variation of the Christoffel symbol w.r.t. g
6.4 Hessian of the gravitational functional 191
yielding
( )
gρσ (dv Γ
λ
ρσ ) = g
λ µDν − 1 µν λ2g D (dv gh h µν)
Both become relevant, when we insert an artificial inverse metric in K that lowers the indices of
the normal vector field such that the above equation applies:
(d K) = (d gρσD n ) =−gρµgσν(d g )D n +gρσv D (d ρσ λh vh ρ σ vh µν ρ σ ρ v n )−g nh σ λ (dv Γh ρσ )
While the first term reduces to the extrinsic curvature Kµν = Dµnν the second and third contri-
bution are expanded in terms of the above identities, i.e.
( ) ( )
(d K) =−Kµν(d g )+ 1Dµ nν(d g) +D bρ − nµDν − 1nλgµνv v µν 2 v µν ρ 2 Dλ (dh h h v g )h µν( )
=− 1 µν 1 µ ν λ µν ρ2K (dv gµν)− 2 n D −n g Dλ (dv g )+D bh h µν ρ
In the last equality we used the Leibniz rule and replaced Dµnν by the extrinsic curvature tensor
Kµν . The second term is exactly the boundary contribution that appears in the variation of
the Einstein-Hilbert functional however with opposite sign. Hence, adapting the coefficients of
Gibbons-Hawking-York and the Einstein-Hilbert functional cancels all the boundary integrals
containing derivatives of the metric fluctuations, implying that Dirichlet conditions are sufficient
to generate a well behaved variational principle. However, there is an additional term D bρρ
which seems to spoil this nice relation but it turns out to be purely of Dirichlet type with an
additional total derivative on the boundary. To see this explicitly, let us integrate this part over
the boundary manifold ∂M:
∫ ∫
dd−
√ √ (
1 µ
)
x H Dρb
ρ = dd−1x H ∂ ρρb +Γ
ρ
µρb
∂M ∫∂M ( ) ∫ ( )
= dd−
√ √ √
1x ∂ Hbρ − dd−1x H √1 bρρ ∂ρ H− µΓ ρH µρb∂M ∂M
The first term represents a total derivative and thus can be converted to a boundary integral over
∂∂M ≡ 0/ , thus vanishes√. For the√remaining integral notice that
µ
Γ = 1gµν√ µρ 2 ∂ρgµν holds true
and furthermore that ∂ 1 µν 1 µν µρ H = 2 HH ∂ρHµν ≡ 2 HH ∂ρgµν due to n Hµν = 0. Hence
we obtain:
∫ √ ∫− − √ ( )dd 1x H D bρ =− dd 1x H 1ρ 2Hµν∂ρHµν − 1 µν ρ2g ∂ρgµν b
∂M ∫∂M √
=− dd−1x H 1 (Hµν −gµν2 ) (∂ρg
ρ
µν)b
∫∂M √
=+ dd−1x H 1 µ ν2n n (∂ρgµν)b
ρ
∂M
Apparently, D bρρ really stands for some contribution that vanishes whenever Dirichlet bound-
ary conditions are imposed, and in fact using the orthogonality of H and n again, we can write
(H)
Dρb
ρ ≡ D bρ +(nλρ ∂ρnλ )bρ . √
Going back to the outcome of the variation for H, the first field derivative of the Gibbons-
Hawking-York functional is thus given by
∫ √ ( )
( gravd d−1 1 µνv Γ(∂ )[g]) =−2 · d x H 2H K(dv g)µν +(dv K)h B h h
∫ ∂M √ { ( )}
=+ dd−1x H (Kµν −HµνK)+ nµDν −nλgµνDλ (dv g)h µν
∫ ∂M √
+ dd−1x H (nλ ∂ρnλ )H
ρµnν(dv g)h µν
∂M
192 Chapter 6. The Hessian operator
In order to deduce the Hessian we have to compute the symmetric version of the functional
derivative of this equation, explicitly:
∫
( gravd d−
√
1
w dv Γ(∂ )[g]) = +dw d x H (K
µν −HµνK)(dv g)µν ‖ =̂NVIII(v,w)h h B h h
∫∂M ( )
+d dd−
√
1x H nµDνw −nλgµνDλ (dv g)h h µν ‖ =̂ −NVI(v,w)
∫∂M √
+d dd−1x H (nλ ∂ n )Hρµw ρ λ n
ν(dv g)µν ‖ =̂ NIX(v,w)h h
∂M
Notice that the second contribution, NVI(v,w), was already determined for the Einstein-Hilbert
functional and represents exactly the boundary contribution consisting of only derivatives of
the metric variation which under a suitable matching of coefficients is the reason to add the
Gibbons-Hawking-York term in order to obtain a well defined variational principle. Hence,
we only have to consider NVIII(v,w) and NIX(v,w) in what follows and reuse the result of the
previous subsection for NVI(v,w).
Aiming at NVIII(v,w), we first present another variation of K which is equivalent to the
above construction but more appropriate for the purpose of deriving the Hessian:
( )
(d K) =− 1D nλw 2 λ n
ρnσ (d 1 κ ρσ
h w g)h ρσ + 2n g Dκ(dw g)h ρσ( )
=− 12 n
ρnσK+2nρnλK
λσ −HρσnλDλ (dw g)h ρσ
In addition, we have to compute the variation of the non-contracted form of the extrinsic curva-
ture which involves a further inverse metric:
(dw K
µν) = (d µλw g D
ν
λn ) =−gµρ(Dσnν)(dw g) µρσ +D (dw nν)+gµλ (d Γν αh h h h wh λα)n
Upon substituting all partial results and expressing Dαnβ as the extrinsic curvature, we arrive at
( )
(d µν µρ σνw K ) =− g K +nµnρKνσ + 12K
µνnσnρ (dw g)h h ρσ
− 1nνnσnρDµ2 (dw g) 1
νρ µσ λ
( h ρσ
+ 2g g n Dλ (dw g)h ρσ )
=− (gµρ +nµnρ)Kσν + 1Kµνnσnρ + 1(nνnσnρDµ −gνρgµσnλ2 2 Dλ ) (dw g)h ρσ
The integral kernel ofNVIII(v,w) is based on these ingredients and has a lengthy but very simple
structure:
dw (K
µν −HµνK) = (dw Kµν)−Hµν(dw K)− (dw Hµν)Kh ( h h h )
=− (gµρ +nµnρ)Kσν + 1(Kµν −HµνK)nσnρ +(nνnµnσnρ µρ νσ( ( 2 ) ( −g g )))K (dw g)h ρσ
+ Hµν nρn Kλσ − 1HρσnλD − 1 nνnσnρDµ −gνρgµσnλλ 2 λ 2 Dλ (dw g)h ρσ
Notice that only the last three terms of the second contribution contains derivatives of the metric
fluctuation. Including the field derivative acting on the volume element of the integration we
arrive at
∫ √
N (v,w) = + dd−1VIII x H (dw g)
1 ρσ µν µν
h ρσ 2H (K −H K)(dv g)h µν
∫ ∂M √
+ dd−1x H (dv g)
µν µν
h µν dw (K −H K)h
∫∂M √
+ dd−1x H (Kµν −HµνK)(dw dh v g)h µν
∂M
6.4 Hessian of the gravitational functional 193
Hence, we are ready to compute the final piece of the full Hessian, in fact a contribution that
we are keen to remove in the very end by imposing suitable boundary conditions. While we
have already encountered several boundary terms in the Hessian that are non-vanishing under
Neumann conditions for the metric fluctuation, the very nature of our truncation ansatz ensures
that Dirichlet conditions remove all these obstructions on the level of the Hessian. Similarly, for
NIX(v,w) we find some contributions that contain derivatives of metric fluctuations, and some
which do not:
∫ √
N (v,w) = + dd−1 1 ρσ αIX x H (dw g)ρσ 2H (n ∂ n )H
λ µ
λ α n
ν(dv g)h h µν
∫ ∂M
+ dd−
√
1x H d (Hλ µnνnαw ∂λnα)(dv g)h h µν
∫∂M √
+ dd−1x H (nα ∂λn )H
λ µnνα (dw dh v g)h µν
∂M
We omitted to present the explicit form of the variation of the second term, but emphasize at
this point that it is proportional to Dλdw g. Notice however that being multiplied by anotherh
fluctuation (dv g)µν here again Dirichlet conditions appear to be a proper choice for a fullh
cancellation of all boundary contributions.
6.4.5 Hessian for the gravitational functional
The gravitational constituent of the effective average action is independent of the ghosts func-
tional that hence contributes only in the metric block to the Hessian operator. Without specify-
ing boundary conditions and/or the explicit form of the exponential map contained in (dv g)h µν
no Hessian operator can be deduced.
Thus, let us first present the full Hessian of the gravitational functional and then proceed by
defining a concrete truncation ansatz with a simple choice for dv gµν . Collecting the previoush
results from the volume and curvature functionals the Hessian is given by
[ ]
Hess Γgravϕ [Φ,Φ̄] (v,w)∫
d √ { ( )= d x g(dv g) − (ūD λ̄D) · gρµgσν − 1gρσgµνh µν k k 2
M }
+ 12(ū
D) · {−Kµνρσ + Uµνρσk } (dw g)h ρσ
∫ √ ( )( )
+ ddx g 1 (ūD) ·Gµν − (ūD λ̄D) ·gµν 12 k k k 2(dw dv +dv dw )g (6.13)h h h h µν
M
A treatment of the full Einstein-Hilbert functional requires a geometrical connection for which
(dv g) would be still a function of g due to a non-trivial field space connection in the metrich
component. Choosing a different, for example flat, connection corresponds to some truncation
which is only equivalent to the Einstein-Hilbert functional for small (effective) perturbations
around g. This is a very important remark, for it allows to simplify the Hessian at the expense
of loosing some of its interpretation. While in this thesis, we are going to consider a truncation
resulting from the Einstein-Hilbert functional for (dv g) = vh future work can draw on equationh
(6.13) and insert different geodesic solutions which may fully reproduce the Einstein-Hilbert
case. Under the choice (dv g) = vh we see that the last term actually does not contribute toh
the Hessian for then (dw dv g) = (dw vh) = 0. If the field space connection – implicitly presenth h h
in (dv g) – is ultra-local, thus does not contain any derivatives, eq. (6.13) always gives riseh
to a Hessian operator as long as the boundary terms vanish under suitable conditions on the
194 Chapter 6. The Hessian operator
fluctuation fields:
∫ √ {( )
Igrav =−(ū∂D ∂Dk λ̄k ) · dd−1x H (dw g)σρ HµρHνσ − 1HρσHµνh 2
∂M }
+Hµρnνnσ +Hνσnρnµ (dv g)h µν
∫ √ ( )
− (ū∂D λ̄ ∂D) · dd−1x H Hµν 1k k 2(dw dh v +dh v d )gh wh µν
( ) ∂M
+ ūD− ū∂Dk k ·NVI(v,w)+ 12(ū
D
k ) ·NVII(v,w)+ (ū∂Dk ) · (NVIII(v,w)+NIX(v,w))
The very nature of the present truncation assures that all these contributions disappear for Dirich-
let boundary conditions. The reason is encoded in the maximal number of covariant derivatives
that constitutes the various invariants. On the bulk, we restricted ourselves to two derivatives
and on the boundary we consider only field monomials with at most one covariant derivative.
Each boundary contribution that originates from some bulk invariant, due to integration by parts,
converts one of its derivatives to a normal vector field. Hence, any boundary term irrespectively
of its origin contains at most one derivative. For the Hessian this implies that out of the two fluc-
tuations there is always at least one fluctuation field without a derivative acting on it: therefore
any boundary term vanishes under Dirichlet conditions in the present truncation.
This simple argument naturally does not generalize to truncations including higher order
derivative invariants, still a thorough treatment of manifolds with boundary in the FRGE setting
requires a Hessian operator. In these cases matching of coefficients along with some choice of
– possibly non-standard – boundary conditions would be appropriate to continue the evaluation
of the RHS of the FRGE. Surely, in the very end the RG solutions reveal whether or not the
chosen matching condition is consistent with the RG flow on theory space.
In what follows we always assume Dirichlet boundary conditions to hold, i.e. for the entire
(effective) field space we consider a fixed field all over ∂M:
∣∣ ∣∣ ∣Φ = Ψ ≡ Φ̄ and ϕ∣ = 0
∂M ∂M ∂M
The related effective fluctuation fields thus have to vanish on the boundary.
6.4.6 Linear field parametrization
In most of the truncations for theories of quantum gravity field space was approximated to be
flat resulting in what is called the linear field parametrization of metric fluctuations:
(dv g) = vh (6.14)h
As already mentioned at several steps in the derivation, this transition from dynamical fields Φ
to fluctuation fields ϕ is unavoidable in the presence of field variations. All the supplementary
functionals – the gauge-, ghost-, and cutoff-action – that are necessary to obtain a well behaved
quantum field theory, are by definition formulated in terms of ϕ using some background field
Φ̄. For the gravitational contribution we could have started at first place with this set of depen-
dencies using the exponential map for a transition between both functionals:
Γgrav[Φ,Φ̄]≡ Γgrav[expΦ̄(ϕ(Φ,Φ̄)),Φ̄]≡ Γ
grav[ϕ ;Φ̄]
Notice that on the effective level we in general encounter a bi-field functional, being individually
dependent on the dynamical and the background field, Φ and Φ̄, respectively. While on the
fundamental level we usually assume ρ([Φ̂]) being background independent, the gauge fixing
and cutoff functional contribute with background non-invariant insertions to the effective action
resulting in the bi-field nature of Γk. When approximating exp (ϕ(Φ,Φ̄)) in a linear fashion,Φ̄
6.4 Hessian of the gravitational functional 195
i.e. ϕ = Φ− Φ̄, the nice correspondence between Γgrav[ϕ ;Φ̄] and Γgrav[Φ,Φ̄] will in general be
lost. This is not problematic at all in case of truncations on the effective level for then we are
just considering a modified ansatz for Γgravk [ϕ ;Φ̄] which is still compatible with the symmetry
and field content, though one may spoil gauge invariance. The mere disadvantage in using non-
geometrical relations for the geodesics on 〈[F]〉 is the status of interpretation and the systematic
generalization of the ‘new’ truncation ansatz. So one may start at first place with an Einstein-
Hilbert like truncation ansatz for Γgravk [Φ,Φ̄] but substituting Φ = Φ̄+ϕ only approximately
reproduces its analog Γgravk [Φ,Φ̄]. Nevertheless, this linear parametrization is at least suitable
for almost flat regions in the vicinity of normal geodesic coordinates of Φ. In other words
restricting the effective fluctuations ϕ to be small, i.e. considering small derivations of the
associated measure, retains the Einstein-Hilbert interpretation while larger fluctuations will in
general result in some modified class of measures.
Combining all the previous results for the gravitational functional and insert the linear
parametrization (dv g) = vh we obtain a Hessian operator of the following form:h  
grav
[ ] Ĥesshh 0 0
Ĥessϕ Γ
grav[Φ,Φ̄] ≡k 0 0 0 (6.15)
0 0 0
Since the gravitational contribution is a functional of only metric field content, the Hessian
is degenerated in the ghost sector and contains only one non-vanishing component, explicitly
given by a combination of second-order differential and ultra-local terms:
( ) √
grav ρσ g ( )
gµν A Ĥess D D √ ρµ σν 1 ρσ µνhh =−(ūk λ̄k ) · g g − 2g g
A ḡ√
+ 1(ūD) · √g {−Kµνρσ + Uµνρσ2 k } (6.16)ḡ
It is part of the propagator for the effective gravitational fluctuation field, for which ghost and
gauge fixing corrections have to be added. The later analysis will depend on this Hessian
structure employing the linear parametrization.
6.4.7 Geometric field parametrization
We have already mentioned that the linear parametrization may spoil the original interpretation
of the truncation ansatz , for one actually considers a (non-geometrically) reduced functional
that only corresponds to the initial object for small effective fluctuation fields h. Nevertheless,
it is not a perturbative approximation at all, for it only gives rise to a different truncation ansatz
with a possible different interpretation for large h. A more sophisticated and natural procedure
relies on a geometric parametrization of the dynamical field g with h and the background metric
ḡ, usually based on some specific connection on field space. This general treatment that actually
corresponds to the covariant formalism of field space we introduced earlier, goes back to work
of DeWitt [39, 68, 69] and has found attention in the mathematical literature on metric field
spaces. It is the geometric method of background field covering that relates the dynamical field
g with the background one ḡ by geodesics, corresponding to the fluctuation fields. The problem
is merely on a practical level, for the associated exponential map is usually non-local and highly
non-linear, which adds another difficulty into the list of challenges one encounters in evaluating
the FRGE.
Recently, there has been progress towards a more geometric parametrization which is local
but still non-linear [84]. The underlying field connection is ultra-local on F and the exponential
map reduces to matrix exponentiation given by
( )
h ρgµν = ḡµρ e ν
196 Chapter 6. The Hessian operator
The non-linear character of this parametrization yields in particular additional contributions
from terms dw dv g which vanish for the linear choice. Future works should in addition employh h
this geometric ansatz and a comparison between the linear and this geometric parametrization
might reveal the interpretation status of the former one. The here derived Hessian is thus a
suitable starting point to proceed in both directions.
6.5 Hessian of the full truncation ansatz
Finally, we combine the results of the preceding sections to obtain the full Hessian of our trunca-
tion ansatz within the linear parametrization. While the cutoff-, the ghost-, and gauge-functional
a purely defined in terms of fluctuation h and background fields ḡ, the gravitational part is a
functional of the dynamical field g. Our truncation thus relies on a specific connection on field
space and in the present non-geometric identification, we actually compute the FRGE for the
following ansatz:
∞ ∣
Γk[ϕ ;Φ̄] = Γ
gh
k [ϕ ;Φ̄]+Γ
gf
k [ϕ ;Φ̄]+ ∑ ∂
n
s Γ
grav
k [Φ̄+ sϕ ,Φ̄]
∣ (6.17)
s=0
n=0
In a covariant formalism the (flat) Taylor expansion has to be replaced by the exponential map
on field space. We are going to simplify this ansatz even further by choosing a suitable linear
gauge fixing condition and neglect the running of the ghost sector later on.
6.5.1 Boundary conditions
In order to extract a Hessian operator for the above truncation, the first step is to insert (dv g) =h
vh into the Hessian and impose a set of boundary conditions on the fluctuation fields in order
to get rid of the impeding boundary terms. However, different to the usual constraints in order
to obtain a suitable variational principle, we rather focus our attention on the second variation
in order to find the FRGE applicable. In total we have three sources of boundary contributions
associated to each of the functionals in the truncation ansatz:
I= Igh[v,w]+Igf[v,w]+Igrav[v,w]
The very existence of a Hessian operator for Γk[ϕ ;Φ̄] requires the vanishing of I. Due to the
restricted number of at most two derivatives in the invariants of our ansatz, Dirichlet boundary
conditions are sufficient to cancel all terms individually, as long as F is linear and contains at
most two D̄. In the presence of a non-vanishing boundary of spacetime constraining field space
on ∂M is mandatory to obtain ellipticity for the set of operators defining the RG steps. Though
in the general case one is interested in preserving ellipticity and symmetries of the formalism, in
particular BRST invariance, there are indications that it might be difficult to retain both in the
case of QG. However, on the level of truncations ellipticity is far more important property, for it
assures a proper heat kernel expansion, or in more general terms a well behaved cutoff operator
based on the Laplacian. In this respect, Dirichlet conditions are a very natural choice and are
also admissible in the sense of Faddeev-Popov as stated by Vassilevich [61]. Nevertheless,
quite different boundary constraints may be imposed and the current status of derivation is such
general that future investigations based on Neumann-, or boundary conditions can be applied as
long as I is removed from the Hessian. While Dirichlet boundary conditions actually cancel
each of the arising boundary terms separately, other constraints might additionally assume a
matching of certain bulk and boundary couplings which then has to be justified by the RG
evolution. In fact, for local, linear gauge fixing conditions, we have a total set of seven equations
constituting the boundary operator, whose kernel yields the valid field space.
B∂M ..(D);Ψ(Φ) = (Φ−Ψ)(•)δ∂M(•) with ker(B∂M(D);Ψ)≡ {Φ ∈ Y |Φ|∂M = Ψ|∂M} (6.18)
6.5 Hessian of the full truncation ansatz 197
This boundary operator is equivalent to the previously described condition:
∣∣ ∣ ∣Φ = Ψ ≡ Φ̄∣ and ϕ∣ = 0 (6.19)
∂M ∂M ∂M
Notice that once we implement the background field covering on field space Ψ is encoded in Φ̄
and thus redundant.
The only source for the ghost sector and its coupling to the metric field content is the ghost
functional coming along with boundary terms:5
Igh[v,w]
√ ∫ √ ( ( ) ( ) )
≡− 2 ghū ddx ḡ FA ξ̄ ḡµνd d g +(v ) ḡµνk ν µ w v Lξ ξ̄ µ dwL gA ξ A
√ ∫M √ ( )( )
− 2 ghū dd−1 ρσk x H F ḡµνν (vξ̄ )µ nσg λλρ wξ +nρgλσ wλξ +nλ ξ λ (dwg)ρσ
√ ∫∂M √ ( )( )
− 2 gh dd−1 ρσū x H F µν λ λ λk ν ḡ ξ̄µ nσ (vg)λρ wξ +nρ(vg)λσ wξ +nλ vξ (dwg)ρσ
∂M
For suitable choices of gauge fixing condition, it is in fact sufficient to impose Dirichlet con-
ditions on ghost and metric fields to get rid of these terms, i.e. for all Φ ∈ ker(B∂M( ) andD);Ψ
v,w ∈ TΦF we have Igh[v,w] = 0.
The gauge fixing functional gives rise to a single contribution restricting only the metric
part of field space, i.e.
∫ √ ( )
Igf[ , ]≡ gfv w ūk ddx ḡ FA µν Bµ ḡ (wh)AFν (vh)B
M
We are going to implement a class of gauge fixing conditions in a moment, however under the
above mentioned assumptions of locality, linearity, and at most second order in the derivatives,
any Fwill result in a contribution that vanishes on the boundary for Φ ∈ ker(B∂M( ); ).D Ψ
The remaining part originates from the gravitational functional and is either related to some
integration by parts of certain bulk terms or by the implemented variations of the Gibbons-
Hawking-York like terms in the truncation ansatz. Its total contribution to the Hessian reads
∫ √ {( )
Igrav[v,w] =−(ū∂D λ̄ ∂D) · dd−1k k x H (dw g) HµρHνσσρ − 1 ρσ µνh 2H H
∂M }
+Hµρnνnσ +Hνσnρnµ (dv g)h µν
∫ √ ( )
− (ū∂D λ̄ ∂D) · dd−1x H Hµν 1k k 2 (dw dv +dv dh h h w )gh µν
( ) ∂M
+ ūD− ū∂D N 1 D ∂Dk k VI(v,w)+ 2(ūk )NVII(v,w)+ (ūk )(NVIII(v,w)+NIX(v,w))
Notice that for the truncation ansatz (6.17) the second integral vanishes identically for any
choice of boundary conditions. Furthermore, while Dirichlet constraints yield a cancellation of
every term irrespective of the coefficients, a matching of ūD and ū∂Dk k as required for the construc-
tion of field equations by means of the variational principle will at least cancels NVI(v,w).
In conclusion we have to either restrict field space by some adapted choice of boundary
conditions for the fluctuation fields that cancel I or imposing certain boundary conditions and
adapt the explicit form of Γk[ϕ ;Φ̄] such that the non-vanishing terms in I cancel each other.
In the present work, we follow the first approach for the structure of the truncation ansatz favor
Dirichlet boundary conditions and allows for an application of the heat kernel technique. Thus,
5Notice that for brevity we neglect are not presenting the symmetric version of the boundary terms that appears
in the Hessian. The actual result is given by 1Igh2 [v,w]+
1
2I
gh[w,v].
198 Chapter 6. The Hessian operator
as already mentioned in the gravitational part, in the sequel we impose the boundary constraints
of Dirichlet type that gives rise to a cancellation of the entire functional I, as required. This
step marks the end of the general status of our results, so that from now on the statements are
only valid within the assumption of Dirichlet conditions.
6.5.2 The Hessian operator
Now that we have imposed suitable boundary conditions for which I vanishes identically, the
full Hessian gives rise to a Hessian operator consisting of the gauge fixing, ghost, and gravita-
tional contributions:
[ ] [ ] [ ] [ ]
Ĥessϕ Γk[ϕ ;Φ̄] = Ĥessϕ Γ
gf
k [ϕ ;Φ̄] + Ĥess Γ
gh
ϕ k [ϕ ;Φ̄] + Ĥessϕ Γ
grav
k [ϕ ;Φ̄]
Its matrix structure in field space has only two vanishing entries, the diagonal elements of the
ghost block. The remaining components give rise to the following matrix form:
 
[ ] Ĥesshh Ĥesshξ Ĥesshξ̄ 
Ĥessϕ Γk[ϕ ;Φ̄] ≡Ĥess ξh 0 Ĥessξ ξ̄ (6.20)
Ĥessξ̄h Ĥessξ̄ ξ 0
All its entries are L2-operators that in general will not commute with each other. For the later
truncation the relevant contributions are those which are independent on the ghost and anti-ghost
fields, thus the metric component consisting of the gravitational and the gauge fixing Hessian:
√
µν B ρσ − D D · √g
( )
g (Ĥess ) = (ū λ̄ ) gρµgσν − 1gρσgµνhh B k k ḡ 2
√
g
+ 1 D √2(ūk ) · {−K
µνρσ + Uµνρσ}
ḡ ( )
− gf · µν αβ ρσūk Fα [ḡ] g Fβ [ḡ] (6.21)
In addition, due to the linearity of the ghost functional in ξ and ξ̄ the entries in the ghost block
are also independent on the ghost fluctuations. They exclusively originate from Γghk [ϕ ;Φ̄] and
assume the form:
√
µ λ ( ) ρσ
(Ĥessξ ξ̄ )α =+ 2
gh
ūk (g
−1) D̄λgρσ + D̄ g
µν
ξ̄ξ σ λρ
+ D̄ρgλσ +gλρ D̄σ +gλσ D̄ρ Fν ḡ
√ α{ }α ( )
(Ĥess αξ̄ξ )λ =− 2
gh ρσ
ūk (g
−1) ḡµνFν D̄ξ ξ̄ λgρσ +gλρD̄σ +gλσ D̄ρ (6.22)µ
The components that interrelate metric fluctuation with the ghost sector all depend on either ξ̄
and ξ . They emanate from the ghost functional and can be summarized as follows:
√ gh √ gh
( gh ghĤess αβξh) =− 2ūk · (Ĥess αβξh) (Ĥesshξ )λ =− 2ūk · (Ĥesshξ )λ (6.23a)√ gh √ gh
(Ĥess )µ =− 2 ghhξ̄ ūk · (Ĥess µhξ̄ ) (Ĥess αβξ̄h) =− 2
gh
ūk · (Ĥess )αβξ̄h (6.23b)
Their explicit form is found in equation (6.9). Later on, neglecting those contributions simplifies
the RHS of the FRGE significantly, which corresponds to projecting out the running of the
ghost sector.
6.6 First variation formula
The original motivation to include the Gibbons-Hawking-York term to the Einstein-Hilbert func-
tional was motivated by the variational principle in the presence of boundaries [79]. In the
6.6 First variation formula 199
present context we expect that this correspondence is only satisfied under additional assump-
tion, since quantum corrections in principle spoil this classical matching.
Even though our main focus is the derivation of the RG flow, it is worth considering the
conditions under which a well stated variational principle can be established. In the metric
sector the gauge fixing functional produces the following boundary term under first variation:
∫
! √ ( )
0= gfū ddx ḡ FAk µ ḡ
µν(vh)
B
AFν (h)B
M
Whenever Fis linear in the covariant derivative D̄ Dirichlet and Neumann boundary constraints
make this term disappear. In particular the class of gauge fixing conditions we finally implement
satisfies this property. Furthermore, we have to assure that the gravitational contribution also
vanishes, for which however mixed boundary conditions are necessary:
{ ( )}
!
0=+ (ū∂D λ̄ ∂D) ·Hµν − (ū∂D) · (nλ ∂ n )Hρµnν +Kµν µνk k k ρ λ −H K (dv g)h µν
{ ( )}
+ (ūD − ū∂D µ νk k ) · n D −gµνnρDρ (dv g)h µν
In the linear field parametrization we employ here, this requirement translates into
{ ( )}
!
0=+ (ū∂D λ̄ ∂D) ·Hµνk k − (ū∂Dk ) · (nλ ∂ ρµρnλ )H nν +Kµν −HµνK (vh)µν
{ ( )}
+ (ūD − ū∂D) · nµDν −gµν ρk k n Dρ (vh)µν
Notice that the first (second) term is zero for Dirichlet (Neumann) constraints for the metric
fluctuations thus an additional matching of the couplings would be necessary. Apparently, the
‘classical’ choice of ūDk ≡ ū∂Dk suffices to get rid of the entire boundary contribution in case of
Dirichlet type constraints.
Finally, the variation of the ghost functional generates two more boundary integrals that
however vanish for the above restrictions of the gauge fixing condition and imposing Dirichlet
constraints on the fluctuation fields:
∫
! − √ ρσ ( )( )0= + ghū dd 1x H F ḡµν λk ν ξ̄µ nσgλρ vξ +nρgλσ vλ λξ +nλ ξ (dvg)ρσ
∫ ∂M √ ( ( ) )
+ ghū ddx ḡ FA ξ̄ ḡµνk ν µ dvLξg A
M
However, for gauge fixing conditions that are of degree two or higher in the derivatives D̄, the
vanishing of these boundary terms is not in general given and a case by case study is demanded.
6.6.1 Stationary point solutions
Assuming that all boundary contributions dissolve under a suitable choice of constraints on field
space, stationary point solutions for the effective fluctuations can be deduced from a variational
principle. They read as follows:
◮ Anti-ghost field variation
{ ( )}
0= ḡµν ρσFν D̄λgρσ +gλρ D̄
λ
σ +gλσ D̄ρ ξ ≡M[g, ḡ](ξ )µ
Thus, solutions ξ which satisfy the stationary point equation should be orthogonal to the
image of the Faddeev-Popov operator M[g, ḡ]. Due to the linearity of the field equation
and the lack of further constraints on ξ , the trivial solution ξ = 0 exists. Since the kernel
of M[g, ḡ] is assumed to be trivial this is actually the only possible solution.
200 Chapter 6. The Hessian operator
◮ Ghost field variation
( )
0 ρσ= D̄λg
µν
ρσ + D̄σgλρ + D̄ρgλσ +gλρD̄σ +gλσ D̄ρ Fν ḡ ξ̄µ
This equation represents the adjoint of the Faddeev-Popov operator acting on the anti-
ghost field. Similar arguments as in the previous case apply, hence favoring the trivial
solution ξ̄ = 0.
◮ Metric field variation
√ {( ( )) }
0 2 2 gh α β α β α β λ λ ρσ=+ ūk D̄σ ξ δ
µν
ρ + D̄ρξ δσ +δρ δσ D̄λ ξ +ξ D̄λ Fν ḡ ξ̄µ
{ ( ) }
− gf αβ+ ū F ḡµνFAk µ ν hA (6.24)
√ { ( )}
+ √
g
(ūD λ̄D) ·gαβ +(ūD) · Rαβ 1 αβ
ḡ k k k
− 2g R (6.25)
The variation w.r.t. metric fluctuations obtains contributions form each functional in our
truncation ansatz, except the purely background terms inserted to study Background In-
dependence. Here the linear parametrization was employed giving rise to the modified
Einstein field equations in the last line. The remaining contribution to the metric field
equations are due to the gauge fixing and ghost functional. Inserting the solutions from
the previous field equations we get
{ ( ) } √ { ( )}
0 − gf αβ= ū F ḡµνFA √gh + (ūD λ̄D) ·gαβk µ ν A k k +(ūDk ) · Rαβ − 12g
αβR
ḡ
Notice that self-consistent background fields satisfying (dvΓk[ϕ ;Φ̄])ϕ=0;Φ̄=Φ̄s.c. = 0 are
solutions to the modified Einstein field equations only, for in this case the gauge fixing
term drops out. Contrary to the ghost sector, we are considering a mixed field equations
based on the fluctuation and the dynamical field h and g, respectively. While h = 0 is
allowed and exactly gives rise to the self-consistent background solutions, setting g = 0
would solve the equation as it stands, however is not a metric field anymore and thus
prohibited. Rather, we may consider instanton solutions for either ḡ or g, satisfying the
above modified Einstein equations.
Let us emphasize again that we are studying the effective action and thus the effective dynam-
ical fields and fluctuations, for which the above derived stationary point equations hold true.
Whenever we are interested in interpreting those results, we have to keep this subtlety in mind.
In particular the imposed boundary conditions apply to ϕ rather than ϕ and will constraint the
space of measures yielding expectation fields with, for instance, vanishing boundary values.
From this perspective, even requiring Dirichlet and Neumann conditions to be satisfied simul-
taneously, is not over-determining but in fact amounts to a restriction of the class of theories
keeping the underlying field space perfectly well-defined.
6.7 Implementation of the gauge fixing condition
In this section we give a more restricted representation for the missing ingredient in our trun-
cation ansatz: the gauge fixing condition. So far, all results only depend on a few conditions
on F[Φ̄], the commutativity with the background metric, linearity, and the vanishing of the
following term under Dirichlet conditions:
∫ √ ( )
Igf[ , gfv w]≡ ū ddx ḡ FA ḡµνk µ (wh)AFBν (vh)B
M
We already mentioned that a sufficient condition is a gauge fixing condition linear in D̄.
6.7 Implementation of the gauge fixing condition 201
In the sequel of this discussion we work with a generalized harmonic gauge condition which
is compatible with the Faddeev-Popov method:
( )
ρσ
F[ḡ] αρ βσ αβ ρσµ = ḡ ḡ −ϖ ḡ ḡ ḡαµD̄β (6.26)
Notice that the first factor has the same tensor structure than the field space metric in the gravi-
tational sector.
A particular important, and well studied case of this family of gauge fixing conditions is the
harmonic one with√ϖ = 1/2, sometimes referred to as de Donder gauge. In a local chart this
corresponds to ∂µ( ḡḡµν) = 0 and defines the harmonic coordinates, a kind of analog to the
Lorentz gauge condition for Yang-Mills connections. Harmonic coordinates can be ascribed
to any Riemannian manifold at least in the local vicinity of any spacetime point p ∈M. Their
existence is by no means a specific feature of a certain class of geometries but in fact a general
result obtained in the theory of elliptic differential operators associated to the In addition, the
elliptic regularity theorem and the Hölder condition ensure that these coordinates are in fact
regular and thus have a special status in a mathematical description of gauge fixing conditions.
Nevertheless, for a moment we are going to use the family of gauge fixing conditions de-
fined in eq. (6.26) to reduce the Hessian operator and present an explicit form for the effective
field equations.
6.7.1 Hessian operator
Before we are going to simplify the Hessian operator, let us first check the validity of the
generalized harmonic gauge condition in the present context. Apparently, eq. (6.26) defines a
linear operator, which – due to the metric compatibility of D̄ – commutes with the background
metric, i.e.
ρσ · · ρσ and ρσF[ḡ]µ (c h)ρσ = c F[ḡ]µ (h)ρσ [F[ḡ]µ , ḡ] = 0
Next, we verify the vanishing of Igf[v,w] under Dirichlet conditions:
∫ √ ( ) ( )
Igf[v, ] = gfw ū ddx ḡ ḡνρk ḡ
βσ −ϖ ḡνβ ḡρσ D̄β (w ) FBh A ν (vh)B
∫M ( )
− √ ( )= gfūk dd 1x H ḡνρ ḡβσ −ϖ ḡνβ ḡρσ nβ (wh)AFBν (vh)B = 0
∂M
In the first step the total derivative is transformed into a boundary integral, containing an undif-
ferentiated wh which vanishes therefore.
Hence, the family of gauge fixing conditions allows for a Hessian op[erator of t]he form given
in eq. (6.20). For later reasons, We focus only on the entries of Ĥessϕ Γk[ϕ ;Φ̄] independent
on the ghost fields. Therefore, consider the contraction of the gauge fixing condition occurring
in the Hessian part of the gauge fixing functional:
( ) ( )
ν µ
F [ḡ] gαβ
σρ
F [ḡ] = (ḡνα ḡµκ −ϖ ḡµν ḡκα) ḡ σβ ρλ ρσ λβα β αβ D̄κ ḡ ḡ −ϖ ḡ ḡ D̄λ
= ḡσν D̄µD̄ρ −ϖ (ḡρσ D̄νD̄µ + ḡµνD̄σ D̄ρ)+ϖ2ḡµν ḡρσ D̄2
It turns out convenient to express the symmetrized version of this operator – w.r.t. (µν) and
(ρσ) – in terms of the kinetic operator appearing in the Hessian of the gravitational functional:
K(g)µνρσ ≡ (gµνDρDσ −2gνσDρDµ +gρσDµDν)− (gµνgρσ −gµρgνσ )D2
202 Chapter 6. The Hessian operator
Let us denote K̄ ..=K(ḡ), then we have the following identity
( )
ν µ
F [ḡ] gαβ
σρ
F [ḡ] =− 1K̄µνρσ +(1 −ϖ)(ḡµνD̄ρ σα β 2 2 D̄ + ḡ
ρσ D̄µD̄ν)
( )
+ ḡνσ (D̄µD̄ρ − D̄ρD̄µ)+ 1 ḡµρ ḡνσ2 +(ϖ
2− 1 µν2)ḡ ḡ
ρσ D̄2
Here again, an implicit symmetrization in both (µν) and (ρσ) is understood. Notice that
ḡνσ [D̄µ , D̄ρ ]v ≡−(ḡνσ R̄µρ + R̄ρµνσρσ )vρσ holds true for any symmetric 2-tensor field.6 Thus,
the final form of the squared gauge fixing operator assumes the form
( )
ν µ σρ
Fα [ḡ] g
αβ F 1 µνρσ 1 µν ρ σβ [ḡ] =− 2K̄ +(2 −ϖ)(ḡ D̄ D̄ + ḡ
ρσ D̄µD̄ν)
( )
− ḡνσ R̄µρ − R̄ρµνσ + 1 ḡµρ ḡνσ 2 1 µν ρσ 22 +(ϖ − 2)ḡ ḡ D̄
The first line contains uncontracted covariant derivatives while the remaining contributions are
either ultra-local operators or combine to the Laplacian. Remarkably, for ϖ = 1/2, the harmonic
gauge condition, the additional uncontracted derivatives vanish. Next, we insert this result into
the metric-metric entry of the Hessian operator which is now explicitly given by
√
µν B ρσ − D D · √g
( )
g (Ĥesshh)B = (ūk λ̄
ρµ
k ) g g
σν − 1gρσgµν
ḡ 2
+ gfū · (ḡνσk R̄µρ + R̄ρµνσ)√
+ 1(ūD) · √g {−Kµνρσ2 k + Uµνρσ}+ 1
gf
2 ūk · K̄µνρσḡ
− 1 gf2 ūk · (1−2ϖ)(ḡ
µν D̄ρD̄σ + ḡρσ D̄µD̄ν)
( )
− 1 gf2 ūk · ḡ
µρ ḡνσ − (1−2ϖ2)ḡµν ḡρσ D̄2 (6.27)
We omit to explicitly compute the ghost sector contributions to the Hessian and retain their
general form for a moment, i.e.
√
µ λ ( ) ρσ
( ) = + 2 ghĤessξ ξ̄ α ūk (g
−1) D̄λgρσ + D̄σgλρ + D̄ρgλσ +g
µν
ξ̄ξ λρ
D̄σ +gλσ D̄ρ Fν ḡ
√ α{ ( )}
( )α =− 2 ghĤess ū (g−1 α ρσξ̄ξ λ k ) ḡµνFξ ξ̄ ν D̄λgρσ +gλρD̄σ +gλσ D̄ρµ
Retrospectively, this turns out to be much more convenient, since most of the terms drop out in
the projection of the RHS of the FRGE.
6.7.2 First variation formula
We are now going to revisit the metric part of the stationary point solutions using the explicit
form of the gauge fixing condition. We have already stated that for the effective field equations
to emerge from a variational principle the following terms have to vanish:
∫
! √ ( )
0=+ ddx ḡ FA ḡµνµ (v
B
h)AFν (h)B
∫M
! √ ( )( )
0= + dd−1 ρσx H F µνν ḡ ξ̄ n
λ
µ σhλρ vξ +nρhλσ v
λ
ξ +n ξ
λ
λ (vh)ρσ
∫ ∂M √ ( ( ) )
+ ddx ḡ FAν ξ̄
µν
µ ḡ dvLξh A
M
Fortunately, in the present context Fis chosen linear in D̄ and thus all boundary terms contain at
least one non-differentiated fluctuation field which vanishes due to Dirichlet conditions. Thus,
6 In general the commutator of covariant derivatives acting on a 2-tensor field vρσ is given by [Dµ , Dν ]vρσ =
−Rλ λρµνvλσ −R σµνvρλ .
6.8 The level expansion 203
in particular the metric field equations are well defined. Inserting the result from the previous
subsection into eq. (6.25) yields
√ {( ( )) }
0 2 2 gh α β β β ρσ=+ ūk D̄σ ξ δρ + D̄
α α λ
ρξ δσ +δρ δσ D̄λ ξ +ξ
λ D̄ µνλ Fν ḡ ξ̄µ
{ ( ) }
− gf αβ+ ū F ḡµνk µ FAν hA ( )
+ 1 gfū · K̄αβρσ − 1 gfh ū · (1−2ϖ) ḡαβ D̄ρD̄σ + ḡρσ D̄αD̄β2 k ρσ 2 k hρσ( ) ( )
− gfū · ḡβσ R̄αρ + gfR̄ραβσ h − 1 ū · ḡαρ ḡβσ − (1−2ϖ2)ḡαβ ḡρσ 2k ρσ 2 k D̄ hρσ
√ { ( )}
g
+ √ (ūD λ̄D) ·gαβ +(ūD) · Rαβ − 1gαβR
ḡ k k k 2
Again, the harmonic gauge condition is quite special for it leads to the cancellation of uncon-
tracted derivatives in the field equations. Furthermore notice that all terms proportional to gfūk
are linear in h and thus indeed vanish for self-consistent background solutions.
6.8 The level expansion
In this section we give a representation of the current state of our truncation in the level expan-
sion. Geometrically, this corresponds to expanding the exponential map, whereby the dynami-
cal field is expressed in terms of background- and fluctuation fields. The only functional that is
defined by means of the dynamical fields is the gravitational term, thus the present discussion
involves only Γgravk [Φ,Φ̄]. Besides a more consistent description in which the arguments of all
functionals agree, it also helps in clarifying the several modifications we already applied to the
initial truncation ansatz. Thereby, the separation between simplifications in determining the
Hessian operator and evaluating the functional trace becomes more transparent.
The Hessian operator is the central ingredient we need in evaluating the RHS of the FRGE.
What remains is essentially a successive application of mathematical techniques to simplify the
operator structure of the Hessian. Nevertheless, in these final steps we have to severely adapt the
truncation ansatz, since the mathematical apparatus to evaluate the non-linear operator product
is very limited. However, already at the present stage we have implicitly modified the initial
truncation ansatz by imposing Dirichlet conditions on the fluctuation fields, use a specific class
of gauge fixing conditions, and – more severally– choose a linear parametrization scheme.
The first modification actually affects theory space itself, namely the set of boundary con-
ditions we imposed on field space. It turns out that Dirichlet boundary conditions are particular
suitable in the present context, for they give rise to an strongly elliptic Laplacian and a well-
defined Hessian operator. In general, we can impose different boundary conditions as long as
they are consistent with the gauge condition and allow for the construction of a Hessian operator.
But in any case field space has to be restricted by fixing the behavior of the fields on ∂M. For the
choice of Dirichlet conditions, i.e. h|∂M ≡ 0, the set of boundary invariants reduces significantly,
in particular the boundary volume element is identical to the background counterpart:
∫ ∫ √
dd−
√ ∣
1x H∣ ≡ dd−1x H̄ since gµν |(D) ∂M = ḡµν |∂M
∂M ∂M
On the other hand, the Gibbons-Hawking-York invariant contains the extrinsic curvature, a
function of the derivatives of g, which in general differ from those of ḡ, i.e. ∂ρgµν |∂M 6=
∂ρ ḡµν |∂M. While g agrees with ḡ on the boundary it will differ on the manifold interior and
thus
( ) ( )
K(g) = Dµn
µ = D̄µn
µ + ḡµκ ∂ρgµκ −∂ ρρ ḡµκ n = K̄(ḡ)+ ḡµκ ∂ρgµκ −∂ ḡ nρρ µκ
204 Chapter 6. The Hessian operator
Hence, the Gibbons-Hawking-York field monomial based on ḡ and g give rise to different basis
invariants that can be distinguished under Dirichlet boundary conditions.
The second adaption concerns the gauge fixing functional which is now based on the class
of generalized harmonic gauge conditions. However, this is mainly in view of the upcoming
analysis, thus due to the trace evaluation. In order to construct a suitable Hessian operator it suf-
fices that F is consistent with the boundary conditions of the fluctuation fields, that commutes
with ḡ and is linear in h.
The so far presented adaption of field space and the initial truncation ansatz, seemed natural
and in fact do not alter the interpretation of the gravitational sector in terms of an Einstein-
Hilbert and Gibbons-Hawking-York functional. However, for the sake of simplicity we have
applied the linear parametrization dv g = vh h, instead of a geometric one, consistent with the
fiber bundle structure of field space. Nevertheless, it simply describes a different truncation
ansatz compared to the geometrical expansion of Γgravk [Φ,Φ̄].
Now, let us consider the level expansion based on the linear parametrization scheme. The
general description reflects a Taylor expansion in the fluctuation fields and formally reads
∞ 1 (n)
Γk[Φ,Φ̄] = ∑ Γk [Φ̄] (ϕ)
n ≡ Γk[ϕ ;Φ̄]
=0 n!n
Focusing on the gravitational functional we obtain the following structure of the level expansion
up to second order in h:
∣ ∣
Γgrav[ , ]≡ Γgrav[ , ]+ ( Γgrav ∣ 1 gravk g ḡ k ḡ ḡ dh k [g, ḡ]) + 2(dhdhΓk [g, ḡ])∣ +O(h3)g=ḡ g=ḡ
Based on the linear parametrization that we applied in the previous sections, i.e. dhg= h, we can
explicitly derive the various terms in the expansion. In the first contribution both the background
invariants and the dynamical ones enter, which yields
∫ √ ( )
Γgravk [ḡ, ḡ] = +2 · ddx ḡ (ūD λ̄D B Bk k )+ (ūk λ̄k )
∫ M
− √ddx ḡ R̄((ūD)+ (ūBk k))
M∫ √ ( )
+2 · dd−1x H̄ (ū∂D λ̄ ∂D)+ (ū∂B λ̄ ∂Bk k k k )
∫∂M √ ( )
−2 · dd−1x H̄ K̄(H̄) (ū∂Dk )+ (ū∂Bk ) (6.28)
∂M
Notice that either λ̄ ∂Dk or λ̄
∂B
k is superfluous, since the dynamical boundary volume element
agrees with its background counterpart due to Dirichlet conditions.
Now, all the remaining terms in the expansion are entirely due to the dynamical invariants.
The term linear in h comprises the gravitational contribution to the metric field equations with
additional boundary terms:
∣ ∫ √ { ( )}
( gravd Γ ∣ d D D µν Dh k [g, ḡ]) = + d x ḡ (ūk λ̄k ) · ḡ +(ūk ) · R̄µν − 1 µνg=ḡ 2 ḡ R̄ hµν
∫M √ { ( )}
− dd−1x H̄ (ūDk − ū∂Dk ) · n̄µD̄ν − ḡµν n̄ρ D̄ρ hµν (6.29)
∂M
Apparently, the coefficient of the boundary volume invariant disappears, which holds true for all
higher orders in h. Hence, there is no chance to distinguish between λ̄ ∂D and λ̄ ∂Bk k for Dirichlet
conditions.
6.8 The level expansion 205
Finally, the second order in h is given by the gravitational contribution to the Hessian op-
erator, listed in eq. (6.13). Evaluated at vh = wh = h and for g = ḡ it assumes the following
form:
∣ [ ] ∣
1( grav2 dhdhΓk [g, ḡ])
∣ = 1Hess Γgrav2 ϕ̂ [Φ,Φ̄] (h,h)∣∫ g=ḡ { g=ḡ√ ( )
= 1 d4 · d x ḡ h −2(ū
D λ̄D) · ḡρµ ḡσν − 1 ḡρσ ḡµνµν k k 2 (6.30)
M { }}
+(ūD) · −K̄µνρσ + Ūµνρσk hρσ
In combination with the gauge fixing- and ghost functional, which are already quadratic and
linear in h, respectively, this concludes the level-expansion up to second order.
In this setting the coefficients of the invariants are combinations of the original couplings,
related to the background or dynamical functionals. For the level-(0) monomials we introduce
the following new coefficients:
(0)λ̄ (0)ū ..k k = (ū
D
k λ̄
D
k )+ (
B λ̄ B) (0)ū ū ..k k k = (ū
D
k )+ (ū
B
k) (6.31a)
∂ (0)
ūk λ̄
∂ (0) ..
k = (ū
∂D λ̄ ∂Dk k )+ (ū
∂B λ̄ ∂B) ∂ (0)k k ū .
.
k = (ū
∂D
k )+ (ū
∂B
k ) (6.31b)
For the invariants of higher orders in h, there is no contribution from the background sector.
This√will be different if we would include mixed field monomials in the truncation ansatz, as
e.g. gR̄. Thus, we have the following correspondence between the previous- and the level-
coefficients:
(p)λ̄ (p)ū ..k k = (ū
D λ̄D (p) .. Dk k ) ūk = (ūk ) ∀ p≥ 1 (6.32a)
∂ (p)λ̄ ∂ (p)ū ..k k = (ū
∂D ∂D
k λ̄k )
∂ (p)
ū ..k = (ū
∂D
k ) ∀ p≥ 1 (6.32b)
Notice, that again due to Dirichlet conditions for the fluctuation fields, boundary monomials
vanish in the higher order terms of Γk[ϕ ;Φ̄] and thus do not define basis directions in the field
space of Dirichlet constraints.

7.1 Inversion of the Hessian operator
Inversion of a matrix with non-commuting en-
tries
7.2 Decomposition of symmetric 2-tensor
fields
Hodge theorem for k-forms with ∂M 6= 0/
Decomposition theorem for symmetric 2-
tensors for ∂M 6= 0/
The transverse-traceless decomposition
7.3 Projection techniques for truncations
Fixing the ghost sector
Conformal projection technique
The Hessian in the trace-traceless decompo-
sition
Projection to maximally symmetric spaces
7.4 The ‘Ω deformed’ α = 1 harmonic gauge
7.5 Heat kernel
Mellin transform
Heat kernel expansion
Tensor space traces
7. TRACE EVALUATION
This chapter describes in great detail the technique applied in order to evaluate the functional
traces. While the derivation of the Hessian operator was performed for very general settings, it
is in this chapter that we have to specify the truncation ansatz and employ certain modifications
that ultimately lead to the difference between the preliminary version and the final form of
truncated theory space.
We start in section 7.1 with the derivation of a general prescription to invert matrices of
operators. In section 7.2 we present the generalization necessary to employ a decomposition of
symmetric 2-tensor fields in the presence of boundaries. While wemake use of only a fraction of
the full decomposition, future work can make use of these results to study extended truncations.
In the next section 7.3 we actually modify the truncation ansatz by exploiting the freedom in
the gauge and ghost sector. Furthermore, we describe in detail the various projection techniques
that become useful when attacking the functional trace on the RHS of the flow equation.
In section 7.4 we develop a new computational strategy for deriving the explicit beta-
functions which is based on the conformal projection technique and on a specific choice of
the gauge sector. Thereby, we can circumvent the problematic aspects encountered when study-
ing bi-metric truncations. The problem thereby resides in the dependence of the FRGE on two
independent metrics. For this case there exist basically no standard tools (such as general heat
kernel expansions, etc.) to evaluate the functional trace on the RHS This is one of the reasons
why, despite their obvious significance, there are still almost no results on bi-metric truncations
available today. In a previous work [172] a transverse-traceless (TT) decomposition of the fluc-
tuation field hµν was used, which compared to the present technique amounts to an considerable
increase of computational effort.
We conclude this chapter using the heat kernel expansion in section 7.5 to project the re-
maining L2-traces onto the basis invariants of the truncation ansatz.
208 Chapter 7. Trace evaluation
7.1 Inversion of the Hessian operator
In the derivation of the Hessian operator, assuming Dirichlet boundary conditions, we finally
arrived at a field space operator of the following form:
 
[ ] Ĥesshh Ĥesshξ Ĥesshξ̄ 
Ĥessϕ̂ Γk[ϕ̂;Φ̄] ≡Ĥessξh 0 Ĥess ξ ξ̄
Ĥessξ̄h Ĥessξ̄ ξ 0
Every[entry represents a L
2
] transformation in the respective component on field space, thus
Ĥessϕ̂ Γk[ϕ̂;Φ̄] constitutes a matrix with non-commuting entries in general. This requires a
more elaborated mechanism to invert the Hessian operator necessary to determine the RHS
of the FRGE, for besides an inversion on the level of its L2-components we have to invert its
associated matrix on field space. Different to the standard case of commuting entries additional
terms will appear that turns this simple procedure into a very lengthy one. However, as we will
see, already some slight modification of the truncation ansatz will cure most of the occurring
difficulties with only a small price to pay.
7.1.1 Inversion of a matrix with non-commuting entries
In this section we derive the matrix inverse for a symmetric n×nmatrixMn with non-commuting
entries. If it exists, the defining property for an inverse is twofold, namely under matrix multi-
plication it should map Mn to the identity from the left and form the right:
(Mn)
−1 −1
Mn = 1n , Mn (Mn) = 1n
In particular these conditions ensure that ( )−1Mn and Mn trivially commute.
Now, in order to determine the general form of ( )−1Mn let us start with the simple two-
dimensional case and then iteratively deduce the structure in the higher dimensional case. A
generic 2×2 matrix and its inverse ( )−1M2 M2 are given by:
( ) ( )
α1 α2 β1 β2
M2 = , and (
−1
M2) =
α3 α4 β3 β4
The defining conditions invoke constraints on the elements of ( )−1M2 with respect to those of
M2 and give rise to eight relations:
( ) ( ) ( )
β1α1+β2α3 β1 α2+β2α4 1 0 α1 β1+α2β3 α1 β2+α2β4
= =
β3α1+β4α3 β3 α2+β4α4 0 1 α3 β1+α4β3 α3 β2+α4β4
The ordering of the L2-operators is important and has not to be changed without care.
The following procedure now depends on what entities we assume to be invertible. For
our purpose it is most practicable to assume α1 and α4 to be invertible also in the sense of
L2-operator(s and we denote)their inverse by α
−1 and α−11 4 , respectively. Later we will see that
in addition α −α α−14 3 1 α2 has to be invertible. In the next steps, we successively express the
components of ( )−1M2 in terms of the known entries of M2, i.e. the non-commuting operators
α1, α2, α3, α4, α
−1
1 , and α
−1
4 .
◮ At first, we are going to solve β3 α1+β4α3 = 0 for β3, resulting in β3 =−β4α3 α−11 and
then substitute the outcome into β3α2+β4α4 = 1 yielding:
( )
1= β3α2+β α
−1 −1
4 4(−β4α3 α1 )α2+β4α4 = β4 α4−α3 α1 α2
7.1 Inversion of the Hessian operator 209
( )
At this point we have to assume that α4−α3α−11 α2 to be also invertible, as mentioned
above. Thus, we arrive at the following intermediate result:
( )−1
β4 = α4−α −13α1 α2
◮ This immediately leads to an expression for β3 invoking the previously derived relation
β3 =−β4α α−13 1 :
( ) ( )
β3 =− α4− −1
−1
α3 α1 α2 α
−1
3 α1 =−α4 1rk(α )−α α−13 1 α −12 α4 α3 α−14 1
Hereby, in the second step, we used (AB)−1 = B−1A−1 for invertible matrices A and B.
Furthermore, 1rk(α ) denotes the identity matrix of rank equal to the rank of α4, here4
identical to idL2 .
◮ Concerning the remaining two entries, we start with the identity α1 β1 +α2β3 = 1 and
solve for β1. Then, inserting the previously derived result for β3 leads to
( )
β = α−1
−1
1 1 (1− α2 β ) = α−1+α−13 1 1 α α −α α−12 4 3 1 α2 α α−13 1
◮ Finally, we can use any of the remaining equalities involving β2, for instance α1 β2 +
α2 β4 = 0, to complete the set of expressions for the entries in the inverse matrix (M2)
−1.
We thus find
− −
( )
1 − −1 − −1 −1β2 = α1 α2 β4 = α1 α2 α4 α3α1 α2
( )
Notice that α −14−α3α1 α2 appears in every component and its invertible character is
crucial in the derivation of the final result.
Combining all these results ultimately gives rise to the full expression for ( −1M2) in terms of
M2 for non-commuting entries:
(
− − ( − )−1 − − ( )1 1 − 1 1 1 −1 −1
)
−1 α1 +α(1 α2 α4 α3α α2 α3 α −α α2 α4−α3α α2(M2) = )1 1 1 ( 1− α −α α−1 −1α α α−1 α −α α−
)
1 −1
4 3 1 2 3 1 4 3 1 α2
Though derivation was based on a 2× 2 matrix, we actually never made any assumptions
on the substructure of its entries, and in fact kept the formalism very general. Generalization
from ( )−1M2 to higher rank matrices (
−1
Mn) are thus straightforward. Identifying α1 and α4 to
be invertible (n− r)× (n− r) and r× r matrices A and D, respectively, requires α2 and α3 to be
of (n− r)× r or r× (n− r) rank. For this setting, we obtain the following statement: Given
( )
A B
Mn =
C D
( )
a rank n≥ 2 matrix, if A, D and D−CA−1B are invertible then its inverse is given by
(
− − ( − )−1 − − ( − ) )1 1 1 1 1 1 −1
( )−1
A +A B D−CA B CA −A B D−CA B
Mn = ( − )−1 − ( − )−1 (7.1)− D−CA 1B CA 1 D−CA 1B
Notice that we do not have to concern about inverses of non-squared matrices, since our only as-
sumptions affect the diagonal elem(ents. For n>) 2 one then repeat this procedure two derive the
inverse for A and D, as well as of D−CA−1B , which in general yield an iterative description.
210 Chapter 7. Trace evaluation
Field space matrix of rank 3
While applicable for any rank matrix satisfying the above criteria, in the present context field
space operators are given by 3×3 matrices. Thus, we exemplify the described procedure for a
rank 3 matrix and use the result for the Hessian operator derived previously.
( )
A B
M3 =
C D
For later convenience we choose the following decomposition reflecting the separation of metric
and ghost fluctuations and corresponding to r = rk(D) = 2:
( ) ( ) ( ) ( )α4 α6 α7
A= α1 , B= α2 α3 , C = , D=
α5 α8 α9
Hereby, α j represents L2-operators. The necessary conditions that have to be fulfilled in order
to apply the inver(sion rule for )non-commuting matrices affects only the entries A, D, and the
combination N≡ D−CA−1B , which all have to be invertible. In fact, on the level of operators
we demand α1, α7, and α8 to have well defined inverse. Under this assumption, the inverted
matrix reads
( )
α−1+A−1BN−1Cα−1 −α−1BN−1
( )−1M = 1 1 13 −N−1Cα−11 N−1
The remaining evaluation is simple matrix multiplication except for the inversion of N which
again is a matrix of non-commuting components, this time of rank 2. Since we for reasons that
become apparent later on, we have chosen α7 and α8 to be invertible, the following identity
turns out to be very useful:
( ) ( ) ( )
α6−α −14α1 α α −1 −1 −12 7−α4α1 α3 0 1 · α8−α5α1 α2 α9−α5αN = = 1 α3
α −α α−1α α −α α−1 −1 −18 5 1 2 9 5 1 α3 1 0 α6−α4α1 α2 α7−α4α1 α3
Since inversion is a kind of right action that acts on each matrix factor separately while reverting
their order, N−1 can be computed stepwise. Therefore, notice that the artificial traceless matrix
factor is its own inverse, i.e.
( ) ( ) ( )−1 ( )
0 1 · 0 1 0 1 0 112 = thus =
1 0 1 0 1 0 1 0
This construction has the advantage that it becomes evident that for α1, α7, and α8 being invert-
ible in general also N will have an inverse, namely
( )−1 ( )− − −11
−1 α8−α5α1 α2 α9−α 15α1 α3 · 0 1N =
α −α α−1 −16 4 1 α2 α7−α4α1 α3 1 0
( ) ( )
α −α α−
−1
1
8 5 1 α2 α9−α α−15 1 α3 · 0 1=
α −α −1 −16 4α1 α2 α7−α4α1 α3 1 0
Its explicit form can now in principle be deduce with the above prescription, now for 2× 2
matrices. However, for our purpose this form is already sufficient, for we are going to trace this
kind of matrix in the very end. Notice in particular that the off-diagonal entries play introduce
most of the non-standard contributions for the case of non-commuting components.
7.1 Inversion of the Hessian operator 211
The matrix trace of field space
The whole purpose for these derivations is found on RHS of the Functional Renormalization
Group Equation where the trace over an inverted operator on field space has to be computed,
i.e.
[( [ ] )− ]1
STrF Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄]
Thus, we are now going to apply the just derived results to this special case, whereby in the
previous notation the field space operators are 3×3 matrices with non-commuting entries:
[ ] ( )A B [ ]
STr ( −1F M3) ◦∂tRk[Φ̄] with M3 ≡ ≡ Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄]
C D
The aim of this section is the reduction of the functional trace STrF to a mere trace over L
2-
operators STrL2 . Again, we choose a convenient separation of field space operators in blocks
of at most 2×2 matrices and label the operator entries according to the convention used above,
i.e.
( ) ( ) ( ) ( )α4 α6 α7
A= α1 , B= α2 α3 , C = , D=
α5 α8 α9
The cutoff operator is essentially orthogonal in its components, with the following block struc-
ture
( ) ( ) ( ) ( )
(α1) 0 0 (α7) 0 1 (α )Rk 1×2 (D) Rk ≡ ◦ R
8
k 0Rk[Φ̄] = (D) with Rk = (α8) (α0 7
)
2×1 Rk Rk 0 1 0 0 Rk
The last equality will turn out useful in later considerations. Now, the derivative of this operator
w.r.t. the RG time t is composed with the inverse of M3 by means of matrix multiplication.
Employing the explicit form of (M3)
−1 we obtain
( ) ( )
−1 −1 (α )
−1 ◦ A +A BN
−1CA−1 −A−1BN−1 ∂ R 1t
(M ) ∂ k
01×2
3 tRk[Φ̄] = ◦− (D)N−1CA−1 N−1 02×1 ∂tR( )k[ −1 −1 − ]1 −1 (α1) − −1 −1 (D)A +A BN CA ∂tRk A BN ∂tR= k
− −1 −1 (α1)N CA ∂ R N−1 (D)t k ∂tRk
( )
Analog to the previous notation we have introduced an invertible matrix N ≡ D−CA−1B
which is related to the determinant factor appearing in the commuting case. In the resulting
matrix operator the inverse of N has to be inserted in each component which will blow up the
equation even further. However, a slight reduction is obtained when taking the trace of this
matrix, whereby the off-diagonal elements disappear. This is exactly the procedure we have to
apply in order to evaluate the RHS of the FRGE, which in the present case gives rise to
[ ] [( ) ](α )
STr (M )−1 ◦∂ R [Φ̄] = STr A−1+A−1BN−1CA−1F 3 t k 1×1|L2 ∂ 1tR
[ ] k
+ STr N−1 (D)2×2|L2 ∂tRk
Here, STrn×n|L2 denotes the supertrace over n× n matrices with L2-operators as entries, thus it
corresponds to a successive evaluation of matrix trace and STrL2 . Notice that for this truncation
212 Chapter 7. Trace evaluation
STrF ≡ STr3×3|L2 holds true. Hence, the first trace STr1×1|L2 [ · · · ] is identical to STrL2 [ · · · ]. For
the second contribution, before focusing on the operator trace we have to compute the trace of
the 2×2 matrix
( )−1 ( )− −1 ( )1 (α )
−1 (D) α8−α5α1 α2 α9−α −15α1 α3 0 1 0 ∂ 7tRN ∂tR kk =
α −α α−
◦ ◦
1 − −1 (α )6 4 1 α α α α 82 7 4 1 α3 1 0 ∂tRk 0
( )−1 ( )
α −α α−1 (α )8 5 1 α2 α9−α α−1α ∂ R 85 1 3 t= k 0− ◦1 −1 (α ) (7.2)α 76−α4α1 α2 α7−α4α1 α3 0 ∂tRk
Due to the diagonal character of the remaining matrix multiplication and in view of the trace
evaluation, we only have to derive the diagonal elements of the inverse matrix. Proceeding
along the prescription for non-commuting matrices, we thus arrive at
[ ]
(D)
STr N−12×2|L2 ∂tR
[ ( k ( ) ( ) ) ](α )
= STr γ−1+ γ−1 α −α α−1α γ−1 α −α α−1 −1L2 ξ ξ 9 5 1 3 6 4 1 α2 γξ ∂ R
8
t
[ ] ξ̄ k
+ STr γ−1 (α7)L2 ∂tRk (7.3)ξ̄
For this inversion to hold we need two more assumptions, namely that both of the following
operators are invertible:
( )
γξ = α8−α5α−11 α2 (7.4a)(
γ = α −α α−
) ( ) ( )
1
ξ̄ 7 4 1 α3 − α6−α4α−1α γ−11 2 ξ α9−α α
−1
5 1 α3 (7.4b)
Combining this result with the previous one, finally yields the reduction of the functional trace
on field space to operator traces:
[ ]
STrF ( )
−1
M3 ◦∂tRk[Φ̄] (7.5)
[( ) ] [ ]
= STr[h] α−1 (α+α−1BN−1CA−1 ∂ R 1) STr[ξ̄ ]+ γ−1 (α )2 1 1 t k 2 ∂tR
7
L [ ( L ξ̄ k ) ]
[ξ ] ( ) ( )
+STr γ−1 (α )2 ξ + γ
−1
ξ α9−α α
−1
5 1 α γ
−1 α −α α−1α γ−1 8
L 3 ξ̄ 6 4 1 2 ξ
∂tRk
Here, we introduced an additional superscript for the L2-traces indicating over which part of
field space it has to be evaluated. However, due to the domain and target spaces associated to
each operator this is a somehow redundant information, nevertheless it might be helpful.
Actually, for the considered truncation, equation (7.5) can be further simplified due to the
vanishing of the diagonal terms in the ghost sector of the Hessian and the cutoff operator. The
sum of both operators defines M3 which for this special case assumes the form
   
α1
α1 α2 α3 Ĥesshh+Rk Ĥesshξ Ĥesshξ̄
M ≡α 0 α 
 α
3 4 7 ≡ Ĥessξh 0 Ĥess 7ξ ξ̄ +Rk 
α5 α8 0 Ĥessξ̄h Ĥessξ̄ ξ +
α
R 8k 0
Thus, in all derived solutions we can set α6 and α9 to zero. This implies the following reduced
form of eq. (7.4) for the functional trace of M3:
[ ]
STr ( )−1F M3 ◦∂tRk[Φ̄] (7.6)
[( ) ] [ ][h]
= STr α−1 (α ) [ξ̄ ] (α )2 1 +α
−1
1 BN
−1CA−1 ∂tR
1 + STr γ−1 ∂ 7
L k L2 ξ̄ t
R
[ ( ) k( ) ( ) ][ξ ]
+STr γ−1 (α )2 ξ + γ
−1 −1
ξ α5 α1 α3 γ
−1 α α−1α γ−1 8
L ξ̄ 4 1 2 ξ
∂tRk
7.2 Decomposition of symmetric 2-tensor fields 213
Notice that also the abbreviated operators γξ and γξ̄ simplify, for they explicitly depend on the
diagonal elements of the ghost block. Their reduced expression is given by
( )
γξ = α8−α −1( 5
α1 α2 (7.7a)
− ) (γ = α −α α 1α − α α−
) ( )
1
ξ̄ 7 4 1 3 4 1 α2 γ
−1
ξ α5 α
−1
1 α3 (7.7b)
Nevertheless, this seems more to be a drop in the ocean than a true simplification. The entire
expression still consists of several products of operators, some of them being inverted, and there
is thus quite a lot of effort we have to put into the evaluation of the L2-traces.
7.2 Decomposition of symmetric 2-tensor fields
Group theoretical methods are found in various branches of physics and usually come along
with severe simplifications even in very practical applications. Most often the applied tech-
niques have some group theoretical background which however is hidden in the details of the
method. The current status of analyzing the RHS of the FRGE for our specific truncation
ansatz consists of a set of L2-operators acting on the space of either metric or ghost type tensor
fields. While the ghost sector is unconstrained, the metric component of field space is a subset
of symmetric 2-tensor fields fulfilling the non-linear condition of being non-degenerate. For the
fluctuation fields this additional requirement drops out and what remains is an element of the
symmetric subset of T(0,2)M. The aim of this section is a decomposition of these tensor fields
such that the action of the Hessian operator becomes of Laplace type. This technique goes back
to works of Berger and Ebin [176], as well as York [177] and is intrinsically linked with dif-
ferential geometry and group theory. Both combine in the construction of fiber bundles, here
explicitly given by tensor bundles. Furthermore, we consider only operators that arise from a
scalar functional by means of second variation. This implies that the symmetries of the fields
and the invariance of the Γk decide about the variety of allowed operators we may encounter. We
start with the decomposition of a symmetric 2-tensor field based on the irreducible subspaces of
Gl(d;R). Since we have a metric tensor at our disposal, we can further split the resulting com-
ponents using the restricted group action of O(d) and SO(d), for which in particular contraction
is defined. This principle is first applied to a generic k-form in the presence of boundaries and
then presented in detail for the symmetric part of a 2-tensor field. The result is ultimately imple-
mented into our current form of the operator trace. For more details, references, and derivations,
as well as a mathematical discussion we refer to [178, 179] for the Hodge theorem and [176,
177, 180–182] for the symmetric tensor field decomposition, respectively.
As a preliminary remark, let us consider a generic tensor field t ∈ T(p,q)M which by def-
inition has at most dp+q independent components. Imposing symmetry restrictions or other
constraints will in general reduce this number. In the course of this section we will see how
certain symmetry conditions affect the overall structure of tensor fields. For instance, choosing
p= 0 and q= 2, we showed in subsection 1.2.4 that an arbitrary t ∈T(0,2)M consist of two irre-
ducible components w.r.t. the general linear group Gl(d;R): a symmetric and an anti-symmetric
part, thus
t S A S S A Aµν = tµν + tµν with tµν = tν µ ∧ tµν =−tν µ (7.8)
We thus replaced an unconstrained generic tensor field by two constrained tensor fields of the
same degree, whereby the total number of independent components is unchanged. This be-
comes in particular useful if we consider transformations of t, for instance a symmetric, linear
transformation T̂ µν = T̂ ν µ :
T̂ µνtµν = T̂
µνtS µνµν + T̂ t
A ≡ T̂ µνµν tSµν
214 Chapter 7. Trace evaluation
In the last step the main advantage of the decomposition method becomes apparent: the anti-
symmetric component vanishes under the action of a symmetric transformation. Any further
analysis based on this contraction involves only the independent components of tSµν , thus we
have a reduction from d2 to d(d+ 1)/2 independent quantities. This suggests that a decision
about the most efficient decomposition is very sensitive on the symmetry properties of the as-
sociated operator. Hence most of the decomposition theorems rely on a specific setting as for
instance Riemannian geometry with its natural differential operators.
In the sequel we describe a systematic decomposition method built on the covariant deriva-
tive D̄ for a smooth, connected and compact Riemannian manifold with in general non-vanishing
boundary. Its original formulation was given for manifolds without boundary [176], though
generalizations for ∂M are known since decades, for instance [181]. Given a scalar product
on the tensor algebra we can uniquely define the transverse and longitudinal part of a tensor
field:
Definition 7.2.1 — Transverse & longitudinal tensors. Let M be(a sm)ooth manifold and
T(p,q)M the space of tensor fields of type (p,q). Furthermore, let · , · (p,q)
T(p,q)
: T M×
M
T(p,q)M→ R be the inner product on T(p,q)M.
β ···b
A tensor field t ∈ T(p,q)M, in a local chart given by 1 ptα1···α (x), is called . . .q
. . . transverse in α j (β j) :⇔ µ β1···β j ···bp β1···µ ···bpD̄ tα1···µ ···α (x) = 0 resp. D̄µ tα1···α ···α (x) = 0q j q
. . . longitudinal in α (β ) :⇔ ∀v ∈ T(p,q)j j ( ) M transverse in α j (β j) :
v , t
T(p,q)
= 0
M
The longitudinal tensor fields form the orthogonal complement of the transverse sector.
A simple example is given by tensors of type (1,0), i.e. the set of vector fields, which can be
decomposed as t = (tT)+(tL) with D̄ (tT)µµ = 0 and ḡ(tT , tL) = 0. Orthogonality will turn out to
be very useful due to the decoupling of the different contributions in the decomposition.
7.2.1 Hodge theorem for k-forms with ∂M 6= 0/
The general procedure to decompose tensor fields – and thus also suitable for its fully anti-
symmetric subsets the k-forms – is based on the kernel of a certain differential operator. As
mentioned above we are going to employ a Hodge-decomposition associated to the covariant
derivative. In order to keep the notation as general as possible, we introduce a new symbol
d, thus one can replace D̄ with different operators was long as they share similar properties.
Furthermore, in the present context it is worth emphasizing the degree of differential k-form by
k
writing ω ≡ω∈ Λk(M). Using the Riemannian metric ḡ as the generator of the inner products
on T(p,q)M we define
k k k
d : Λk(M)→ Λk+1(M) with d ω= D̄ ω∈ Λk+1• (M) ∀ ω∈ Λk(M)
Equipped with an inner product, there is an associated – in a sense reverted – adjoint operator
which is also known as co-differential:
− (k → k 1 k k-1
) ( k k-1 )
δ : Λ (M) Λ (M) with ω , d υ ≡ δ ω , υ
T(0,k)M T(0,k−1)M
The properties of the inner product can be absorbed in what is called the Hodge-star operator:
( )
k 1 k j1,..., jk
⋆ ω ≡ √ω ḡε j ,..., j ,i ,...,i
i1,...,id−k k!
1 k 1 d−k
7.2 Decomposition of symmetric 2-tensor fields 215
Here, we have introduced the fully anti-symmetric epsilon tensor ε representing the Young
operator of the anti-symmetric component. This allows for a more convenient expression of the
co-differential δ in terms of ⋆ and d:
k
δ ω≡ (− k1)d(k+1) (⋆d⋆) ω
Hodge theorem
In case of manifolds without boundary the ’Hodge theorem’ provides a suitable decomposi-
tion of k-forms into independent constituents based on the image and kernels of d and the
co-differential δ:
Λk(M) = ker (dδ+ d)⊕dΛk−1δ (M)⊕ k+1δΛ (M) (Hodge decomposition) (7.9)
Any k-form can thus be expressed as a sum of some transverse and longitudinal terms, whereby
the first summand represents a harmonic k-form:
{ }
k k
Harmk ≡ ker(∆ ≡ dδ+δd)≡ ω | (dδ+δd) ω= 0
Here, ∆ ≡ dδ+ δd defines the usual Laplacian operator, which is elliptic in case of Dirichlet
boundary conditions.
Hodge-Morrey theorem
The Hodge-decomposition is a very useful result which finds application in many branches
of mathematics and physics, hence generalizations to manifolds with non-vanishing boundary
were of great interest. However, in this latter case neither δ = d∗∗ holds true nor are the split
components orthogonal, due to the appearance of boundary contributions when integrating by
parts. Fortunately, it is sufficient to slightly change the former decomposition and exchange the
set of harmonic k-forms with the set of harmonic k-fields, introduced by Kodaira [183]:1 For
∂M 6= 0/ boundary terms will in general spoil this identity.
{ }
k ≡ kH ω | k kd ω= δ ω= 0
In addition to this replacement one has to implement boundary conditions on the k-forms which
are either of Dirichlet or Neumann type, respectively denoted as
{ } { ( )∣ }
Λk
k k k k ∣
(D)(M)≡ ω | ω | k∂M = 0 and Λ(N)(M)≡ ω | ⋆ ω ∣ = 0
∂M
These modification give rise to the ’Hodge-Morrey decomposition’, the generalization of eq.
(7.9) for the case of manifolds with non-vanishing boundary:
Λk(M) = Hk(M)⊕dΛk−1( ) (M)⊕ Λ
k+1
δ ( ) (M) (Hodge-Morrey decomposition) (7.10)D N
The boundary conditions ensure the orthogonality of this splitting, necessary for a suitable
decomposition of the underlying vector space. Notice, if the boundary of M happens to be
empty, ∂M= 0/ , the Hodge-Morrey theorem reduces to eq. (7.9), as required.
1Notice that the harmonic fields are a subset of harmonic forms, i.e. Hk ⊆Harmk. In case of vanishing boundary,
∂M= 0/ , in fact equality holds which can be seen by
( ) ( ) ( )
k k k k k k k k k k
0= ω ,(dδ+δd) ω = d ω ,d ω + δ ω,δ ω = ‖d ω ‖+‖δ ω ‖⇒ d ω= δ ω= 0 .
216 Chapter 7. Trace evaluation
Hodge-Morrey decomposition for 1-forms
For the final splitting of symmetric 2-tensor fields we will only need the explicit decomposition
of dual vectors, i.e. 1-forms. From the Hodge-Morrey theorem we can deduce the following
mutually orthogonal decomposition with respect to the above defined inner product:
Λ1(M) = H1(M)⊕dΛ0(D)(M)⊕δΛ2(N)(M)
The first constituent of a dual vector is the harmonic field component, i.e. an element of H1(M)
1 1
satisfying the closeness (d υ= 0) and co-closeness condition (δ υ= 0). The next constituent
Λ0( )(M) is equivalent to the space of continuous differentiable functions which in additionD
satisfy Dirichlet-boundary conditions. Its contribution to the orthogonal decomposition is lo-
0
cally given by D̄µ σ dxµ . Finally, the last ingredient is constructed by the application of the
co-differential on a 2-form respecting Neumann conditions on ∂M, i.e.
2 2 2
δ η≡ (−1)d (⋆d⋆) η= (−1)dD̄ν η µµν dx
To obtain the second equality one actually has to determine the explicit form of the co-differential
acting on Λ2( )(M). Therefore, we computation rules for the epsilon tensor are used to arrive atN
2 2
(δ η)µ = (−1)dD̄ν η µν .
Due to the Hodge-Morrey theorem we can thus represent any generic 1-form as a sum of
three constrained constituents
( )
1 1 1 0 2
ω=ω dxµµ = υµ +D̄µ σ +(−1)dD̄ν ηµν dxµ (7.11)
While the first term underlies the harmonic field constraint, the remaining parts only fulfill
boundary conditions, i.e.
1 1 1 1 0 2
d υ= D̄ν υµ= 0= δ υ= D̄
µ υµ ∧ σ |∂M = 0 ∧ D̄ν ηµν |∂M = 0 (7.12)
In the sequel, a simplified notation involving only two constituents will be sufficient. Therefore,
let us combine the harmonic field and the image of the co-differential into a new – kind of
1 1 2
longitudinal – term ξ=υ +δ η :
( )
1 1 0 1 1 2 1
ω= ξ µ +D̄µ σ dx
µ with ξ= υ + ◦ η= 0= D̄µδ δ δ δ ξ µ ∧ σ |∂M = 0 (7.13)
1 1 2 1 1
Since ξ µ |∂M = (υµ +D̄ν η µν)|∂M =υµ |∂M holds true, the new field ξ is unconstrained on the
boundary of M.
7.2.2 Decomposition theorem for symmetric 2-tensors for ∂M 6= 0/
At the very beginning of this section, we have stated a result from group theory that a generic
2-tensor field t ∈ T(0,2)M can be partitioned into its irreducible components w.r.t. Gl(d;R),
namely tµν = tS Aµν + tµν , or in more general terms:
{ }
T(0,2)M≡ Sym2(M)+Λ2(M) with Sym2(M)≡ s ∈ T(0,2)M |sµν = sν µ (7.14)
This is a first, however in general not orthogonal, decomposition based on the general linear
group. Once an inner product is at our disposal we can further decompose the symmetric and
anti-symmetric tensor field looking for the irreducible parts of the restricted representations
7.2 Decomposition of symmetric 2-tensor fields 217
for the orthogonal group O(d) which resulted in the famous Hodge-Morrey theorem for the
anti-symmetric tensor tA ∈ Λ2(M).
In the sequel of this section we present a similar orthogonal decomposition for Sym2(M),
whereby the previous results for k-forms turn out to be very useful. Instead of directly introduc-
ing the operator d(S) that provides the decomposition scheme for Sym
2(M), we take a look at
the more intuitive co-differential, or adjoint operator. It represents the counterpart of the fully
2
anti-symmetric δω in that we have to replace the Young operator of Gl(d;R) with its symmetric
analog yielding
( )
∗ 1 1 1 1d( ) ω ≡ 2 D̄µ ων +D̄ν ωµ dxµ ⊗dxν = 2L• ω ∈ Sym (M)S
Remarkably, it is the Lie derivative of a 1-form that generates a symmetric tensor field. Using
the inner product related to the field space metric, the associated differential operator reads
( ) ( )
d s≡ dxβ −2gρσ µν(S) ḡβν D̄ ρµ sρσ ≡−2 D̄ sρβ +ϖD̄ ρσβ ḡ sρσ dxβ
Since the manifold is endowed with a boundary, it will in general be necessary to impose bound-
ary conditions in order to establish the close bond of adjointness between d ∗(S) and d( ). In partic-S
ular Dirichlet constraints suffices to get rid of the boundary terms that occur when integrating
by parts.
For this natural choice of operators there exists a Hodge-Morrey like theorem for Sym2(M):
Again, let us assumeM to be a compact Riemannian manifold with regular, orientable boundary
∂M 6= 0/ . Then, the decomposition with respect to d(S) for symmetric 2-tensor fields satisfying
Dirichlet boundary conditions is given by:
Sym2(D)(M) = d
∗
( )Λ
1
(D)(M)⊕ker(d(S)) (7.15)S
Thus, based on this theorem a generic symmetric 2-tensor field s ∈ Sym2(D)(M) that vanishes on
∂M can be decomposed as follows
( )
s= s µ ν µ νµνdx ⊗dx = D̄µων + D̄νωµ + zµν dx ⊗dx
Hereby the left-hand-side (LHS) is solely constrained to be symmetric and of Dirichlet type,
while the RHS contains tensor fields obeying additional conditions, namely
( ) 1
zµν = z ∧ gρσ µνν µ ḡβν D̄µ zρσ = 0 ∧ ω |∂M = 0
1
Finally, we can apply the Hodge-Morrey decomposition (7.13) to ω which yields a representa-
tion based on two more restricted tensor fields ξ and σ . Due to the identity D̄µD̄νσ = D̄νD̄µσ
for scalar fields this gives rise to
( )
s= s dxµ ⊗dxν = D̄ ξ + D̄ ξ +2D̄ D̄ σ + z dxµ νµν µ ν ν µ µ ν µν ⊗dx (7.16)
Instead of a single symmetric 2-tensor field s we have thus a total number of three constrained
tensors of different degree, each carrying a certain aspect of s. In detail, we have the following
1 0
list of conditions that have to be satisfied by z, ξ and σ :
1
δ ξ= D̄µξµ = 0 ∧ (ξν |∂M =− D̄ν)σ |∂M ∧ σ |∂M = 0
z = z ∧ gρσ µνµν ν µ ḡβν D̄µ zρσ = 0 (7.17)
1 1 1 2
Notice that, if necessary, ξ can be further orthogonally decomposed as ξ=υ +δ η .
218 Chapter 7. Trace evaluation
7.2.3 The transverse-traceless decomposition
There are several ways to reduce the tensor structure of the Hessian to some simple scalar
operator, one using a full decomposition of the metric fluctuation as presented in eq. (7.16).
However, this projection technique implies a whole sequence of traces over constrained fields
which then have to be individually evaluated using different heat kernel expansions. Later on
we are going to introduce another method that at least in the present truncation simplifies the
trace evaluation a lot. Basically, it employs not the full metric field decomposition, but only a
split into trace and traceless part, i.e.
s noTr
1 Tr
µν ≡ sµν + ḡµνs (7.18)d
whereby sTr = bgµνsµν denotes the trace part of s and snoTr the remaining traceless component,
satisfying ḡµνsnoTrµν = 0. It is also very instructive to combine both decompositions which re-
sults in the transverse-traceless decomposition. Therefore notice that from eq. (7.16) using the
1 0
constraints of (7.17), we express the trace part of s in terms of the tensor fields z, ξ and σ , i.e.
sTr = ḡµνsTr = 2D̄µξ +2D̄2σ + ḡµνµν µ z
2
µν = 2D̄ σ + z
Tr
1
In the second step we made use of the transverse property of ξ , i.e. D̄µξµ = 0. Hence, the
traceless part assumes the form
( )
snoTr
1 1
= s − ḡµνsTr = D̄ ξ + D̄ ξ +2D̄ D̄ σ + z − ḡ 2D̄2σ + zTrµν µν µ ν ν µ µ ν µν µνd ( ) ( d )
= D̄µξν + D̄νξµ +2 gρµgσ − 1g g D̄ρD̄λ σ + g 1 ρλd µν λσ ρµgσλ − gd µνgλσ z
Remarkably, the brackets in the last two terms are reminiscent of the local field space metric,
gµνρλ , for ϖ = −1/d. Whenever this matching holds true, zTr is not only traceless but in fact
also transverse as is (D̄µξν + D̄νξµ) however for any choice of ϖ .
In the present case, we follow a path that circumvents the full decomposition of the tensor
fields by using the freedom of the gauge fixing condition and only split the metric fluctuations
into its trace- and traceless part. However, this new, time-saving technique might be suitable
only for a class of truncations, in particular it might be limited to an ansatz with a sufficient
small number of derivatives in the invariants. Thus, further investigations for manifolds with
non-vanishing boundary may rely on the full decomposition procedure presented here.
7.3 Projection techniques for truncations
The Functional Renormalization Group Equation combines the difficulties of functional differ-
ential equations with the problem of non-linearity:
[
1 ( [ ] ) ]−1
∂tΓk[ϕ ;Φ̄] = STr Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄] (7.19)
2
Exact as it stands, the usual employed simplifications are based on a projection of theory
space T to its coordinatized version, making use of a full set of independent basis invariants
{Pa[ϕ ;Φ̄]}. Formally, an action functional living on theory space T can be expanded in this
basis, whereby the coefficients are associated to combinations of couplings:
T∋ A[ϕ ;Φ̄]≡ ∑ ūa ·Pa[ϕ ;Φ̄]
a
7.3 Projection techniques for truncations 219
At least on a formal level, we can use this expansion to rewrite the single functional differ-
ential equation in terms of an infinite number of ordinary differential equations for the coef-
ficients. Therefore, we insert the expanded form of the effective average action, Γk[ϕ ;Φ̄] ≡
∑a ū
a
k ·Pa[ϕ ;Φ̄], into the FRGE.
( )
LHS : ∂tΓk[ϕ ;Φ̄] = ∂t ∑ ūak ·Pa[ϕ ;Φ̄] = ∑ (∂ at ūk) ·Pa[ϕ ;Φ̄]
a a
Furthermore, since we assume {Pa[ϕ̂;Φ̄]} to form a complete, linearly independent basis, the
RHS of eq. (7.19) can be projected onto the basis invariants {Pa[ϕ̂;Φ̄]} in a unique way, which
defines the dimensionful beta-functions β̄ :
[( [ ] ) ]1 −1
RHS : STr Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄] =.. ∑ β̄ a({ūbk};k) ·Pa[ϕ ;Φ̄]2 a
In order to obtain the differential equations for the (dimensionful) coefficients, we have to com-
bine both sides yielding
∑ (∂ at ūk) ·Pa[ϕ ;Φ̄] = ∑ β̄ a({ūbk};k) ·Pb[ϕ ;Φ̄] ⇐⇒
a ( )a
∑ ∂ ūa− β̄ a({ūbt k k};k) ·Pa[ϕ ;Φ̄] = 0
a
Since the P’s are linearly independent, this implies that each bracket has to vanish identically.
This is the aforementioned infinite number of differential equations for the ū’s, each describ-
ing the RG evolution along a certain direction in theory space, associated to a specific basis
invariant.
∂t ū
a
k = β̄
a({ūbk};k) ∀a
Due to the second functional variation that constitutes the operator on the RHS, the resulting
differential equations are in general coupled in an hierarchical order, which is indicated by
the explicit dependence of β̄ on the set of coefficients. Furthermore, due to the connection
between the cutoff operator and the background metric, there is an additional contribution in the
dimensionful version of the EAA which gives rise to a non-autonomous system of differential
equations, thus the explicit dependence of β̄ on k.
In this way, we get rid of the functional nature of the non-linear differential equation, how-
ever we are now plagued by the problem to simultaneously solve an infinite number of coupled
ordinary differential equations. As long as we do not have a systematic technique to, for instance
iteratively, attack this mathematical problem, we have to rely on approximations where only a
(possibly still infinite) subset of field monomials is considered. However, such a truncated the-
ory space is in general not invariant under the RG flow, since the non-linearity together with the
second variation of the basis invariants will lead to a generation of terms on the RHS which are
not present in the original ansatz for Γk. Nevertheless, once we consider a subset which exhibits
an additional symmetry, it might happen that the FRGE preserves this symmetry and the RG
evolution closes for the most general truncation ansatz of this kind.
To get a better understanding of the approximation involved when truncating theory space,
let us introduce a projection operator
( )
ΠP (A)≡ Π ūaP ·P ..= ūb ·Pb b ∑ b a
a
220 Chapter 7. Trace evaluation
Surely, if it exists, ΠP is idempotent and thus in fact a projection. The crucial point here isb
the infinite dimensional character of theory space. Assume that we can find a suitable linearly
independent basis of field monomials on T, in the general case this space will not be endowed
with an inner product and projection operators are difficult to obtain. Still, let us proceed on a
formal level and apply ΠP to the FRGE:b
( ) ( [( [ ] )− ])1 1
ΠP ∂tΓk[ϕ ;Φ̄] = ΠP STr Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂b tRk[Φ̄]2 b
Notice that the basis invariants are k-independent in the dimensionful description. Thus, ∂t and
ΠP commute and we can change the order on the LHS. Again, we insert the coordinatizedb
version of the effective average action Γ [ϕ ;Φ̄] = ∑ ūak a k ·Pa[ϕ ;Φ̄], now on both sides, yielding
( [( )− ])1 [ ] 1
(∂t ū
b
k) ·Pb[ϕ ;Φ̄] = ΠP STr Ĥess ∑ ūaϕ k ·Pa[ϕ ;Φ̄] +Rk[Φ̄] ◦∂b tRk[Φ̄]2 a
The supertrace gives rise to the dimensionful beta-functions and thus the application of ΠPb
reproduces the differential equation for the coefficient ūbk :
( )
(∂t ū
b
k) ·Pb[ϕ ;Φ̄] = Π ∑ β̄ a({ūbP k};k) ·Pa[ϕ ;Φ̄] = β b({ūbk};k) ·Pb[ϕ ;Φ̄]b
a
Hence, even if we are only interested in the RG evolution of a single basis invariant, we still
have to compute the Hessian of a generic action functional in full theory space, for the projection
is applied only after evaluating the trace.
In order to actually reduce theory space in a practical manner, we have to simplify the RHS
before expanding the trace or even before computing the Hessian operator. This kind of trunca-
tion is in general only an approximated solution to the exact FRGE but it is so far unavoidable.
Thus, one of the crucial, yet very complicated, parts in the evaluation of truncations is an es-
timate of the validity of the obtained results. We come back to this point when studying the
generalized C-function like proposal. For now, let us choose a specific truncation in the shape
of a projection operator
ΠSTrunc(Γk) = Π
S a .. a
Trunc(∑ ūk ·Pa[ϕ ;Φ̄]) = ∑ ūk ·Pa[ϕ ;Φ̄]
a a∈S
Usually S is a finite set, though even in theory spaces of metric quantum gravity there systematic
treatments of some infinite subsets of Twere successfully carried out. The approximation arises
once we substitute ΠSTrunc(Γk) for Γk in the Hessian operator on the RHS. Thereby, we neglect
the additional contributions to the functional trace which originate from the omitted invariants.
Technically, this approximation can be understood as a ‘sandwiched’ type of projection, i.e.
[( [ ] )
S 1 −1 ]∂tΠ (Γk[ϕ ;Φ̄]) = ΠS (STr Ĥessϕ ΠS (Γk[ϕ ;Φ̄]) +R [Φ̄] ◦∂ R [Φ̄] )Trunc Trunc Trunc k t k2
+ corrections
The corrections appear not due to the outer projection but the inner one, which in general ne-
glects an infinite number of contributions. In the present case the RHS of the FRGE contains
invariants with higher powers of R, R̄, (ξ ξ̄ ), · · · , for instance, while its LHS consists of only
a few of them, namely those invariants retained in the Einstein-Hilbert and Gibbons-Hawking-
York functional. Thus, in order to complete the specification of the truncation it is necessary to
define a projection of the RHS onto exactly those field monomials.
7.3 Projection techniques for truncations 221
Besides a physical motivated ansatz for Γk, the main focus in the development of suitable
truncation schemes is the simplification of the Hessian operator appearing on the RHS. Since it
its difficult to predict issues that may arise during the evaluation process, the general procedure
is to initially start with a certain class of action functionals that have some physical or mathemat-
ical relevance. Then one has to compute and invert the Hessian operator before finally taking
the trace over the composed endomorphism. Whenever severe problems in one of these steps
occur one adapts slight modifications of the truncation ansatz such that standard techniques
can be applied, as for example heat kernel techniques. Usually, these modifications change the
truncated theory space by fixing the k-dependence of certain coefficients and thus modify the
projection operator in the following sense:
(S,S
Π fix
) (S,S )
Trunc (Γk[ϕ ;Φ̄])≡ Π fixTrunc (∑ ūak ·Pa[ϕ ;Φ̄])
a
≡ ∑ ūak ·Pa[ϕ ;Φ̄]+ ∑ ca({ūbk};k) ·Pa[ϕ ;Φ̄]
a∈S a∈Sfix
Notice that the fixed coefficients ca({ūbk};k) are in general artificially matched to a combination
of the running couplings and still k-dependent. The price one has to pay for simplifying the
RHS of the FRGE is reflected by the reduced dimensionality of truncated theory space, which
is determined by dim(S) only. With new techniques one then can study the validity of the
previously considered truncation by including the fixed basis invariants and look for solutions
that stay close or even agree with the choice ca({ūbk};k).
Apart from an actual modification of the truncation ansatz, we can make use of all simplify-
(S,S )
ing techniques that commute with the truncation projection Π fixTrunc (Γk[ϕ ;Φ̄]) and the functional
trace. We will encounter a particular example of this method applied to the ghost sector. We
have seen that the inversion of a matrix with non-commuting entries results in a very compli-
cated expression, which however reduces significantly once the matrix assumes a block diag-
onal form. All the problematic contributions are in fact proportional either to ξ̄ or ξ . Thus
fixing and thereby modifying the truncation such that the ghost invariants have a fixed (not inde-
(S,S )
pendently running) coefficient results in a projection operator Π fixTrunc (Γk[ϕ ;Φ̄]) that commutes
with ξ̄ = 0 = ξ . Even changing invariants on intermediate steps is acceptable and does not
reduce the considered truncated space as long as the included basis invariants can be uniquely
resolved (see for instance the maximally symmetric projection technique).
In the following, we are going to simplify the very general form of the present truncation
ansatz such that we can ultimately apply the heat kernel expansion to determine the dimension-
ful beta-functions. First of all, based on the results of the matrix inversion formula, we will fix
the ghost sector by omitting the k-dependence of its coefficient.
7.3.1 Fixing the ghost sector
Let us now consider truncations for which the ghost sector is kept fixed, that is it contributes
to the RG evolution of the metric fluctuations but does not feel any RG effect for its own. For
the exact flow, one expects there is no apparent reason why we should assume a trivial RG
behavior of the ghost sector, however when approximating the exact Effective Average Action
it is always a question which contributions to omit. In fact, there have been investigations
of the ghost sector and its RG dependence [132–134], however in most cases the community
focuses on the gravitational part alone. The fixation of the ghost sector can be easily arranged
by keeping its coefficient artificially independent, meaning ghk ū ≡ ūghk = const. Up to now, all
results are such general that neglecting the ghost sector in the RG evolution might seem most
unnecessarily. The reason is closely related to the complicated structure of the trace which we
are going to evaluate in the final step using heat kernel methods. While the derivation of the
222 Chapter 7. Trace evaluation
Hessian operator, the inversion of the matrix operator, and the evaluation of the matrix trace
were shown for a large class of gauge fixing functionals and including the full ghost sector,
this final step in computing the remaining L2 traces requires a list of further simplifications.
Fortunately, we thus have a lot of freedom by either adapting the gauge fixing condition to fit
our needs or modifying the truncation ansatz.
R Notice that keeping the ghost sector fixed under RG evolution does not correspond to a
vanishing of Γgh[ϕ ;Φ̄]. In the latter case, being equivalent to ghū = ūghk k ≡ 0, there will be
no contributions from the ghost sector to the gravitational RG solutions. For general ūgh
however, we simply do not resolve the running which is perfectly fine in the vicinity of a
possible fixed point and in regions where ghūk is almost k independent.
Hence, let us have a look at what kind of reduction might appear once we keep ghū ≡ ūghk
fixed. While the RHS of the FRGE is almost unaffected by this choice, all ghost invariants
disappear on the LHS, since ∂t now acts on a constant ghost sector. Thus, the LHS, ∂tΓk[ϕ ;Φ̄],
is independent on ξ and ξ̄ . This in particular implies that all monomials that are generated in
the RG procedure and containing ghost fields are projected out. In other words, the projection
of the RHS onto our truncation ansatz does not coincide with the identity map, indicating that
the truncation is not closed and further invariants have to be added.2 The great advantage once
we purely focus on the RG evolution of the gravitational sector is thus the irrelevance of field
monomials on the RHS containing ξ or ξ̄ . As a natural consequence all contributions to the
Hessian operator which are ghost field dependent can be neglected, for they are finally projected
out. In the present case this amounts to the following reduction:
 ∣∣  α1
Ĥesshh+Rk 0 0 ∣∣ α1 0 0 
M3|ξ̄=0=ξ ≡  0 0
α
Ĥess 7ξ ξ̄ +Rk ∣∣ ≡ 0 0 α ∣ 7
0 αĤess 8ξ̄ ξ +Rk 0 ∣ 0 α8 0ξ̄=0=ξ
At this point it becomes apparent why we have chosen α1, α7, and α8 to be invertible while all
other operators were kept unconstrained. This constellation is indeed necessary, for all other
operators vanishes under projecting out ghost invariants, in particular the operator matrices
B and C, which become non-invertible. The great virtue of being ignorant against the RG
evolution for the ghost invariants appears on the level of the L2-trace. Adducing eq. (7.6), the
functional trace reduces quite efficiently to
[ ]∣∣
STr ( )−1F M3 ◦∂tRk[Φ̄] ∣ (7.20)
[ ξ̄=0=]ξ [ ] [ ]
[h] (
= STr −
) (α ) [ξ̄ ] ( − )1 1 1 (α7) [ξ ] ( − )1 (α8)
L2
α1 ∂tRk + STr 2 α7 ∂tRk +STr 2 α8 ∂tRL L k
Hence, the orthogonality of the Hessian and cutoff operator within this class of truncations
yields a sum over three non-entangled operators. It is for this severe simplification that we
reduce our truncation ansatz such that it becomes insensitive of the RG flow of the ghost sec-
tor. Nevertheless, it should be stated that all truncations including ghost invariants have not
altered the qualitative results, in particular they agree with the Asymptotic Safety conjecture for
Quantum Gravity [132–134].
R While
gh
ūk ≡ ūgh corresponds to an actual reduction of the truncated theory space, thus
modifies the truncation ansatz, the evaluation of ξ̄ = 0 = ξ describes a projection tech-
(S,S )
nique Π fixξ̄=0=ξ that commutes with the modified projection operator ΠTrunc (Γk[ϕ ;Φ̄]).
2Nevertheless, this also happens in the metric sector due to the appearance of higher derivative terms which we
are not going to resolve.
7.3 Projection techniques for truncations 223
7.3.2 Conformal projection technique
The remaining trace evaluation is only concerned with L2-operator based on the covariant deriva-
tives w.r.t. ḡ and g. However, the preferred structure of these operators would be a function of
a single Laplacian only, since otherwise we have to deal with the non-commuting operators D̄
and D. The general procedure would be to expand the Hessian operator, which is a function of
ḡ and g, in powers of h and then evaluate the trace level by level in the fluctuation field h:
∣
: ∂ Γgrav[ , ]≡ ∂ Γgrav[ , ]+∂ ΓgravLHS g ḡ ḡ ḡ d [g, ḡ]∣ +O(h2t k t k t h k )g=ḡ
RHS : STr [F[g, ḡ] ]≡ STr [F [ḡ, ḡ]+dhF[g, ḡ]|g=ḡ+O(h) ] (7.21)
For the present truncation ansatz, in which all level-(p) with p ≥ 1 coefficients are identified,
it suffices to compare only the first two orders in h. Thus at each order of the expansion D is
identified with D̄ at the expanse of an additional fluctuation field h and a more difficult tensor
structure. While this cures the problem of non-commutation, we will depart further from a
purely Laplacian operator form.
Thus, we are need of a different technique that solves both problems at once but still re-
tains the essential information of the truncation ansatz. This is actually the main problem in
simplifying the Hessian operator, namely that the same method has to be applied also on the
LHS which may in general alter the truncation ansatz. As described in the introduction of
this section, a suitable projection technique has to commute with ΠTrunc(•) and in case it does
not modifications of the truncated subset arise. In the current analysis, the gravitational sector
should remain unaltered, while adaptions to the ghost and gauge fixing functionals are accept-
able. This was already utilized in setting ghūk to a fixed value in order to apply the zero-ghost
projection technique.
The single-metric projection
First, let us consider a very simple projection namely Πg=ḡ(F[g, ḡ]) ≡ F[g, ḡ]|g ≡ ḡ which cor-
responds to the zeroth order in the expansion of eq. (7.21). Then, naturally all covariant deriva-
tives D are replaced by D̄ while the original tensor structure remains unchanged, thus yielding
a RHS that is perfectly suitable for the further analysis. However, this projection does not com-
mute even with the reduction onto the gravitational part of the truncation ansatz, which consists
of seven basis invariants, either purely constructed w.r.t. ḡ or g:
∫ √ ∫ √ ∫ √ ∫ √
ddx ḡ R̄ ddx ḡ dd−1 H̄ K̄ dd−1 H̄
∫M ∫M ∫ ∂M √ ∂M
dd
√ √
x gR ddx g dd−1x H K
M M ∂M
Under the action of Πg=ḡ onto the FRGE the Hessian operator gets sufficiently simplified, how-
ever all dynamical invariants reduce to their background counterparts and thus cannot be re-
solved in the RG evolution:
Πg=ḡ(LHS) : Π
grav grav
g=ḡ(∂tΓk [g, ḡ])≡ ∂tΓk [ḡ, ḡ]
Πg=ḡ(RHS) : STr [Πg=ḡ(F [g, ḡ]) ]≡ STr [F[ḡ, ḡ] ]
While at the beginning we distinguished between seven basis invariants constructed with the
background and dynamical field ḡ and g, respectively, the projection projg=ḡ retains only four
distinguishable monomials. In particular for the Einstein-Hilbert functionals we have
( ∫
1 · √
∫ ) ( ∫ ∫ )
Πg=ḡ ∂t ūk gR+∂
2
t ūk ·
√ √
ḡR̄ = Π 1 2
√
g=ḡ β̄ · gR+ β̄ · ḡR̄
=⇒ ∂t(ū1k + ū2k) = (β̄ 1+ β̄ 2)
224 Chapter 7. Trace evaluation
This demonstrates that Πg=ḡ is incapable of resolving the background and dynamical RG evolu-
tions and thus inappropriate for bi-metric truncations.
The bi-metric conformal projection
We now demonstrate that for ∂M= 0/ a slight modification of Πg=ḡ is sufficient to extract the full
set of beta-functions described by eq. (7.21) while keeping the tensor structure as simple as for
g = ḡ. The cure that circumvents both problems at once is the conformal projection methods,
which in the case of vanishing boundary was successfully applied in ref. [172, 173] to study bi-
metric truncations. The essential step is instead of identifying g and ḡ one conformally relates
the two metrics as follows: gµν(x) = e2Ωḡµν(x). Then, after having eliminated gµν in favor of
Ω and ḡµν , the associated projection operator ΠΩ is used to expand the supertrace in the flow
equation in powers of the x-independent conformal factor Ω. For Ω = 0 we obtain the invariants
built from ḡ, whereas the level-(1) invariants ∝ h are identified as those linear in Ω. All terms
which are at least quadratic in Ω are irrelevant in the present context, since we identified all
couplings of level-(1) and higher.
Indeed Π . 2ΩΩ(F [g, ḡ]) .= F[e ḡ, ḡ] separates the contributions to background and dynamical
invariants. When applied to the FRGE it results in the following LHS and RHS:
Π (LHS) : Π (∂ Γgrav[g, ḡ])≡ ∂ Γgrav[e2ΩΩ Ω t k t k ḡ, ḡ][ ]
Π (RHS) : STr [Π (F[g, ḡ]) ]≡ STr F[e2ΩΩ Ω ḡ, ḡ] (7.22)
In fact, expanding both sides in powers of Ω yields an expression quite similar to eq. (7.21)
with hµν ≡ (e2Ω −1)ḡµν :
Π ( ) : ∂ Γgrav[ , ]+∂ (∂ ΓgravLHS ḡ ḡ [e2ΩΩ t k ḡ, ḡ]| ) ·Ω+O(Ω2)[ t Ω k Ω=0 ]
Π (RHS) : STr F[ḡ, ḡ]+∂ F[e2Ωḡ, ḡ]| ·Ω+O(Ω2Ω Ω Ω=0 ) (7.23)
On intermediate steps Ω thus acts as a separating device that keeps the terms associated to
level-(0) or higher orders in h, since the former are independent of h and hence of Ω. For the
Einstein-Hilbert functionals this separation reads as follows:
( ∫ ∫ ) ( ∫ ∫ )
Π ∂ ū1 · √gR+∂ ū2 √ 1 √ 2 √Ω t k t k · ḡR̄ = ΠΩ β̄ · gR+ β̄ · ḡR̄
( ) ( ) ( ) ( )
=⇒ ∂ 1t ūk + ū2 +Ω · (d−4)ū1k k +O(Ω2) = β̄ 1+ β̄ 2 +Ω · (d−4)β̄ 1 +O(Ω2)
Hence, while Ω = 0 gives rise to the single-metric projection including the background and
dynamical beta-functions, the first order in Ω contains only the dynamical terms and thus we
have two independent equations to solve for ū1 and ū2, separately.
While the gravitational functional on the bulk is preserved under conformal projection, the
gauge fixing functional is projected out on the LHS of the flow equation:
( )
Π (F [ḡ](h)) = (e2ΩΩ µ −1)Fµ [ḡ](ḡ) = 2Ω ḡαρ ḡβσ −ϖ ḡαβ ḡρσ ḡαµD̄β ḡρσ = 0
Hence, the gauge fixing functional which is quadratic in F[ḡ](h) vanishes ΠΩ(Γ
gf
k [h; ḡ]) = 0, a
consequence of F[ḡ] being a function of D̄ only, which is a metric compatible derivative. Once
we apply the conformal projection technique there the RG evolution of gfūk is not resolvable any-
more. We come back to this point after considering a much severe issue concerning boundary
invariants.
R One should keep in mind that we introduced this bookkeeping constant Ω only as a mere
projection tool which is ultimately set to zero and has no physical meaning. All other
techniques that are compatible with the truncation ansatz can be used and should result in
the same set of beta-functions.
7.3 Projection techniques for truncations 225
Conformal projection in the presence of boundary invariants
While this conformal projection method works very well for ∂M= 0/ , there is a subtlety involved
when considering manifolds with non-vanishing boundary. Only if Ω = 0 the dynamical g and
the background ḡ field fulfill the same boundary conditions and thus share the same field space.
In particular, the conformal projection requires to set h| ≡ (e2Ω∂M − 1)ḡ|∂M on intermediate
steps which corresponds to a combination of Dirichlet and Neumann condition rather than pure
Dirichlet boundary conditions:
hµν | 2Ω∂M ≡ (e −1)ḡµν |∂M =⇒ D̄ρhµν |∂M ≡ 0
These modified boundary condition for h will in general spoil the arguments that led to the
Hessian operator we derived for Dirichlet conditions and more severely does not preserve the
gravitational truncation ansatz when applied to the dynamical boundary invariant. This basically
corresponds to imposing Dirichlet and Neuma√nn conditio√ns simultaneously, while at the same
time distinguish between contributions from HK and H̄K̄. However, these are mutually
exclusive constraints and thus ΠΩ is not compatible with a dynamical Gibbons-Hawking-York
term in the presence of Dirichlet conditions on field space:
( )
Π (K| ) = Π K̄− 1(nµ D̄ν − ḡµνnλΩ g=ḡ Ω 2 D̄λ )hµν = K̄
In the final step we used the fact that D̄ḡ= 0 and Ω is an x-independent constant. Thus, solving
for both ū∂B and ū∂D becomes impossible once the conformal projection technique is used.
At this point there are basically two possibilities how to continue to resolve the RG evo-
lution of the dynamical Gibbons-Hawking-York term: either choose different boundary condi-
tions for field space, for instance Neumann conditions which are perfectly compatible with the
conformal projection technique, or consider a different projection method. Concerning the first
option, notice that Dirichlet conditions were found to be essential in deriving a well-defined
Hessian operator. For instance, in the case of Neumann conditions, D̄h|∂M = 0, several bound-
ary terms remained in the second variation of Γk and thus no Hessian operator can be deduced.
Only when evaluated for the conformal projection, i.e. additionally imposing h = (e2Ω − 1)ḡ,
these terms result in negligible contributions of second order in Ω:
∫ √ ( )
IΩ ≡+2(d−1)(d−3) · (ū∂D λ̄ ∂Dk k ) · dd−1x H Ω2+O(Ω3)
∫ ∂M√ ( )( )
+ 1 ∂D d−1 22(ūk ) · d x H (d +2d)K̄+2(d−2)K̄
µν n̄µ n̄ Ω
2
ν +O(Ω
3)
∂M
Here, we have introduced n̄µ ≡ eΩnµ as the normal vector of the boundary that is normalized
with respect to the background metric ḡ. The first term originates from the volume element
on the boundary constructed by the dynamical field and in fact is absent in case of Dirichlet
conditions, Ω = 0. Similarly, for the second term which is associated to the Gibbons-Hawking-
York functional. But at the point we utilize the conformal projection technique we already
should have a suitable Hessian operator. Thus, even though different boundary conditions are
compatible with ΠΩ they introduce much severe and fundamental problems on the level of the
FRGE. This suggests that we should rather look for a different projection technique.
An alternative, that at first sounds very promising, is an x-dependent conformal projection.
Thus we replace the original constant parameter Ω with a function Ω(x) that respects Dirichlet
boundary conditions:
Π .Ω(x)(•) .= (•)|g=e2Ω(x) ḡ with Ω(x)|∂M = 0
226 Chapter 7. Trace evaluation
Obviously, now g and ḡ fulfill the same boundary conditions and thus are elements of the same
field space. In particular, the fluctuation field h vanishes on ∂M but has a non-trivial derivative
in general:
Π 2Ω(x) 2Ω(x)(hµν)|∂M = (e −1)ḡµν |∂M = (2Ω(x)+O(Ω ))ḡµν |∂M = 0 and
Π 2Ω(x) 2Ω(x)(D̄ρhµν)|∂M = D̄ρ(e −1)ḡµν |∂M = 2ḡµν(D̄Ω(x))+O(Ω )|∂M 6= 0
This seems encouraging since an x-dependent conformal projection can be consistently com-
bined with Dirichlet boundary conditions. By construction, it will preserve the dynamical
Gibbons-Hawking-York term on the LHS of the flow equation, which in the level expansion
reads
∣ ∣
∂ Γgrav[ , grav grav gravt k g ḡ]≡ ∂tΓk [ḡ, ḡ]+∂t(dhΓk [g, ḡ])∣ + 12∂t(dhd Γ ∣ 3g=ḡ h k [g, ḡ]) +O(h )g=ḡ
For the present truncation, for which only the zeroth and first order couplings differ, it is actually
sufficient to reduce the FRGE up to first order in h. Once we employ the x-dependent conformal
projection method, this corresponds to focusing on the zeroth and first order in Ω(x) only, i.e.
∣
ΠΩ(x)(∂tΓ
grav
k [g, ḡ]) = Π
grav
Ω(x)(∂tΓk [ḡ, ḡ])+ΠΩ(x)(∂ (
grav
t d Γ ∣ 2h k [g, ḡ]) )+O(Ω )∣ g=ḡ
= ∂ Γgrav[ , ]+∂ ( Γgravt k ḡ ḡ t d2Ω(x)ḡ k [g, ḡ])
∣ +O(Ω2)
g=ḡ
To obtain the second equality, we notice that the first term is independent on h and thus invari-
ant under ΠΩ(x). For the second term, representing the h-linear contribution, we expanded the
exponential exp(2Ω(x)) = 1+ 2Ω(x)+O(Ω2) to first order. Next, we insert its explicit form
given in eq. (6.29) that leads to
( ∣ )
Π ( ΓgravΩ(x) dh k [g, ḡ])
∣
∫ g=ḡ√ { ( )}
=+ ddx ḡ (ūDk λ̄
D) · ḡµν +(ūD) · R̄µν − 1k k 2 ḡ
µν R̄ Ω(x)ḡµν
∫M √ { ( )}
− dd−1x H̄ (ūD − ū∂D) · n̄µD̄ν − ḡµν n̄ρk k D̄ρ Ω(x)ḡµν
∫∂M √ { }
=+ ddx ḡ Ω(x) (ūDk λ̄
D D
k ) ·2d − (d−2)(ūk ) · R̄ (7.24)
M ∫ √ { }
+(d−1) dd−1x H̄ (ūDk − ū∂D ρk ) · n̄ D̄ρΩ(x)
∂M
This equation illustrates the problem of choosing an x-independent conformal projection. While
the bulk invariant does not feel any difference for Ω and Ω(x), the boundary contribution that
originates from the dynamical Gibbons-Hawking-York functional survives only for Ω(x). In
combination with the zeroth order term, eq. (6.28), that contains also background couplings,
the x-dependent conformal projection of the LHS of the flow equation yields, to the required
order
( ) ∫ √ ( )
Π gravΩ(x) Γk [g, ḡ] = +2 · ddx ḡ (1+d ·Ω(x))(ūD λ̄D)+ (ūB Bk k k λ̄k )
∫ M √
− ddx ḡ R̄((1+(d−2)Ω(x))(ūDk )+ (ūBk))
M∫
− √ ( )+2 · dd 1x H̄ (ū∂D λ̄ ∂D)+ (ū∂Bk k k λ̄ ∂Bk )
∫∂M
−2 · dd−
√ ( )
1x H̄ K̄(H̄) (ū∂Dk )+ (ū
∂B
k )
∂M ∫ √ { }
+(d−1) dd−1x H̄ (ūD − ū∂Dk k ) · n̄ρD̄ρΩ(x) +O(Ω2) (7.25)
∂M
7.3 Projection techniques for truncations 227
This indeed is a sufficient generalization of the conformal technique that comprises all seven
basis invariants.
∫ √
R The the dynamical boundary volume element, ∂M H, agrees with its background coun-
terpart due to Dirichlet conditions. Thus either λ̄ ∂Dk or λ̄
∂B
k can be resolved but not both,
independent on the conformal projection technique.
Notice that even in the case of a generalized conformal projection the gauge fixing action
does not contribute. While for Ω(x) = Ω constant, gfΓk [e
2Ωḡ, ḡ] = 0 is identical zero, in the
present case the linearity of F[ḡ](h) in the fluctuation field h implies that the gauge fixing
functional is of second order in Ω(x):
∫
gf √
Γk [2Ω(x)ḡ; ḡ] =
1 gf
2 ūk d
dx ḡ Fµ(2Ω(x)ḡ)ḡ
µνFν(2Ω(x)ḡ) = O(Ω(x)
2)
M
Thus, the running of the gauge parameter gfūk cannot be resolved with this method, even in the
x-dependent case.
Finally, in the ghost sector the LHS of the flow equation reduces under the x-dependent
conformal projection to
gh
∂tΓk [2Ωḡ,ξ , ξ̄ ; ḡ] (7.26)
√ ∫ √ { }
=− 2(∂ gh dt ūk ) d x ḡ (1+2
µ µ
Ω(x)) ξ̄µ δν D̄
2+(1−2ϖ)D̄µD̄ν + R̄ νν ξ
√ ∫
− √
{ ( )
2 2(∂ ghū ) ddx ḡ ξ̄ (2−d ·ϖ) ξ λ D̄µt k µ D̄λ Ω(x)+ (D̄µξ λ )D̄λ Ω(x)
}
+(D̄ ξ µ)D̄λλ Ω(x)−2ϖ(D̄ ξ λ )D̄µλ Ω(x)
Incidentally, note that for constant Ω(x) = Ω and the harmonic gauge condition, corresponding
to ϖ = 1/2, the ghost action, and therefore the entire truncation ansatz on the bulk, does not
contain any covariant derivative D̄ 2 µνµ that would not be contracted to a Laplacian D̄ ≡ ḡ D̄µD̄ν .
However, since ghūk is chosen k-independent the ghost sector vanishes on the LHS.
The drawback of an x-dependent conformal method appears on the RHS of the flow equa-
tion. Initially, the conformal projection technique was introduced to circumvent a h-expansion,
since in the latter case additional not contracted derivatives are generated preventing a straight-
forward heat kernel expansion. Though in principal there are methods available that treat this
general case, we would need a full decomposition of the fluctuation fields and ultimately rely
on off-diagonal heat kernel techniques to evaluate the trace. Already in the case of manifolds
with vanishing boundary this turns out to be quite sophisticated and time-consuming, whereas
for non-vanishing ∂M additional complication arises. For instance the supplementary boundary
constraints have to be implemented and furthermore additional heat kernel coefficients appear
that generate the boundary invariants.
Thus, simplified projection techniques are desirable as for instance the just described x-
dependent conformal projection method. In fact this technique can be understood as keeping
track of the hTr ≡ ḡµνhµν part only, corresponding to the trace contribution in the h-expansion,
which for the present truncation is sufficient. While this resolves a number of unwanted terms
in the Hessian, there is still a sufficient amount of not fully contracted derivatives acting on Ω(x)
that requires the full apparatus of off-diagonal heat kernel techniques described above. One can
further specify the function Ω(x) to best fit the needs of the present truncation. Nevertheless,
this turns out cumbersome since the requirement to retain the boundary term in question already
228 Chapter 7. Trace evaluation
excludes the most simple choices. Basically, the following conditions should be satisfied by
Ω(x):
g ρµν = ḡµν +Ω(x)ḡµν n̄ D̄ρΩ(x) 6= 0 Ω(x)|∂M = 0
Besides these requirements, it is desirable to a obtain a simple as possible Hessian operator w.r.t.
Ω(x).
R It should be noted that the appearance of D̄Ω(x) in the Hessian is in fact an indication
for a non-trivial RG evolution of ū∂D, though for computational purposes it makes things
unnecessary complicated.
The bi-metric-bulk – pure-background-boundary truncation
In this thesis, we have prepared the ground for an in depth study of a bi-metric Einstein-Hilbert–
Gibbons-Hawking-York truncation. We have derived the Hessian for general boundary condi-
tions and arbitrary parametrization schemes and only afterwards we have specified to a partic-
ular class of gauge fixing conditions, the linear parametrization, and Dirichlet boundary condi-
tions. Similarly, we have provided the general inversion rule for the matrix in field space, while
only later we ignored the running of the ghost sector. Thus, future work may enter at any step
and change or generalize the setting at will to explore the RG evolution of boundary terms in
quantum gravity to its full beauty. In the sequel, we consider the a single-metric ansatz for the
boundary gravitational part in order to utilize the standard Ω-expansion. In a next step, one
might proceed in using the prescribed x-dependent generalization to resolve the dynamical and
background running independently.
For now, we employ the projection ΠΩ instead of ΠΩ(x) for which the Effective Average
Action on the LHS of the flow equation reduces to
∫
· √
[ ]
Π dΩ (Γk[g, ḡ]) = +2 d x ḡ (1+d ·Ω)ūD λ̄Dk k + ūB λ̄ Bk k
∫ M √
− ddx ḡ R̄ [(1+(d−2)Ω)ūDk + ūBk ]
M∫
− √ [ ]+2 · dd 1x H̄ ū∂D λ̄ ∂D+ ū∂B ∂Bk k k λ̄k
∫∂M
−2 · dd−
√ [ ]
1x H̄ K̄(H̄) ū∂D ∂Bk + ūk +O(Ω
2) (7.27)
∂M
Thus, a total of six independent couplings is left, that we are going to resolve in what follows.
Notice that the ghost as well as the gauge fixing coefficients drop out, however for different
reasons. While the former is set to a constant value, the latter basis invariant turns zero under
conformal projection with gfūk unspecified so far. From now on, there will be no more reduction
of the truncated theory space, but nevertheless we will exploit the freedom in choosing gfūk to
get rid of certain undesirable terms.
The Hessian operator on the RHS of the flow equation has to be equally projected by means
of ΠΩ. Therefore, let us first list the conformal projection of several tensor fields appearing in
the Hessian operator:
Π (g ) = e2Ωḡ Π (gµν) = e−2Ωḡµν
√ dΩ√
Ω µν µν Ω ΠΩ( g) = e ḡ
Π (RµνρσΩ ) = e
−6ΩR̄µνρσ Π (Rµν) = e−4ΩΩ R̄
µν ΠΩ(R) = e
−2ΩR̄
In the gravitational sector of the Hessian operator, the conformal projection technique only
affects the contributions due to Γgravk , since the additional terms of the gauge fixing functional
7.3 Projection techniques for truncations 229
are already independent on h. For a constant parameter Ω we obtain
µν B ρσ
( )
Π (g (Ĥess ) ) =−(ūD λ̄D) · ḡρµ ḡσνΩ hh B k k − 1 ρσ µν2 ḡ ḡ · e
(d−4)Ω
+ gfūk · (ḡνσ R̄µρ + R̄ρµνσ){ }
+ 1(ūD2 k ) · e
(d−6)Ω −K̄µνρσ + Ūµνρσ + 1 gfū · K̄µνρσ2 k
− 1 gf2 ūk · (1−2ϖ)(ḡµνD̄ρD̄σ + ḡρσ D̄µD̄ν)( )
− 1 gfū · ḡµρ ḡνσ2 k − (1−2ϖ2)ḡµν ḡρσ D̄2
Since on the LHS of the flow equation it is sufficient to expand up to first order in Ω and retain
all gravitational invariants (except the dynamical Gibbons-Hawking-York term), we can neglect
all higher order contributions in the Hessian, too. This gives rise to the following simplification
µν B ρσ
[ ]( )
ΠΩ(g (Ĥesshh)B ) =− (ūD Dk λ̄k ) · (1+(d−4)Ω) ḡρµ ḡσν − 1 ḡρσ2 ḡ
µν
+ gfū · (ḡνσ R̄µρ + R̄ρµνσk )
+ 1 [(1+(d−6)Ω)(ūD)] · Ūµνρσ2 [ k ]
− 12 (1+(
gf
d−6)Ω)(ūD)− ū K̄µνρσk k
− 1 gf2 ūk · (1−2ϖ)(ḡ
µνD̄ρD̄σ + ḡρσ D̄µD̄ν)
( )
− 1 gf2 ūk · ḡ
µρ ḡνσ − (1−2ϖ2)ḡµν ḡρσ D̄2 (7.28)
The first three terms are ultra-local tensor structures, and are almost in shape for a heat ker-
nel expansion, while the remaining terms contain free derivatives. Out of them, only the last
contribution is of Laplacian type and needs no more further modification.
Next, consider the ghost sector in the Hessian operator. Since ΠΩ(hµν) = 2Ωḡµν holds true
all contributions that involve a derivative of h w.r.t. D̄ vanish. This leads to a strong reduction
under which the ghost part of the Hessian operator assumes the form:
√
µ − λ ( )(Ĥessξ ξ̄ )α =+ 2(1 2 ρσ+ Ω) · ūgh(g 1) ḡ D̄ + ḡ D̄ F µνξ̄ξ λρ σ λσ ρ ν ḡ
√ α ( )
=+ 2(1+2Ω) · ūgh(g−1 λ µ) δ D̄2+(1−2 µϖ)D̄µD̄
ξ̄ ξ λ λ
+ R̄λ (7.29a)
√ α{ }
α − · gh −1 α ρσ
( )
(Ĥessξ̄ ξ )λ = 2(1+2Ω) ū (g ) ḡ
µνFν ḡλρD̄σ + ḡλσ D̄ξ ξ̄ ρ
√ µ ( )
=− 2(1+2Ω) · ūgh(g−1 α µ µ) δλ D̄
2+(1−2ϖ)D̄µD̄ + R̄ (7.29b)
ξ ξ̄ λµ λ
Hereby, on an intermediate step, we used twice [D̄µ , D̄ ]v = R̄µλλ µ vµ for dual vector fields, to
simplify the result. A close look at the structure of these equations already gives a hint what
choices for the gauge fixing condition and ᾱk are most favorable. For the moment, we turn our
attention to the ultra-local contributions and come back to the issue of uncontracted derivatives
at the very end.
R On has to keep in mind that the constant Ω is an artificial parameter which is only in-
troduced in order to distinguish between purely background contributions and terms of
dynamical origin. It has no physical relevance and does not alter field space or the trunca-
tion ansatz at all.
7.3.3 The Hessian in the trace-traceless decomposition
Previously, we described a technique to decompose symmetric 2-tensor fields with respect to a
derivative operator D̄ and a scalar product, here given by ḡ. We are now going to apply only
230 Chapter 7. Trace evaluation
its simplest form, namely the trace-traceless splitting of the metric part of the Hessian operator.
The corresponding projection operators are given by
µν
(Π ) .Tr ρσ .=
1 ḡ µνρσ ḡ (7.30a)d
µν ( µν µν) ( µ µ )
(Π . 1 νnoTr)ρσ .= 1ρσ − (ΠTr)ρσ ≡ 2 δρ δσ +δ
ν 2 µν
ρ δσ − ḡρσ ḡ (7.30b)d
With the above normalization, both operators are idempotent and thus define projections. Very
important is the decomposition behavior of the field space metric under the trace-traceless split
induced by eq. (7.30). It turns out that gAB is orthogonal w.r.t. (ΠTr) and (ΠnoTr) and hence it
commutes with the projections:
(Π ) gABnoTr A (ΠTr)B = 0= (ΠTr)Ag
AB(ΠnoTr)B ∧ gAB(Π C A BCTr/noTr)B = (ΠTr/noTr)B g
The traceless and trace component of the gravitational part of the field space metric assumes the
form:
( ) ( )
gµνB
ρσ ρσ
(Π ) = ḡµρ ḡνσ − 1 −2ϖ ḡµν ḡρσ ∧ gµνB(Π 1 µν ρσnoTr B d Tr)B = −ϖ ḡ ḡd
Notice that for ϖ ≡ 1/d the metric vanishes in the trace sector and thus is not invertible for
general symmetric 2-tensor fields. Avoiding this special case, let us consider the projection of
the inverted field space metric g−1µνρσ ≡ ḡµρ ḡ ϖνσ + (1− ·ϖ) ḡµν ḡρσ :d
− αβ ( )g 1µνB(Π BnoTr)ρσ = ḡµρ ḡ 1νσ − ḡµν ḡρσ ≡ (Πd noTr)µν · ḡαρ ḡβσ (7.31a)
( ) ( )
g−1 B 1 αβµνB(ΠTr)ρσ = −ϖ · (ḡµν ḡρσ )≡ (ΠTr)µν · 1(1− ḡ ḡ (7.31b)d d d·ϖ) αβ ρσ
This is a quite important result for it shows the decomposition of the field space metric under
the trace-traceless split. Its relevance in the present case is due to the fact that in the previous
expressions for the Hessian operator we used the matrix gAB(Ĥesshh)
C
B rather than the operator
Ĥesshh itself.
However, before we are going to derive the projections of the Hessian operator, let us first
decompose the gravitational cutoff operator gravRk [ḡ]. By defi(nition its tensor)structure is given
by the identity of the respective field subspace, here µν = 1 µδ δ ν µ+δ δ ν1ρσ 2 σ ρ ρ σ . Thus, the cutoff
operator commutes with the trace-traceless decomposition and its orthogonal components are
given by:
grav
Rk [ḡ] = (Π +Π
grav
Tr noTr)Rk [ḡ] (ΠTr+ΠnoTr)
= Π ◦ gravTr Rk [ḡ]◦ΠTr+ΠnoTr ◦
grav
Rk [ḡ]◦ΠnoTr (7.32)
Next, consider eq. (7.28) that describes the Hessian operator conformally projected and
with a family of generalized harmonic gauge fixing conditions inserted. Let us contract this
Hessian matrix on the left with the inverse metric g−1AB . While the LHS of the FRGE is com-
pletely unaffected by this split due to the disappearance of metric fluctuations under conformal
projection, the RHS decomposes in an non-orthogonal way.
A noTr D D
Π ( C A C
Tr A
Ω Ĥesshh)B = (ΠnoTr)B (Ĥesshh )C (ΠnoTr)D +(ΠTr)B (Ĥesshh)C (ΠTr)D (7.33)
noTr−Tr D Tr−noTr D
+(Π CnoTr)B ( ) (Π )
A+(Π ) C AĤesshh C Tr D Tr B (Ĥesshh )C (ΠnoTr)D
Since the field space metric is diagonal in the trace-traceless split, we make use of eq. (7.31)
and stepwise evaluate the operator parts in this decomposition. Therefore consider first the
7.3 Projection techniques for truncations 231
contribution to the Hessian operator originating from the volume functional:
g−
{ } ( ){ }
1 ρµ σν 1 ρσ µν −1 B B ρµ σν 1 ρσ µν
αβ µν ḡ ḡ − 2 ḡ ḡ = gαβ (ΠB noTr)µν +(ΠTr)µν ḡ ḡ − 2 ḡ ḡ{ }
=+(Π )κλ ḡ ḡ ḡρµ ḡσν − 1 ḡρσ ḡµνnoTr αβ κµ λν { 2 }
+(Π )κλ 1 ḡ ḡ ρµ σν 1 ρσ µνTr αβ d(1−d·ϖ) κλ µν ḡ ḡ − 2 ḡ ḡ
Utilizing the idempotency of the projection operator and further simplify the expressions yields:
g−
{ } { }
1 ρµ σν 1 ρσ µν κλ ν µ ρσ
αβ µν ḡ ḡ − 2 ḡ ḡ =+(ΠnoTr)αβ · δκ δλ · (ΠnoTr)µν{( ) }
+(Π )κλ 2−d ν µ ρσTr αβ · 2−2d·ϖ δκ δλ · (ΠTr)µν
Notice that this part of the operator is diagonal, in fact proportional to the identity, in the trace-
traceless decomposition and the pre-factor of the trace part turns to one if ϖ = 1/2.
The next term in (Ĥesshh) contains a Ricci- and a Riemann tensor. The first of these con-
tributions is based on the curvature tensors which originally were associated to the dynamical
metric. We abbreviate the calculation and only state the result:
− { }g 1 {ḡνσ R̄µρ + R̄ρµνσ}= (Π )κλ · µ ν ρσαβ µν noTr αβ R̄κ δ ν µλ + R̄ κλ · (ΠnoTr)µν
Both, mixed and purely trace invariants vanish due to the symmetry properties of the Riemann
tensor. Thus the result describes an operator restricted to the traceless part of metric field space
only and therefore is also diagonal in the trace-traceless decomposition.
Now, the missing ultra-local term Ūµνρσ is evaluated for trace-traceless split. It contains
combinations of Ricci- and scalar curvature tensors and decomposes as follows
− { } {( )( ) }g 1 Ūµνρσ µ ρσαβ µν =+(Π κλ 2−d d−4 νTr)αβ ·{ 2−2 ·ϖ
R̄δκ δλ · (Πd d } Tr
)µν
ρσ
+(Π κλ (d−4) µνTr)αβ · (1− ϖ) ḡκλ R̄ · (Π ){d d
noTr µν
}
ρσ
+( )κλ · (d−4)Π µνnoTr αβ R̄ ḡ · (Π )d κλ Tr µν
{
κλ ν µ µ
} ρσ
+(ΠnoTr)αβ · R̄δκ δλ −4R̄κ δ
ν
λ · (ΠnoTr)µν
Here, for the first time we encounter off-diagonal contributions given by the second and third
term. Similarly, the decomposition of K̄µνρσ splits non-diagonally whereby these terms in
addition consist of uncontracted derivatives:
− { } { ( )( ) }g 1 K̄µνρσ µ=+(Π )κλ · 2 2−d d−1 δ νδ D̄2 ρσαβ µν Tr αβ 2−2 ·ϖ κ λ · (Π ){ d d } Tr µν
+( )κλ · (d−2) ρσΠTr αβ (1− ϖ) ḡ D̄
µD̄ν · (Π )
{d d
κλ noTr µν
}
+( )κλ · (d−2) µν ρσΠnoTr αβ ḡ D̄κD̄λ · (Πd Tr)µν
{ µ }
+(Π κλ ν µ ν 2
ρσ
noTr)αβ · −2δλ D̄ D̄κ +δκ δλ D̄ · (ΠnoTr)µν
However, we wont be concerned with these terms for now, since they are treated later quite
effectively using the freedom in the gauge fixing condition.
We are left with two contributions associated to the gauge fixing functional, the first which
reads
g−1 {(1−2ϖ)(ḡµνD̄ρD̄σ + ḡρσ D̄µ ναβ µν D̄ )}{ ( ) }
=+(Π κλTr)αβ · 2 1−2ϖ δ ν
µ 2 · ρσδ D̄ (ΠTr)µν
{( 1−dϖ− )
κ λ }
+(Π )κλ · 1 2ϖ ḡ D̄µD̄ν ρσTr αβ 1− ϖ κλ · (ΠnoTr)d µν
+(ΠnoTr)
κλ
αβ · {(1−2ϖ)ḡµνD̄κD̄λ} ·
ρσ
(ΠTr)µν
232 Chapter 7. Trace evaluation
While this expression does not contain a traceless-traceless term in the decomposition further
off-diagonal contributions appear. Notice that for the harmonic gauge condition this entire
expression vanishes.
Finally, the remaining term is again related to Γgfk . While it is not proportional to (1−2ϖ),
it is of Laplacian type and needs no further simplification:
{( ) } {( ) }2 µ ρσ
g−1 µρ νσ 2 µν ρσ 2 κλ 2dϖ −d+1 ν 2αβ µν ḡ ḡ − (1−2ϖ )ḡ ḡ D̄ = (ΠTr)αβ · 1− δ δ D̄ · (Π )dϖ κ λ Tr µν
{
κλ · ν µ
}
+(Π 2
ρσ
noTr)αβ δκ δλ D̄ · (ΠnoTr)µν
Here again, we observe a diagonal decomposition under the trace-traceless splitting.
We can now read off the components in the trace-traceless decomposition of the Hessian
operator defined in eq. (7.33).
◮ Traceless-Traceless term:
noTr µν [ ] ( )
(Ĥess ) =− ūD λ̄Dhh κλ k k · (1+(d−4)Ω) · δ
ν µ
κ δλ (7.34)( µ ν) ( µ )
+ gf gfū · R̄ δ ν + R̄µ − 1 ū · δ ν 2k κ λ κλ 2 k κ δ( λ
D̄
)
+ 12 [(1+(d−6
µ µ
)Ω)ūD] · R̄δ νk κ δλ −4R̄ δ
ν
[ ] ( κ λ )
− 1 (1+( −6)Ω) D− gf µ2 d ūk ūk · −2δ
ν µ
λ D̄ D̄κ +δ
νδ D̄2κ λ
◮ Trace-Trace term:
Tr µν [ ] ( ) µ
(Ĥesshh)κλ =− ū
D D
k λ̄k · (1+(d−4)Ω) · 2−d ν( 2−2d)·ϖ
δ δ
( κ) λ
+ 12 [(1
µ
+(d−6)Ω)ūD] · 2−d d−4 · R̄δ νk 2−2 ·ϖ κ δ[ ] (d( d)( ) λ)
− (1 µ+(d−6)Ω) gfūD − ū · 2−d d−1 D̄2 δ ν δ
( )k k 2−2d·ϖ d κ λ
− 1 gfū · 2dϖ2−d+1 ν µ 22 k 1− ϖ δκ δλ D̄ (7.35)d
◮ Traceless-Trace term:
noTr−Tr µν
(Ĥess ) = + 1hh κλ 2 [(1+(d−6)Ω)ū
D
k ] · (d−4) R̄ µν[ d ] κλ
ḡ
− 12 (1+( −6) )
gf (d−2)
d Ω ūD − ū · ḡµνk k D̄d κD̄λ
− 1 gf2 ūk · (1−2ϖ)ḡ
µνD̄κD̄λ (7.36)
◮ Trace-Traceless term:
− ( )Tr noTr µν
(Ĥesshh ) = +
1
κλ 2 [(1+(d−6)Ω)ū
D] · d−4 µνk d(1− ϖ) ḡd κλ R̄
[ ] ( )
− 12 (1+(d−6)Ω)
D − gfū ū · d−2 µk k (1− ϖ) ḡκλ D̄ D̄
ν
( d d
− 1 gfū · 1−
)
2ϖ µ ν
2 k 1− ϖ ḡκλ D̄ D̄ (7.37)d
Except for the terms related to K̄ and those proportional to (1− 2ϖ), there are only two ad-
ditional contributions in the off-diagonal sector. Whereas the former two expression can be
taken care of using the freedom in gfūk and ϖ , the latter which is ultra-local requires a different
projection technique that we are going to exploit in the next subsection.
7.3.4 Projection to maximally symmetric spaces
The Hessian operator contains terms dependent on the Ricci- and Riemann-tensors, which –
similar to the scalar curvature– are of second order in the covariant derivatives D̄. Since this
7.3 Projection techniques for truncations 233
corresponds to the maximal degree of D̄ contained in the present truncation ansatz, these contri-
butions either lead to higher order basis invariants or condense to the scalar curvature invariant
which is the only basis monomial of degree O(D̄2).
A very convenient way to extract these contributions in the projection of the functional
trace is based on maximally symmetric spaces. It uses the fact that the basis invariants on the
LHS transform trivially when background geometries are inserted for which the Riemann- and
Ricci-tensors are proportional to the constant scalar curvature, i.e.
( )
Π 1 2 1m.s.(R̄µνλσ) = ( −1) ḡµλ ḡνσ − ḡνλ ḡµσ C and Πm.s.(R̄µν) = ḡ
2
µνC (7.38)d d d
This is the definition of a maximally symmetric spaces, whereby C2 ≡ Πm.s.(R̄) is a parameter
representing the constant curvature. Notice that for higher derivative invariants the projection
operator does not in general commute with the restriction to maximally symmetric spaces and
thus this technique will be insufficient for a full resolution of the truncation, i.e.
( )
Π Sm.s. Π (Γk[ϕ ;Φ̄]) =6 ΠS (ΠTrunc Trunc m.s.(Γk[ϕ ;Φ̄])) in general
This is already the case for the basis invariants of O(D̄4) which are at most four in number.
Under projection onto maximally symmetric spaces a generic functional of these four field
monomials condenses to a single term given by
( ) ( )
Πm.s. ū
1 · D̄2R̄+ ū2k k · R̄2+ ū3k · R̄µν R̄ + ū4 · R̄µνρσµν k R̄µνρσ = ū2+ 1 ū3+ 2 ū4k k ( −1) k ·C
4
d d d
This map is no isomorphism and thus inappropriate to distinguish between the RG evolution of
the four couplings. Notice that the invariant associated to ū1k even vanishes identically under the
projection Πm.s., due to the fact that C is constant.
However, for the family of f (R̄) truncations, of which the current Einstein-Hilbert func-
tional is a special case, an isomorphism can be established and thus the trace evaluation for
maximally symmetric spaces is just a simple procedure to extract the scalar part contained in
R̄µνλσ and R̄µν . As an example, consider a polynomial-like function in R̄ and apply the maxi-
mally symmetric space projection:
( )
∞ (n) n ∞ (n)Πm.s. ∑ ūn=0 k · R̄ = ∑ ū ·C
2·n
n=0 k
Instead of a polynomial in the scalar curvature R̄ we have now a series of same degree in a con-
stant value C0, in which one can straightforwardly expand. This demonstrates the isomorphism
property of Πm.s. in case of f (R̄)-theories.
Concerning the boundary invariants, we need a further specification of the embedded bound-
ary to connect the extrinsic curvature tensor toC. A suitable choice would be the d-dimensional
half-sphere that has Sd−1 as its boundary. From the Gauß-Codazzi equations, for instance in the
form of eq. (3.11), that we obtain [184]
√
(d−1)
Π 1m.s.(K̄µν) = √ H̄ C Π (K̄) = C
d(d− µν m.s.1) d
For the present truncation, this additional projection can be omitted, since t√he Hess√ian operator
is of even power inC and will only contribute to boundary contributions of H̄- or H̄K̄2-type,
for instance. Nevertheless, we adopt this general projection technique in what follows.
The Πm.s.-isomorphism applied to the LHS of the flow equation amounts to a simple re-
placement of the scalar- and extrinsic curvature tensor by powers of the constant C. Explicitly,
234 Chapter 7. Trace evaluation
the projection of eq. (7.27) using the prescribed isomorphism results in
( ) ∫ √ ( )
Π ◦Π Γgrav dm.s. Ω k [g, ḡ] = +2 · d x ḡ (1+d ·Ω)ūD λ̄D+(ūB λ̄ Bk k k k ) (7.39)
∫ M
− √ddx ḡ C2 ((1+(d−2)Ω)ūDk +(ūBk))
M∫
− √ ( )+2 · dd 1x H̄ (ū∂D λ̄ ∂Dk k )+ (ū∂B ∂Bk λ̄k )
∫∂M √
−2 · dd−
√
1 (d−1) ( )x H̄ C (ū∂Dk )+ (ū∂Bk ) +O(Ω2)d
∂M
On the RHS we the Riemann- and Ricci-tensor reduces to a combination of metric ten-
sor structures. Thus, in particular the contraction with the traceless component yields further
simplifications. In the gravitational sector, the application of Πm.s. gives rise to the following
structures:3
◮ Traceless-Traceless term:
noTr µν [ ] ( µ)
Π D Dm.s.(Ĥesshh )κλ =− ūk λ̄(k · (1+(d−4))Ω) · δ
ν
κ δλ (7.40)( )
+ gf 1 ν
µ 2 1 gfū ν
µ 2
k · (d−1)δκ δλ C − ū( 2 k
· δκ δλ D̄)
+ 1 [(1+( −6) ) D] · (d−4) µ2 d Ω ū νk δκ δλ C
2
[ ]d ( )
− 12 (1+(d−6)Ω)ū
D − gfū · −2δ νD̄µD̄ +δ ν µk k λ κ κ δ 2λ D̄
◮ Trace-Trace term:
Tr µν [ ] ( )
Π (Ĥess ) =− ūD λ̄Dm.s. hh κλ k k · 1
µ
( + (d−4)Ω) · 2−d δ ν δ
( 2−2d)·ϖ( κ) λ
+ 12 [(1+(d−6)Ω)ū
D
k ] · 2−d d−4 2 ν
µ
[ ]2−(2d(·ϖ
·C δ δ
d)( ) κ )λ
− 1 −6 D− gf · 2−d d−1 2 ν µ( + (d )Ω)ū ū
( )k k 2−2d·ϖ
D̄ δ
d κ δλ
− 1 gf · 2dϖ2ū −d+1 ν µ 22 k (1−dϖ) δκ δλ D̄ (7.41)
◮ Traceless-Trace term:
noTr−Tr µν [ ]
Πm.s.(Ĥesshh )κλ =−
1
2 (1+(d−6)
(d−2)
Ω) gfūDk − ūk · ḡµνD̄d κD̄λ
− 1 gf2 ū · (1−2ϖ)ḡ
µν
k D̄κD̄λ (7.42)
◮ Trace-Traceless term:
− ( )Tr noTr µν [ ] (d−2)
Π gfm.s.(Ĥess
1 D µ ν
hh )κλ =− 2 (1+(d−6)Ω)ūk − ūk · ḡ D̄ D̄ ḡ( ) d(1−dϖ)
κλ µν
− 1 gfū · 1−2ϖ µ ν2 k (1− ϖ) ḡκλ D̄ D̄ (7.43)d
Notice that the existence of off-diagonal terms in the trace-traceless decomposition is now
closely linked to the appearance of uncontracted derivatives.
Finally, the ghost sector appears a slight modification in that the contained Ricci-tensor is
simply replaced by its maximally symmetric version, yielding
√
µ λ ( µ ( ) )
Πm.s.((Ĥess ) ) = + 2(1+2Ω) · ūgh −1ξ ξ̄ α (g ) δλ D̄
2+ 1C2 +(1−2ϖ)D̄µD̄
ξ̄ ξ d λα
(7.44a)
√
α − · gh −1 α
( µ ( ) )
Πm.s.((Ĥessξ̄ ξ )λ ) = 2(1+2Ω) ū (g ) δλ D̄
2+ 1C2 +(1−2ϖ)D̄µD̄
ξ ξ̄ d λµ
(7.44b)
3From now on we usually omit O(Ω2) which has to be understood implicitly.
7.4 The ‘Ω deformed’ α = 1 harmonic gauge 235
In order to apply the standard heat kernel expansion to evaluate the L2-operator trace, we fi-
nally have to solve the problem of consistently removing the uncontracted derivatives in the
Hessian operator. This will remove both remaining issues, namely the off-diagonal terms in the
trace-traceless decomposition will vanish and the Hessian operator becomes a function of the
Laplacian D̄2 only.
R Notice that we are not fixing the background metric at all, but only use the prescribed
technique to evaluate the functional trace. Different methods, which make no use of any
special form of ḡ at all, will give rise to the same results.
7.4 The ‘Ω deformed’ α = 1 harmonic gauge
C This entire section is conceptual and partly verbatim equivalent to subsection 2.5 of [173]
with only minor modifications related to the non-trivial boundary while adapting the no-
tation to the present work.
In this section we exploit the gauge fixing freedom to finalize the Hessian operator such that
standard heat kernel techniques are applicable.
The outstanding task is to compute the RHS of the truncated FRGE, thereby retaining only
the field monomials which are contained in the ansatz for the EAA. We want to utilize the
known asymptotic expansions for the heat kernels of the operators involved, which is also well-
studied in the case of ∂M. This would be particularly easy if those operators were functions
of the Laplacian D̄2 only. A priori this is not the case, however. There are also operators such
as D̄µD̄ν with uncontracted indices. One strategy to deal with them is to further decompose
the fields hµν , ξ µ , and ξ̄µ into a sum of differentially constrained fields along the lines of the
transverse-traceless (TT)- or York-decomposition. We already presented the extension to space-
times with non-vanishing boundary where the fields satisfy Dirichlet conditions, see section 7.2.
For further details and the complete story of the strategy we refer to ref. [172].
However, in this thesis we instead make use of the yet to be fixed parameters in the gauge
fixing functional, a new technique introduced in ref. [173] that for the present truncation cancels
all non-Laplacian differential operator terms. For reasons which will become apparent in a
moment, we denote it the ‘Ω deformed α = 1 harmonic gauge’.
The properties
We have already noticed that the harmonic gauge condition ϖ = 1/2 reduces the operators
occurring in the flow equation, in the ghost sector, to a function of the fully contracted Laplacian
operator only. In the truncation ansatz, all operators of the type D̄µD̄ν are proportional to
(1−2ϖ) and thus drop out for ϖ = 1/2.
This choice, interpreted as a projection operator Πϖ also affects the ghost contributions on
the RHS of the flow equation as can be seen from eq. (7.44b).
√
µ − λ ( µ ( ))Πϖ ◦Πm.s.((Ĥessξ ξ̄ )α) = +2 2Ω · ūgh(g 1) δλ D̄
2+ 1C2 (7.45a)
√ ξ̄ ξ α
d
( ( ))
Πϖ ◦ α gh −1
α µ
Π 2 1 2m.s.((Ĥessξ̄ ξ )λ ) =−2 2Ω · ū (g ) δλ D̄ + C (7.45b)ξ ξ̄ µ d
It is the only choice of ϖ that leads to a full cancellation of these terms in the ghost sector.
Hence, from this point of view the harmonic gauge condition is necessary if the Hessian be-
comes of Laplacian type in the ghost sector.
The gravitational functional Γgravk itself is free of uncontracted derivatives, but after a second
variation they appear in the off-diagonal and the traceless-traceless component of the associated
236 Chapter 7. Trace evaluation
Hessian operator. Quite remarkably, the harmonic gauge condition ϖ = 1/2 is also the best
choice for the gravitational contributions. Employing ϖ = 1/2 all those terms that are propor-
tional to some function of ϖ and contain uncontracted differential operators indeed vanish.
However, there is another source of troublesome covariant derivatives, namely those con-
tained in Ūµνρσ that in the trace-tracele[ss decomposition contribu]te to the off-diagonal and the
traceless part. Both are proportional to (1+(d−6)Ω)( gfūDk )− ūk . Under the following choice
of the gauge parameter these remaining terms disappear, leaving only Laplacian-type operators:
gf
ūk = (1+(d−6)Ω)(ūDk )+O(Ω2) (7.46)
Before continuing note that gfūk has k dimension (d−2) which in particular agrees with the one
of ūDk . Introducing the dimensionless gauge fixing parameter α ≡ ūD
gf
k k/ūk , the preferred choice
(7.46) corresponds to the value, up to O(Ω2) terms,
αk = 1− (d−6)Ω ≡ α (7.47)
Note that this α happens to be scale independent and in fact its full RG evolution is matched to
the one of the dynamical Einstein-Hilbert functional.
Indeed, in all single-metric (‘sm’) calculations following ref. [122] the coefficient of the
gauge fixing functional has been parametrized as
gf
ū = (α sm)−1ūsmk k (7.48)
and it was then the choice α smk ≡ α = 1 that removed the uncontracted derivatives.
In order to achieve the same effect in the dynamical sector of the bi-metric setting we would
have to generalize this choice in a Ω-dependent way, namely α = e(6−d)Ω. Since only terms
linear in Ω are relevant, we can think of
α = 1− (d−6)Ω+O(Ω2) (7.49)
as being infinitesimally close to unity. It is this choice, eq. (7.49), which we refer to as the ‘Ω
deformed α = 1 gauge’.
The justification
It should be clear that in general disposing of α as in (7.49), i.e. making it Ω dependent, is by
no means legitimate from the conceptual point of view: The factor Ω represents the dynamical
metric, g= e2Ωḡ, it cannot appear in the final answer for the beta functions, but is rather a book
keeping parameter we should expand in to disentangle terms with different powers of hµν . The
beta functions at levels-(0) and -(1), respectively, are obtained from the Taylor expansion of the
traces:
∣ ∣
Tr[· · · ] = Tr[· · · d]∣ +Ω Tr[· · · ]∣ +O(Ω2) (7.50)
Ω=0 dΩ Ω=0
If we decide to eliminate the uncontracted covariant derivatives by giving the formally Ω-
dependent value (7.49) to α we pay a price, namely we neglect a certain contribution to the
linear term in (7.50), Ω d Tr[· · · ], which arises through the Ω-dependence of α ≡ e(6−d)ΩdΩ . By
the chain rule, it has the form
− d
{ ∣ }∣
(6 d)Ω Tr[· · · ]∣
Ω=0 ∣
∣
(7.51)
dα α=1
Now, the crucial observation is that the trace at Ω = 0 whose α-derivative is taken here is to
justify our choice, equals precisely the trace which appears in the single-metric computation.
7.4 The ‘Ω deformed’ α = 1 harmonic gauge 237
Its α-dependence has been investigated in detail in ref. [135] and found to be rather small. In
particular varying α over the interval from α = 0 to α = 1 causes changes in the flow properties
which are smaller than those due to the cutoff scheme dependence and truncation error which, in
fact, supplied the justification for setting αk equal to a k-independent constant. This implies that,
at the level of accuracy we may expect on ∣the basis of the truncations already made, the term
(7.51) is indeed negligible because Tr[· · · ]∣ has no significant α-dependence near α = 1.
Ω=0
Thus, in a truncation which neglects the running of α also the piece missed by the deformed
gauge, (7.51), may be omitted.
A second, independent justification
Formally using a Ω-dependent α-parameter requires a careful consideration of the resulting
mixing of level-(0) with higher level contributions, which has to be kept as small as possible.
Indeed, there is a way to justify the choice (7.49) from a different perspective. Assume we had
started with a slightly different truncation ansatz in which the gauge fixing functional, Γgfk , is
substituted with
( ) gf ′
(γ) ū
Γ gfk [h; ḡ] = 1− 1 kγ gf Γk [h; ḡ]+ 1γ Γsubsk [h; ḡ] (7.52)ūk
Here γ 6= 0 is a constant introduced for later convenience. Furthermore, Γsubsk [h; ḡ] is a new
functional which under the conformal projection (g= e2Ωḡ), is required to obey
Γsubsk [h= 2Ωḡ; ḡ] = 0 and[ ] [ ]
Π Ĥess ΓsubsΩ h k [h; ḡ] =
γ(d−6)Ω ( gf ′ gf gfe ūk /ūk ) ΠΩĤessh Γk [h; ḡ]
The first condition guarantees that the LHS, i.e. ∂tΓk, remains[unchange]d, whereas the second
requirement assures that the old and the new Hessians Ĥessϕ Γk[ϕ ;Φ̄] on the RHS agree at
linear order in Ω for a certain choice of gf ′ūk . Expanding the Hessian of (7.52) in powers of Ω,
[ γ ] ( ) [ ]
ΠΩĤessh Γk[h; ḡ] = 1+(d−6)Ω (
gf ′
ūk /
gf
ūk )ΠΩĤessh Γ
gf
k [h; ḡ] +O(Ω
2)
we find it proportional to the ‘old’ gauge fixing contribution. The additional pre-factor can be
absorbed by a redefinition of the gauge parameter as follows:
( )
gf ≡ 1+( −6)Ω+O(Ω2) gf ′ūk d ūk . (7.53)
[ ] [ ]
The Hessians are equal then: γΠΩĤessh Γk[h; ḡ] = Π
gf
Ĥess 2Ω h Γk [h; ḡ] +O(Ω ). This in turn
implies that the above steps for removing the uncontracted covariant derivatives work with all
(γ)
choices of Γk , however now with the parameter
gf ′
ū = ūDk k ⇔ α ′k = 1 (7.54)
Within the generalized truncations (7.52), a Ω-independent choice for gf ′ūk is sufficient to ac-
count for the same simplification on the LHS of the flow equation that we used in the original
ansatz with only Γgfk containing the Ω-dependent coupling
gf
ūk . This Ω-dependence of a cou-
pling in the truncation ansatz is thus merely an artifact of the conformal projection on the set
of all possible invariants whereby different level-(p) monomials lead to the same Hessian. In
this sense the Ω-dependence of gfūk can be thought of as a method for implicitly including addi-
tional field monomials in the truncation ansatz that lead to a removal of uncontracted covariant
derivatives without introducing any new effects, otherwise.
There are several suitable candidates for Γsubsk [h; ḡ] of which we consider here only those of
the form
∫ √
Γsubs[ ; ] = gf ′ µν (γ) (γ)k h ḡ ūk ḡ ḡ F̃µ [g, ḡ](h) F̃ν [g, ḡ](h) (7.55)
238 Chapter 7. Trace evaluation
For the choice γ = 2 the pre-factor of the gauge fixing action in eq. (7.52) is 1/2. The ‘missing’
half in the Hessian of the RHS of the flow equation now stems from the level-(2) contribution of
Γsubsk [h; ḡ]. All higher level terms are new, additional field monomials with identical pre-factors
fixed by gf ′ūk . Possible choices of the structure given in (7.55) with γ = 2 include
√
g [ ]
F̃2;Iµ [g, ḡ] = √ gαλgβρgκσ ḡµρ ḡλσ −ϖ ḡµσ ḡλρ D̄κ (7.56a)
√ḡ [ ]
F̃2;II
g
[g, ḡ] = √ gβλ ḡ ḡ gακgσρ −ϖ gαρgκσµ λρ µσ D̄κ (7.56b)ḡ
It is straightforward to verify that all requirements on Γsubsk [h; ḡ] are indeed fulfilled by these
examples. (Notice that additional terms appearing in the Hessian associated to the action (7.55)
are zero when evaluated for the conformally related metrics, g= e2Ωḡ.)
Another interesting option is γ = 1. In this case one actually replaces the gauge fixing
functional (6.26) with a new action containing the gauge fixing condition F̃1µ [g, ḡ](h) = 0. This
choice requires however a new Faddeev-Popov operator µM̃ ν [g, ḡ] that corresponds to this class
of gauge fixing condition. If we want to leave the RHS of the flow equation unchanged, we
have to adopt an additional modification in the ghost sector.
First of all notice that with the choice (7.55) the requirements imposed on Γsubsk [h; ḡ] can be
!
transferred to requirements on γ γF̃µ [g, ḡ](h). We obtain ΠΩF̃µ [g, ḡ](h) = 0 from the vanishing
of Γsubsk [h; ḡ] under the conformal projection, and
γ γ(d−6)Ω/2 γ(dvF̃µ)[g, ḡ](h) = e ΠΩ(dvFµ [g, ḡ](h)
!
follows from the second criterion and by using γΠΩ(F̃µ [g, ḡ](h)) = 0. Neglecting ghost contribu-
tions on the RHS of the flow equation allows us to rewrite the new Faddeev-Popov operator as
being proportional to the original one, namely
µ
Π M̃ [g, ḡ] = eγ(d−6)Ω/2 µΩ ν M ν [g, ḡ] (7.57)
This change of M could be compensated by a Ω-dependent ghost coupling ūgh in the original
truncation. Thus, if there is a gauge fixing condition of the type
(√ )1/2 ( )
F̃1;I
g β
µ [g, ḡ](h) = √ gλκ ḡλκ δµ gαρD̄ρ −ϖgαβ D̄µ (gαβ − ḡαβ ) (7.58)ḡ
[ ]
f[or instance, it]would correspond to the choice
gf
ūk = ū
D 1+ (d − 6)Ω + O(Ω2) and ūgh =
1+ γ gh ′2(d−6)Ω ū for the gauge parameter and the ghost coupling, respectively, in a truncation
with the gauge fixing functional of eq. (11.56). We are not going to continue the analysis of
the various possibilities and simply apply this prescribed technique to remove the undesirable
uncontracted derivative terms in the Hessian operator.
The Hessian simplified
This particular choice of the gauge fixing sector simplifies the Hessian operator significantly.
Thus, we are going to project the FRGE using the deformed α = 1 harmonic gauge (ϖ = 1/2)
with its associated projection ΠαΩ . Thereby, the RG running of the gauge coupling ᾱk adjusts to
the RG evolution of ūDk in precisely the way that technical difficulties of uncontracted covariant
derivatives gets eliminated. In the truncation ansatz, this corresponds to give a precise structure
to the gauge fixing functional:
∫ √
ΠαΩ(Γ
gf
k [h; ḡ]) = (1+(d−6)Ω)ūD ddk x ḡ ḡµνFµ [ḡ](h)Fν [ḡ](h)
M ( )
with F [ḡ](h)≡ ḡαρ ḡβσ − 1 ḡαβ ḡρσµ 2 ḡαµD̄βhρσ
7.5 Heat kernel 239
When considering the bulk bi-metric beta-functions we shall provide another independent justi-
fication of this procedure by comparing our results to those of the calculation in ref. [172] where
the uncontracted derivatives had been dealt with by a full transverse-traceless decomposition of
hµν .
Equation (7.45) already presents the final form of the Hessian operator in the ghost sector,
since it is independent on ᾱk and ΠαΩ corresponds to setting ϖ = 1/2. These contributions
basically consist of a simple differential operator of Laplace type with an additional constant
summand: (D̄2+ 1C2). In the gravitational sector, we experience an enormous reduction in par-
d
ticular because of the vanishing off-diagonal terms in the trace-traceless decomposition. Thus,
what remains are the block diagonal contributions, both –the trace- and traceless-part– being
functions of a single operator, D̄2:
D
ΠαΩ ◦Πm.s. ◦
A C noTrΠ AΩ(Ĥesshh)B =+(ΠnoTr)B (Ĥesshh )C (ΠnoTr)D
Tr D
+(Π C ATr)B (Ĥesshh)C (ΠTr)D (7.59)
It is quite impressive in which way the choice of gauge fixing condition reduces the Hessian
operators. For instance, the traceless-traceless constituent assumes the form:
ΠαΩ ◦Πm.s. ◦
noTr µν
ΠΩ(Πm.s.(Ĥess ) )[ hh κλ ] (
− D D · 1 −4 · ν µ
)
= (ūk λ̄k ) ( + (d )Ω) δ{κ δ(λ ) }
+ 12 [(1+(d−6)Ω)(ū
D µ
k )] ·δ νκ δ d(d−3)+4 C2− D̄2λ ( −1) (7.60)d d
Its trace-trace counterpart, which was already of Laplacian-type, also feels a strong reduction
due to both, the harmonic gauge- and the (α = 1)-choice. In particular the global pre-factor
disappears, since (2−d)/(2−2d ·ϖ) = 1 holds true in the harmonic case. Thus, the final result
reads:
ΠαΩ ◦
Tr µν
Πm.s. ◦ΠΩ(Πm.s.(Ĥess[ hh
)κλ ) ]
=− (ūD λ̄Dk k ) · (1+(d−4)Ω) · ν
µ
δ δ
{( κ )λ }
+ 12 [(1
µ
+(d−6)Ω)(ūD)] · d−4k ·C2− D̄2 δ ν δ (7.61)d κ λ
Notice that both operators have the trivial tensor structure δ ν µκ δλ .
Now all operators occurring under the traces on the RHS of the FRGE are functions of the
Laplacian D̄2 alone, and we can easily apply the standard heat kernel techniques to compute the
beta-functions for the six couplings of interest, ūDk , λ̄
D
k , ū
B
k , λ̄
B
k , and the boundary coefficients
(ū∂B + ū∂Dk k ) as well as (ū
∂Bλ̄ ∂Bk k + ū
∂Dλ̄ ∂Dk k ).
7.5 Heat kernel
Before we continue the evaluation of the functional trace on the RHS of the flow equation, it
is instructive to summarize the so far obtained results. In order to extend the above projection
techniques to affect the entire trace, notice that the cutoff operator Rk[ḡ] is invariant under the
projection Π .all .= ΠαΩ ◦Πm.s. ◦ΠΩ. In particular ΠΩ, as would be ΠΩ(x), leave Rk[ḡ] unchanged.
Hence, we can make use of Πall on the complete RHS with no further modifications.
In the last sections, we first demonstrated that under the action of Πξ=0=ξ̄ the ghost and
240 Chapter 7. Trace evaluation
metric sector contribute separately, with independent traces given in eq. (7.20):
[( [ ] ) ]∣−1 ∣
STrF Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄] ∣∣ (7.62)
[ ] ξ̄=0=ξ( )
[h] −1
= STr 2 Ĥess
grav grav
L hh
+Rk [ḡ] ∂tRk
[( )
[ξ̄ ] −1
] [ ( )−1 ]
+ STr + gh[ ] ∂ gh
[ξ ] gh gh
2 ĤessL ξ ξ̄ Rk ḡ tRk +STr ĤessL2 ξ̄ ξ +Rk [ḡ] ∂tRk
In the next steps, we mainly focused on the metric trace employing various projection tech-
niques, Πall , that ultimately lead to a trivial tensor structure of the Hessian with only Laplacian
type operators. Therefore, we decomposed the identity 1 in its trace and traceless components
and applied this projection on the Hessian operator. It turns out that for the harmonic gauge
fixing (α = 1)-choice Ĥesshh becomes diagonal w.r.t. ΠTr and ΠnoTr. Hence, since
grav
Rk [ḡ] is
proportional to 1, the trace of the metric component decomposes in an orthogonal way4
[( )− ] [ ( ) ][ 1h] [h] noTr −1
STr Ĥess + grav[ ] ∂ grav = STr Π Ĥess + grav grav
L2 hh
Rk ḡ tRk L2 noTr hh Rk [ḡ] ∂tRk
[ ( ) ]
Tr −1
+STr[h] Π Ĥess + grav[ ] ∂ grav
L2 Tr hh
Rk ḡ tRk
In total there are four traces over functions of the Laplacian operator. However, the structure of
the traces is in all four cases quite similar, in particular the Hessian operators exhibit a common
pattern that allows for a general treatment of the heat kernel expansion:
H(ℓ,C;c) ..=−D̄2−2e2Ωℓ+ρP cC2
Here, we introduced an x-independent constant c and an artificial bookkeeping parameter ρP = 1.
The latter is inserted to study the impact of ‘paramagnetic’- or ‘diamagnetic’-like contributions
to the RG evolution of the couplings [165]. Loosely speaking, ‘paramagnetic’ terms appear as
non-minimal contribution in the propagator which here corresponds to the scalar curvature part,
proportional to C2.
The four parts of the Hessian operator can now be related to H(ℓ,C;c), evaluated for partic-
α α
ular values of c and ℓ. Substituting (g−1) = δ α = (g−1) , we arrive at
ξ ξ̄ µµ ξ̄ξ µ
noTr ≡ 1 (d−6)Ω D · (λ̄D, ; ) · with ≡ d(d−3)+4ΠallĤesshh 2 e ūk H k C cT 1[noTr] cT ( −1) (7.63a)d d
Tr
Π Ĥess ≡ 1 e(d−6)Ω Dall hh 2 ūk ·H(λ̄
D
k ,
(d−4)
C;cS) ·1[Tr] with cS ≡ (7.63b)√ d
ΠallĤessξ ξ̄ ≡− 2e2Ωūgh ·H(0,C;cV) ·1[ξ̄ ] with cV ≡− 1 (7.63c)√ d
ΠallĤessξ̄ ξ ≡ 2e2Ωūgh ·H(0,C;cV) ·1[ξ ] (7.63d)
Hereby, we assumed that all O(Ω2) terms are neglected.
The cutoff operator has the same (trivial) tensor structure than the above operators and fur-
thermore it shares the coefficients with the kinetic operator −D̄2 in each field space component
for Ω = 0. In detail we have:
grav
Rk [ḡ]≡+ 1 D ·
(0)
2 ūk Rk (−D̄
2) ·1[h] and√ √
gh
Rk [ḡ]≡− 2 gh ·
(0)
ū Rk (−
gh
D̄2) · gh1[ξ̄ ] , Rk [ḡ]≡+ 2ū ·
(0)
R (−D̄2k ) ·1[ξ ]
4For brevity we omitted the additional projections, Πall , which however are essentially to obtain the diagonal
structure of the Hessian operator.
7.5 Heat kernel 241
Hence, for the present truncation, the general form of the inverse operator, that appears on the
RHS of the flow equation, can be written as follows:5
( ) ( )
Π Ĥess +R•[ḡ] = ū•eγΩ −D̄2−2e2Ωℓ+ cρ C2 + • (0) 2all • k k P 1[•] ūk Rk (−D̄ )1[•]
=+ū•
{( ) }
(0)
k ·1[•] · −D̄2−2ℓ+R (−D̄2) + cρ C2{( (k ) ) P }
+ γ · ū• · · − γ+2D̄2−2 ℓ + cρ C2k 1[•] γ P ·Ω+O(Ω2)
Here, γ denotes the conformal weight of the underlying invariant and in the present case is
γgrav = (d−6) and γgh = 2 for the metric- and ghost-contribution, respectively.
Using the abbreviation A(ℓ) ≡ (−D̄2− 2ℓ+ (0)R (−D̄2k )) for the diamagnetic Ω = 0 contri-
bution, we can expand the inverse of the above operator structure up to first order in Ω:
( )−1 ( )−1
ΠallĤess
•
•+Rk[ḡ] = A(ℓ)+ cρPC
2 (ū•k)
−1·1[•] (7.64)
( )− ( ( ) )− · 2Ω A(ℓ)+ cρ C2 −D̄2−2 γ+2 ℓ+ cρ C2 (ū•P γ P k)−1γ ·1[•]
This result is very useful, since all four traces are covered by its general form. In the end, we
only have to insert the particular values of c, ℓ, and γ .
The operator of eq. (7.64) acts on ∂tR•k and finally, we have to trace these terms using the
heat kernel asymptotics. Hence, let us consider the general form of this missing operator:
∂ R•
( )
[ (0)t k ḡ] = ∂
• 2
t ūk ·Rk (−D̄ ) 1[•]
Equation (7.64) implies that any k-independent scaling of the coefficient ū•k does not affect the
RHS of the flow equation since it appears once in the denominator and as ∂ ū•t k in the numerator.
In particular, the two traces in the ghost sector can be combined, since the Hessian L2-operators
differed only by a factor of −1 which is compensated in combination with ∂ •tRk [ḡ]. Hence, we
only have to deal with three independent traces, namely
[( [ ] )− ]1
ΠallSTrF Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ◦∂tRk[Φ̄] (7.65)
[ ( ) ]
[ −1h] noTr
= Π STr Π + grav[ ] ∂ gravall Ĥess R ḡ RL2 noTr hh k t k
[ ( )− ][h] Tr 1
+ Π grav gravallSTr 2 ΠTr ĤessL hh+Rk [ḡ] ∂tRk
[ ( ) ]
[ξ ] −1
+2ΠallSTr 2 Ĥessξ̄ ξ +
gh gh
Rk [ḡ] ∂L tRk
Let us now introduce one more functional Wp(∆, ℓ), with ∆ = −D̄2, in which the inverted
operator and ∂tR•k [ḡ] are combined:
W (∆;ℓ, ū•p ) ..=
1 −p
k 2A(ℓ) · (ū•)−1 ∂ R•( k t k )
≡ 12A(ℓ)
−p · ∂ (0)tRk (∆)+
(0)
R (∆)∂ ln(ū•k t k) ·1[•] (7.66)
In terms of the functional Wp(∆, ℓ) the Taylor expansion of the functional trace in the constant
C assumes the form:
[ ( ( ) ) ]
1 [•] · · · [•]
[ ] [•]
2STr [ ] = STr
γ+2
W (∆;ℓ, ū• •
L2 L2 1 k
) −STr 2 Ωγ · ∆−2 ℓ ·W2(∆;ℓ, ūk) (7.67)
[ { L ( γ ( ) ) }]
[•]
+STr 2 C
2 · (cρP) (Ωγ −1)W2(∆;ℓ, ū•k)+2Ωγ · ∆−2 γ+2γ ℓ ·W3(∆;ℓ, ū
•)
L k
√
5For brevity, we include the pre-factors 12 or ± 2in the k-dependent coefficient ū•k .
242 Chapter 7. Trace evaluation
Here, the neglected higher order terms are either of order O(Ω2) or O(C4) and thus irrelevant
for the present truncation ansatz.
In the final expansion based on the heat kernel asymptotics, we project the RHS on the
basis invariants contained in our truncation ansatz. In total, there are 3 times 6 (−2) traces that
have to be evaluated. The following provides a list of the sixteen components that ultimately
constitute the beta-functions. We start with the terms of zeroth order inC:
| STr[noTr]
[ ] [ ]
RHS C=0 = 2 W1(∆; λ̄
D, ūDk k ) · STr
[Tr]
1 + W (∆; λ̄D, ūD) ·1
L [noTr] L2 1 k k [Tr]
(7.68a)
[gh][ ]
+ STr 2 W1(∆;0, ū
gh) ·1
L [gh]
(7.68b)
[ [ ]− noTr]STr 2 (d−6)Ω∆W2(∆; λ̄Dk , ūDk ) ·1[noTr] (7.68c)L [ ]
−STr[Tr] D D2 (d−6)Ω∆W2(∆; λ̄k , ūk ) ·1[Tr] (7.68d)L
[ ] [ ]− ghSTr 2 2(2Ω)∆W2(∆;0, ūgh) ·1[gh] (7.68e)L
[ ] [ ]noTr
+STr 2(d−4)Ω λ̄D ·W (∆; λ̄D2 2 , ūD) ·1L [noTr] (7.68f)
[ ] [
k k k ]
Tr
+STr 2 2(d−4)Ω λ̄D D D 2k ·W2(∆; λ̄k , ūk ) ·1[Tr] +O(Ω ) (7.68g)L
The next order is already quadratic in C and thus does not induce any boundary contributions
within our truncation (remember that K̄ is of order C).
| − [ ]
[ ]
noTr
RHS= RHS STr C2C=0 2 · (cT ρP)(1+(d−6)Ω)W D DL 2(∆; λ̄k , ūk ) ·1[noTr] (7.69a)[ ]
− [Tr]STr 22 C · (cS ρP)(1+(d−6)Ω)W2(∆; λ̄D Dk , ūk ) ·1[Tr] (7.69b)L
[ ][ ]−STr gh 22 2 ·C · (cV ρP)(1+2Ω)W (∆;0, ūgh2 ) ·1L [gh] (7.69c)
[ [ ]noTr]
+STr 2(d−6)ΩC2 · (c ρ )∆W (∆; λ̄D, ūD) ·1 (7.69d)
L2 T P 3 k k [noTr]
[ [ ]Tr]
+STr 2(d−6)ΩC2 · (c ρ D D
L2 S P
)∆W3(∆; λ̄k , ūk ) ·1[Tr] (7.69e)
[ ] [ ]gh
+STr 2 8Ω(C
2 · (cV ρP)∆W ghL 3(∆;0, ū ) ·1[gh] (7.69f)[ ]
− [noTr]STr 2 4ΩC2 · (cT ρP)(d−4)λ̄Dk ·W3(∆; λ̄Dk , ūDk ) ·1[noTr] (7.69g)L
− [
[ ]
noTr]
STr 2 4ΩC
2 · (cS ρP)(d−4)λ̄D D D 4 2L k ·W3(∆; λ̄k , ūk ) ·1[Tr] +O(C ,Ω ) (7.69h)
[noTr] [noTr]
Each function in this sum is either of type STr 2 [ f W (∆) ·1] or STr 2 [ f ∆W (∆) ·1], withL L
f ≡ f (C,Ω,d) an x-independent function.
R Notice that for the general conformal projection or in the case of an explicit h-expansion
f ≡ f (x) since Ω ≡ Ω(x) becomes spacetime dependent.
The heat kernel
In the preceding steps we managed to reduce the Hessian operator to a function of the fully
contracted Laplacian ∆ ≡ −D̄2 only. For the chosen Dirichlet boundary conditions and M
being compact with non-vanishing boundary in general, ∆ = −D̄2 is not only elliptic but also
strongly elliptic and self-adjoint with non-negative spectrum and a finite number of zero modes,
dim(ker(−D̄2)) < ∞. For these operators, there exists a unique solution to the heat equation
that can be used to expand the RHS of the flow equation.
Therefore, consider the operator h −s∆s ≡ e which under the above conditions is related to a
unique kernel, the heat kernel, defined by
∫ √
(h d 2s f )(y)≡ d x ḡ hs(y,x) f (x) ∀ f ∈ L (M)
M
7.5 Heat kernel 243
The heat kernel has important applications in physics, since it solves the heat equation, the
‘Wick-rotated’ analog of the Schrödinger differential equation:
∫ √
(∆+∂s)hs(y,x) = 0 with lim d
dx ḡ hs(y,x) f (x) = f (y) ∀ f ∈ L2(M)
s→0+ M
In the presence of (regular) boundaries we have to add a supplementary boundary condition to
uniquely specify the kernel hs(y,x); for Dirichlet constraints this simply reads hs(y,x) = 0 for
all y ∈ ∂M.
Besides the symmetry hs(y,x) = hs(x,y), the heat kernel inherits its homogeneity-, locality-,
and translational invariance properties from the differential equation it satisfies. For non-empty
boundaries the invariance under translations is broken. However, in the present context, its
most important feature is given by the existence of a complete asymptotic expansion in the
limits s→ 0+ and s→+∞. The latter asymptotics results in the projection operator on the zero
modes of ∆, which becomes apparent when expanding the heat kernel in eigenfunctions of ∆:
h (y,x) =∑e−sλ js φ j(y)φ j(x) with ∆φ j(x) = λ jφ j(x) (7.70)
j
If s approaches +∞ the exponential starts dominating the eigenfunctions, which are locally
confined due to the heat equation. What remains are the unsuppressed zero-modes, in particular
we have
lim h (x,x) = dimker(−D̄2s )< ∞
s→+∞
For our purpose more interesting is the opposite limit, where hs(x,y) stays close to the delta-
distribution.
∞
lim h (x,x) = s−d/2 ∑ α (x)s j/2s j (7.71)
s→0+
j=0
Hereby, the coefficients α j are smooth functions and the equality holds uniformly, thus we can
integrate over both sides and still retain the identity:
[ ] ∞ √ j
lim Tr −s∆ −d/2L2 [h→ s
( f ) ] = lim TrL2 e f = s ∑ a j( f ;∆) s with
s 0+ s→0+
j=0
∫ √ j−1∫ √
a ( f ;∆) = ddx ḡ f (x)α (x)+ ∑ dd−1x H̄ (nµD̄ )rj j µ f (x) ·α j;r(x) (7.72)
M r=0 ∂M
We introduced a smooth function f for later convenience that furthermore fulfills Dirichlet
boundary conditions. In the present case f (x) is constant, however in general it contains pow-
ers of h(x) or the conformal factor Ω(x) when ΠΩ(x) is applied. Notice that whenever ∂M is
empty, the sum only runs over even integers, since the odd ones represent boundary monomials.
Explicitly, for Dirichlet conditions, the very general and universally valid expression of the few
Seeley-DeWitt coefficients a j( f ;∆), which were already known from the pioneering work of
McKean and Singer [185], are given by6
∫
d √
a0( f ;∆) = +(4π)
− 2 ddx ḡ f (7.73a)
∫M √ √
a1( f ;∆) =−
d π
(4π)− 2 dd−1x H̄ f (7.73b)
∫∂M 2
d √
a2( f ;∆) = +(4π)
− 2 ddx ḡ 16 f R̄ (7.73c)
∫M √
d ( )
+(4π)− 2 dd−1x H̄ 16 3n
ρ D̄ρ f +2 f K̄ (7.73d)
∂M
6Here str(1) denotes the trace over the space on which ∆ acts.
244 Chapter 7. Trace evaluation
Thus, terminating the asymptotic series at order s3/2 the terms retained are precisely those in-
variants contained in the truncation ansatz, i.e.
[ ] ∫ ∫
sD̄2 · str(1)
{ √ 1√ √
TrL2 e f 1 = d
dx ḡ f (x)− 4πs dd−1x H̄ f (x) (7.74)
(4π )d/2s M 4 ∂M
(∫ √ ∫1 √ )
+ s ddx ḡ R̄ f (x)+2 dd−1x H̄K̄ f (x)
6 M ∂M
∫
1 √ }
+ s dd−1x H̄nρD̄ρ f (x)+O(s
3/2)
2 ∂M
The seemingly additional term containing the normal derivative nρD̄ρ f (x) will contribute to
the dynamical curvature invariant of the boundary, where under conformal projection f (x) is
proportional to Ω(x). Extending the final truncation ansatz to include those derivative terms
allows for deriving the beta-function of ū∂B ∂Dk and ūk independently.
R In the presence of boundary there appear Seeley-DeWitt coefficients a j( f ;∆) of half-
integer order [185, 186] that generate only boundary terms in the trace expansion. Fur-
thermore, notice that in order O(s) the Einstein-Hilbert and Gibbons-Hawking-York terms
appear in precisely the ‘preferred’ combinationwith a relative coefficient of+2, necessary
for obtaining field equations in GR.
The great virtue for our present discussion is the special structure of the asymptotic expan-
sion. By dimensional arguments it is already clear that each order in the expansion includes
higher derivative operators, thus eq. (7.72) can really be understood as a series in basis invari-
ants with increasing order in D̄. This is especially useful to project the flow equation onto the
truncation ansatz consisting of the zeroth, first and second order in D̄. Now the trace over a
general function W (∆) of the Laplacian ∆ = −D̄2 can be expanded in this asymptotic series.
Therefore, consider the Fourier transform ofW (∆) and substitute s=−ıv:
∫ +∞ [ ] ∫ +ı∞
TrL2 [W (∆) f ·1] = dvŴ (v)TrL2 eıv∆ f ·1 ≡−ı dsŴ (ıs)TrL2 [hs( f ) ] str(1)
−∞ −ı∞
In this way, the function specific information becomes separated from the geometrical data,
which is represented by the trace over the heat kernel. Inserting the asymptotic expansion in the
limit s→ 0+ into the last equation, we arrive at a derivative expansion of TrL2 [W (∆) f ]:
∫ [ ]+ı∞ √ j−d
TrL2 [W (∆) f ·1] =−ı dsŴ (ıs) ∑ s a j( f ;∆) str(1)
−ı∞ j≥0
∫ +∞ √ j−d
=+ ∑ a j( f ;∆) · str(1) · dvŴ (v) −ıv (7.75)
j≥0 −∞
Notice that the Fourier decomposition is compatible with the asymptotic expansion s → 0+,
since the domain of integration has real part ℜ(s) = 0. Hence, the above series precisely reflects
a projection of the RHS of the flow equation onto the basis monomials that constitute our
truncation ansatz. Furthermore, we have separated the geometric information contained in the
Laplacian, from the specific shape of the functionW .
7.5.1 Mellin transform
A feature of the heat kernel asymptotics is that the part containing the structure of the function
W (v) can be described by its Mellin transform, defined in [187]:
∫ +∞
Qn[W ]≡ dvŴ (v)(−ıv)−n . (7.76)
−∞
7.5 Heat kernel 245
This gives rise to the same integral transform that appears in the heat kernel expansion of W
in eq. (7.75). Hence, the close relation between hs and the Mellin transform is encoded in the
following asymptotic series:
TrL2 [W (∆) f ·1] = ∑ a j( f ;∆) · str(1) ·Q(d− j)/2[W ] (7.77)
j≥0
The Mellin transform can be rewritten in a more elegant form that involves the functionW
rather than its Fourier transform Ŵ . Therefore, we first insert an identity based on the integral
representation of the gamma function Γ(n).
∫ [ ]+∞ ∫1 v·∞
Q [W ] = dvŴ (v)(−ıv)−n · (−ı)n dwwn−1 ıwn e
−∞ Γ∫ +∞ ∫
n 0
1 v·∞· dw
(w)n−1
= dvŴ (v) eıw
Γn −∞ 0 v v
In the last step we suggestively rearranged the terms in the second integral such that substituting
z= v for v in the measure yields
u
∫
1 +∞
∫ ∞ ∫ ∞ ∫ +∞
Q [W ] = dvŴ (v) dzzn−1eıv·z
1
= dzzn−1 duŴ (v)eıv·zn
Γn −∞ 0 Γn 0 −∞
At this point notice that the v integration represents the inverse Fourier transformation. Hence,
the Mellin transform assumes the form
∫
1 ∞
Qn[W ] = dz z
n−1W (z) (7.78)
Γn 0
Notice that for n= 0 and for functionsW that have proper support on the positive real numbers,
we obtain the limit Q0[W ] =W (0).
The Mellin transform of a function exhibits certain very useful properties that turn out to
be relevant in studying the general properties of the later derived threshold-functions. As an
integral transform the Qn are linear in its argument. On the other hand a change of the degree
n is related to a transformation of the argument. For instance the Mellin transforms of the
functionsW (z) by z ·W (z) are related as follows
∫
1 ∞
Q nn[z ·W (z)] = dzz W (z) = n ·Qn+1[W (z)] (7.79)
Γ(n) 0
This homogeneity property can be extended to integration- and differentiation operations of the
argument in Qn. Again, it turns out that a simple change of the index n provides a mechanism
to modifiedW (z):
∫
1 ∞
∫
1 ∞
∫
zn 1 ∞
Q [W ] = dzzn−1W (z) = dzW (z)∂ =− dzznW ′n z (z)
Γ(n) 0 Γ(n) 0 n Γ(n+1) 0
=−Q ′n+1[W ] (7.80)
In the second step we used integration by parts, assuming that for sufficient large values of z
the functionW (z) drops off rapidly enough, thusW (z) has sufficiently restricted support. This
relation can be equivalently described as Qn[W ] = (−∂z)(−n)W (0) for n< 0.7
7For half-integer negative n the fractional derivative has to be understood as being defined by the integral repre-
sentation of eq. (7.80).
246 Chapter 7. Trace evaluation
7.5.2 Heat kernel expansion
We can now apply the previous results to our truncation ansatz using the projection techniques
described in the section 7.3, whereby in the end we have to insert the explicit form ofW given
by the combination of Hessian- and cutoff operator. The first reduction is obtained by project-
ing out all invariants in the heat kernel expansion that cannot be resolved within the present
truncation, using ΠTrunc(•):
ΠTrunc(TrL2 [W (∆) f ·1]) = + ∑ ΠTrunc(a j( f ;∆) · str(1) ·Q(d− j)/2[W ])
j≥0
=+ ∑ Π (a j( f ;∆)) · str(1) ·QTrunc (d− j)/2[W ] (7.81)
j≥0
In the second step we made use of the fact that the entire geometric information is contained in
the Seeley-DeWitt coefficients a j( f ;∆). In particular the basis invariants in the expansion are
separated from the Mellin transformed W function and thus ΠTrunc(•) affects only the Seeley-
DeWitt coefficients a j( f ;∆). The great advantage of using the heat kernel series for the present
truncation is that it shows a trivial projection in that ΠTrunc(a j( f ;∆)) ≡ 0 for all j ≥ 3 and
ΠTrunc(a j( f ;∆))≡ a j( f ;∆) otherwise. Hence, we obtain
ΠTrunc(TrL2 [W (∆) f ·1]) = +a0( f ;∆) · str(1) ·Qd/2[W ]+a1( f ;∆) · str(1) ·Q(d−1)/2[W ]
+a2( f ;∆) · str(1) ·Q(d−2)/2[W ] (7.82)
Thus, ΠTrunc(•) terminates the series at order O(s3/2) and the trace on the RHS of the flow
equation assumes the form:
ΠTrunc(TrL2 [W (∆) f ·1]) (7.83)∫ ∫
str(1) { √ √ √
= +Q [ πW ] · ddx ḡ f (x)− Q [W ] · dd−1x H̄ f (x)
(4π)d/2
d/2 2 (d−1)/2
M ∂M
(∫ √ ∫ √ )
+ 1Q d6 (d−2)/2[W ] · d x ḡ R̄ f (x)+2 d
d−1x H̄K̄ f (x)
M ∂M
∫ √ }
+ 1Q d−1 ρ2 (d−2)/2[W ] · d x H̄n D̄ρ f (x)
∂M
The next two projection techniques, Πξ=0=ξ̄ and ΠΩ actually do not affect the result of eq.
(7.83), since the function W as well as f do not depend on the ghost fields or the fluctuation
field hµν anymore. In addition, the simplified Hessian operator has no field dependence, except
for the background metric ḡ(x) and thus f (x) is either constant in spacetime, depending only on
the dimension d, the constant C and Ω, or it additionally inherits the x-dependence purely from
the background metric ḡ. This has further implications on the expression of eq. (7.83), since
the last contribution is proportional to the covariant derivative of f (x), which thus vanishes. It
is the consequence of the conformal projection technique, and thus also of its simplified version
the single-metric projection, that the dynamical Gibbons-Hawking-York term disappears and
cannot be resolved within the present truncation.
Furthermore, the trace-traceless decomposition acts as the identity on the outer part of the
heat kernel expansion, since the tensor structure is encapsulated in the trace over the identity
str(1) and hence no mixing of different tensor subspaces takes place.
Under the final projection of the basis invariants onto maximally symmetric spaces, Πm.s.
7.5 Heat kernel 247
the scalar- and the extrinsic curvature get substituted with the constant C2 and C, respectively.
ΠTrunc(TrL2 [W (∆) f ·1]) (7.84)∫
str( {1) √ √ ∫ √
= +Q dd/2[W ] · d x ḡ f (x)− π2 Q(d−1)/2[W ] · d
d−1x H̄ f (x)
(4π)d/2 M ∂M
(∫ √ √ ∫ √ )}
+ 1Q [W ] · ddx ḡC26 (d−2)/2 f (x)+2
(d−1) dd−x1 H̄C f (x)
d
M ∂M
Since f (x) and the Mellin transform ofW in principle contain additional factors ofC and Ω what
remains is an expansion in those constants up to second or first order, respectively. This final
projection and the comparison of coefficients that leads to the beta-functions for our truncation
ansatz concludes the second part of the thesis and is discussed in the next chapter. Before we
turn to this final step, let us consider the remaining traces over the tensor spaces in the following
subsection.
7.5.3 Tensor space traces
The heat kernel expansion takes care of the L2-operator trace. There is a remaining vector
space trace str(1) that factors each term in this series and relates to the tensor character of the
underlying field. The general procedure to evaluate (infinite dimensional) endomorphisms was
presented in subsection 1.4.2. However, in order to compute the remaining traces we exclusively
deal with finite vector spaces, for which we simply have to ‘trace’ over all basis elements.
Though we have decomposed the metric field space into its trace- and traceless part, let us
start with the identity element on the full space of symmetric 2-tensor fields, µν µ(1S)
1 ν
ρσ = 2(δρ δσ +
δ ν
µ
ρ δσ ). Its trace corresponds to the dimension of the associated vector space and thus counts
the number of independent components. It is given by
( )
str(1S) = tr
µν µ µ
(1S) = (1S)
1
µν = 2 δµ δ
ν +δ νν µ δν =
d(d+1)
2
In d = 4 the space of symmetric 2-tensor fields is thus of dimension 10, with exactly the same
number of free components. The same could be obtained by Young diagrams in a straightfor-
ward manner. Each index corresponds to a box and a generic 2-tensor splits into its symmetric-
and anti-symmetric parts, i.e. and with dimension d(d+1) or d(d−1)2 2 , respectively. Now,
what about the traces of the orthogonal decomposition into trace- and traceless part? This is
easily computed by projecting the identity 1S to the respective subspace using either ΠTr for the
trace component or ΠnoTr for its traceless counterpart. Hence, we obtain:
str(1Tr) = tr
µν
(ΠTr1S) = tr(Π ) = (Π
1 µν
Tr Tr)µν = ḡ ḡµν = 1 (7.85)d
str(1noTr) = tr(ΠnoTr1S) = tr(ΠnoTr) = tr(ΠS)− tr( ) = d(d+1)−2 = (d−1)(d+2)1Tr 2 2 (7.86)
In four dimensions, we thus have 10 = 9⊕ 1. Since the metric sector is described by ordinary
tensor fields the supertrace str can be replaced be the usual one tr.
On the other hand, the ghost sector is of Grassmann type and therefore an additional minus
sign occurs in the evaluation. Except of this subtlety, the remaining tensor structure is given
by the tangent space which carries the same dimensionality as its underlying manifold, hence
equals d or simply . Overall, the supertrace over the identity on θ ·Γ (TM) reads:
str(1ξ ) =−tr(1Γ (TM)) =−d (7.87)
In the following we write str(1ξ ) = −dρgh with a bookkeeping parameter ρgh = 1 that allows
to separate the ghost contributions from the metric traces.

8.1 Threshold functions
8.2 Beta-functions for the coefficients
8.3 Beta-functions for dimensionful couplings
8.4 Beta-functions for dimensionless cou-
plings
8.5 The semiclassical approximation
8.6 Beta-functions in d = 4
8.A Expanded beta-functions in d = 2+ ε
8.B Beta-functions in d = 3
8. THE BETA-FUNCTIONS
This chapter is about the beta-functions of the truncated FRGE for metric gravity and it con-
cludes the second part of this thesis. Hence, the results of the detailed calculation performed
in the previous chapters are presented in the following sections, first in general dimensions and
then for d = 4 in particular.
So far we have derived the Hessian operator for a bi-metric truncation ansatz that contains
bulk and boundary invariants up to second and first order in the derivative D̄, respectively. The
dependence on a field space consisting of two (on the bulk) independent metrics and ghost fields
which results in a theory space that at this order already consists of an infinite number of basis
monomials. We focused on the Einstein-Hilbert and Gibbons-Hawking-York functionals that
are either constructed by both, background- and dynamical metric, or purely by ḡ, respectively.
This allows to study Background Independence in the context of the Asymptotic Safety scenario
for QG as well as investigating the impact of spacetime topologies with boundaries on the RG
evolution.
The last chapter presented a detail simplification of the Hessian operator and ultimately used
the heat kernel expansion to project the RHS on the field monomials comprised in the truncation
ansatz of eq. (5.15). In this chapter, we first of all give a summary of the so far obtained results,
introduce the threshold functions and then compare coefficients of the LHS and RHS of the
flow equation. We then briefly discuss the beta-functions for the dimensionful couplings that
are more common in physics. Next, we convert those expressions into dimensionless quantities
that define theory space and allow for the search of fixed points. The results are again presented
for general spacetime dimension d and generic shape-functions. Since most of the numerical
results of part III are obtained employing the optimized shape function, we append the results for
the beta-functions for this special shape function. Finally, we focus on the spacetime dimension
d = 4 which for practical reasons is the most relevant one. The discussions of the beta-functions
in d = 3 and in the limit d = 2+ ε are shifted to the appendix 8.A and 8.B.
C Except for the boundary results, section 8.4 to 8.B follow closely ref. [173].
250 Chapter 8. The beta-functions
8.1 Threshold functions
The current status of the RHS of the flow equation involves a sum over L2-traces which ei-
[noTr] [noTr]
ther are of type STr 2 [ f Wp(∆;ℓ; ū
•
k)] or STr 2 [ f ∆Wp(∆;ℓ; ū
•
k)], with f ≡ f (C,Ω,d) an x-L L
independent function and
( )
(∆;ℓ, •)≡ 1A(ℓ)−p · ∂ (0)(∆)+ (0)Wp ūk 2 tRk Rk (∆)η•k ·1[•] (8.1)
Whereby, A(ℓ)≡ (− 2−2ℓ+ (0)D̄ R 2k (−D̄ )) denotes the denominator and
(0)
Rk (∆)≡ k2R(0)(∆/k2)
the shape function. Furthermore we defined the anomalous dimension η• ≡ −∂ ln(ū•k t k) of the
coupling ū•k .
We start this section by evaluating these type of traces using the threshold-functions, pΦn (w)
and pΦ̃n (w), introduced in [122]. Afterwards, we state some general properties of
p
Φn (w) and
p
Φ̃n (w).
8.1.1 The Mellin transform ofWp(∆;ℓ; ū•k)
In the heat kernel expansion the structure of Wp becomes separated from the geometric infor-
mation encoded in the Laplacian operator, −D̄2, and from the function f . It thus suffices to
consider the Mellin transform ofWp(∆;ℓ; ū•k) independently, i.e. evaluate the following integral
transformation
[ ] ∫1 • ∞[ ]
Qn Wp(∆;ℓ; ū
• n−1
k) = dzz Wp(z;ℓ; ū
•
Γ(n) k
)
0
∫ (
1 ∞ 1− 1η•
)
k2R(0)(z/k2)− zR(0)′(z/k2)
= dzzn−1 2 k (8.2)
Γ(n) 0 (z−2ℓ+ k2 (0)( / 2))pR z k
The prime in R(0)′(z/k2) denotes the derivative of the shape function w.r.t. its argument, i.e.
R(0)
′(z/k2)≡ ∂ R(0)s (s)|s=z/k2 . This term originates from ∂t = k∂k acting on (0)Rk (∆) in the numer-
ator, yielding
( ) ( ) ( )′
∂t k
2R(0)(∆/k2) = k∂ k2R(0)k (∆/k
2) = 2k2 R(0)(∆/k2)− (z/k2)R(0) (z/k2)
Introducing artificial factors of k2 allows to rewrite the integrand in terms of the argument of
the shape function, z/k2:
[ ] ∫
(
∞ 1 •
)
(0) 2 z (0)′ 2
• 2( − ) 1n p [•] 1− η R (z/k )− 2R (z/k )Qn Wp(∆;ℓ; ūk) = k dz( kz/k2)n−1 2 kΓ( pn) 0 ((z/k2)−2k−2ℓ+R(0)(z/k2))
Now, we perform a change of integration variables, y = z/k2 and k2 dy = dz. For this transfor-
mation the integration range does not change, since k ∈R+. We obtain
[ ] ∫ (∞ 1− 11 • η•
)
R(0)• 2(n+1−p) [ ] n−1 2 k (y)− yR(0)
′(y)
Qn Wp(∆;ℓ; ūk) = k dy y (8.3)Γ(n) p0 (y−2k−2ℓ+R(0)(y))
In this way, we can absorbed the k-dependence of the Mellin transform in a global pre-factor
k2(n+1−p) by simultaneously replacing ℓ by k−2ℓ. Equation (8.3) can be rewritten using the
threshold function, a k-independent construction that contains the cutoff-specific information:
∫ ∞ (0) (0)′
Φp
1 R (y)− yR (y)
n (w)≡ dyyn−1 (8.4a)Γ(n) p0 (y+w+R(0)(y))
∫ ∞ (0)
Φ̃p
1
(w)≡ dyyn−1 R (y)n Γ( ) ( p (8.4b)n 0 y+w+R(0)(y))
8.1 Threshold functions 251
In terms of this set of threshold functions, the Mellin transform of W (∆;ℓ; ū•p k) assumes the
form:
[ ] ( ( ) ( ))
Qn Wp(∆;ℓ; ū
•
k) = k
2(n+1−p) Φp −2k−2ℓ − 1η• Φ̃pn 2 k n −2k
−2ℓ · 1[•] (8.5)
It is convenient to introduce p( pq w,η•) ..= Φ (w)− 1n k n 2η•
p
k Φ̃n (w) as abbreviation for this specific
combination of threshold functions, since we will encounter those terms in several expressions
for the beta-functions later on. Hence, eq. (8.5) can be rewritten as
[ ]
Qn Wp(∆;ℓ; ū
•) = k2(n+1−p) qpk n(−2k−2ℓ,η•k ) · 1[•] (8.6)
8.1.2 The Mellin transform of ∆Wp(∆;ℓ; ū•k)
Except for the traces of the above type, there are additional terms with an extra factor of −D̄2.
However, from eq. (7.79) theMellin transform can be directly related to the result ofW (∆;ℓ; ū•p k)
by means of the following identity
∫
1 ∞
Qn[z ·W (z)] = dzznW (z) = n ·Qn+1[W (z)]
Γ(n) 0
Hence, the Mellin transform of ∆Wp(∆;ℓ; ū•k) again gives rise to a combination of the threshold
functions (8.4), which fully cover the contribution from the shape function times a k-dependent
pre-factor:
[ ] ( ( ) ( ))
Qn ∆Wp(∆;ℓ; ū
•
k) =
p
nk2(n+2−p) Φ −2n+1 −2k ℓ − 12η
• p
k Φ̃n+1 −2k−2ℓ · 1[•]
≡ nk2(n+2−p) pq (−2k−2n+1 ℓ,η•k ) · 1[•] (8.7)
All terms in the heat kernel expansion (7.69) can be expressed using the results of eq. (8.6)
and (8.7). What remains is the trace over the identity operator 1[•] over the respective field
subspace. Before we turn to this last ingredient, let us consider some of the general properties
of the threshold functions.
8.1.3 Properties and estimates of the threshold functions
This subsection lists and proves a set of general properties the threshold functions as defined in
eq. (8.4) exhibit. Furthermore, we state certain estimates that might prove useful when deducing
general features of the beta-functions independent on any specific shape function. The results
only depend on the following assumptions on the analytic properties of the shape function that
is usually employed:
R(0)(z= 0) = 1 , lim R(0)(z) = 0 , R(0)(z)≥ 0 ∀z≥ 0 (8.8)
z→∞
Under these assumptions it is possible to constraint the w dependence of the shape functions
p
Φn (w) and
p
Φ̃n (w), derive the limit n→ 0+ and present some general estimates of the p- and
n-dependence.
We start with the following proposition for a general number of derivatives of pΦn (w) w.r.t.
w:
∂mΦp
Γ(p+m)
w n (w) = (−1)m Φp+mn (w) (8.9)Γ(p)
252 Chapter 8. The beta-functions
The proof is straightforward. Using the definition of pΦn (w) in eq. (8.4) we obtain for the first
differentiation
( ∫ )
1 ∞ (0) (0)′
∂ Φp (w) = ∂ dyyn−1
R (y)− yR (y)
w n w Γ(n) (y+w+R(0) p0 (y))
∫
1 1 ∞ R(0)(y)− yR(0)′(y)
= (−p) dyyn−1 = (−p) · Φp+1n (w) (8.10)Γ(n) Γ(n) p+10 (y+w+R(0)(y))
Here we interchanged the order of integration and differentiation which is perfectly fine as long
as the integrand is differentiable. Next, a successive application of this result yields
m p − m Γ(p+m) p+m ≡ − m (p−1+m)!∂ Φ p+mw n (w) = ( 1) Φ (w) ( 1)Γ(p) n (p− Φn (w) (8.11)1)!
Whereby the latter equality holds for p and m positive integers.
A similar expression is found for the second type of threshold functions pΦ̃n (w), namely
∂mΦ̃p − m Γ(p+m)w n (w) = ( 1) Φ̃p+mn (w) (8.12)Γ(p)
Applying the derivative w.r.t. w on the integral representation of the threshold function, we
obtain the same result as in the previous case:
( ∫ )∞
p 1 n−1 R(z)∂wΦ̃n (w) = ∂w dzzΓ(n) 0 (z+R(z)+w)p
∫ ∞
− 1 n−1 R(z)= ( p) dzz = (−p) · Φ̃p+1 (w)
Γ(n) 0 (z+R(z)+w)p+1
n
Again, an iterative application yields the above identity.
∂mΦ̃p − m Γ(p+m) p+m (p−1+m)!w n (w) = ( 1) Φ̃n (w)≡ (−1)m p+mΓ(p) (p− Φ̃1)! n (w)
Notice that since both threshold functions behave the same under differentiation w.r.t. w, the
p
qn -function exhibit the equivalent property.
Assuming p > 1 and n > 0, the threshold functions can be expanded in a Taylor series for
vanishing argument w, resulting in
∞ ( )Γ(p+m)
Φpn (w) = ∑ (−1)m Φp+mn (0) · wm (8.13)
=0 Γ(m+1)Γ(p)m
If pΦn (w) is continuous and ( j+1)-times differentiable on the interval [w,0], we can use Taylor’s
theorem to approximate the error when truncate the Taylor series at order m, say. The remainder
is given by
wm+1 ∣
R (w) = ∂m+1Φp (w) ∣m (8.14)
(m+1)! w n w=ξL∈[w,0]
whereby ξL is some real number in the interval [w,0]. Substituting the maximum value the LHS
can assume for any ξL ∈ [w,0] gives a bound on the contributions from the neglected terms.
Along the same lines one can derive the Taylor expansion of pΦ̃n (w) and
p
qn(w,η•k ). Thus,
setting bounds on the coefficients of these Taylor series by deducing the properties of pΦn (0),
p
Φ̃n (w), and
p
qn(w,η•k ) may give some insight if these expansions converge.
8.1 Threshold functions 253
Based on the above assumptions on the shape functions it is possible to rewrite pΦn (w) into
a more convenient form using the following substitution of integral variables:
( )
v= R(0)(y)y−1 =⇒ dv=−dy y−2 R(0)(y)− yR(0)′(y)
At this point the positivity of R(0)(y) is important to retain the integral limits. Taking care of the
extra minus sign by switching the integral limits, the standard threshold function Φpn (w) can be
cast into the following form:
∫
1 ∞ ′p ≡ n−1 R
(0)(y)− yR(0) (y)
Φn (w) dyyΓ( ) (0) pn 0 (y+w+R (y))
∫
1 ∞ R(v)n+1−p≡ dv (8.15)
Γ(n) 0 (1+w/R(v)+ v)p
with R(v)≡ R(0)(y(v))/v. This turns out to be very useful when evaluating pΦn (0) for vanishing
arguments w.
For instance, let us consider the following estimate for the threshold functions, where α ≥ 0:
Φp+αn+α (0) ≤ Γ(n)p ≤ 1 (8.16)
Φn (0) Γ(n+α)
This expression states that the coefficients in the Taylor expansion decrease and thus for suitable
values of w the series converges. To prove this estimate, we make use of eq. (8.15) as follows
∫
1 ∞ R(v)(n+α)+1−(p+α)p+α
Φn+α (0) = dvΓ(n+α) 0 (1+ v)p+α
∫
1 ∞ R(v)n+1−p
= dv · (1+ v)−α
Γ(n+α) 0 (1+ v)p
Since the integrand is a positive function and the integration interval extends only to positive
values v≥ 0 in the integration interval, the additional factor is bounded as (1+ v)−α ≤ 1 for all
α ≥ 0. Hence, we obtain the inequality
∫ ( )∞ n+1−p ∫ ∞ n+1−p
p+α 1 R(v) Γ(n) 1 R(v)
Φn+α (0)≤ dv = dvΓ(n+α) 0 (1+ v)p Γ(n+α) Γ(n) p0 (1+ v)
The last term can be identified with the threshold function pΦn (0) from which thus the above
estimate follows
Φp+α
Γ(n)
n+α (0)≤ Φpn (0)Γ(n+α)
The same statement also holds for the second type of threshold functions pΦ̃n (0) and thus
also for pq (0,η•n k ):
p+α
Φ̃n+α (0) ≤ Γ(n) (8.17)
p
Φ̃n (0) Γ(n+α)
The proof of this estimate is quite similar to the previous one. We start by separating the extra
power of α from the remaining integrand:
∫
1 ∞p+α
Φ̃ (0) = dzzn+α−1 ( R(z)n+α )Γ(n+α) p+α0 z+R(z)
( ) ∫ ( )α
Γ(n) 1 ∞ R(z) z
= dzzn−1 ( ) ( )
Γ(n+α) Γ( ) pn 0 z+R(z) z+R(z)
254 Chapter 8. The beta-functions
This time we use R(z) ≥ 0 to set an upper bound of the additional factor. For z ≥ 0, which
corresponds to the integration range, we have
( z ) 1= ( ) ≤ 1
z+R(z) 1+R(z)/z
The case z= 0 is a null set of the integration and in fact also the integrand vanishes. Using this
inequality, we arrive at the claimed estimate
( ) ∫ ( )
p+α ≤ Γ(n) 1
∞
Φ̃ (0) dzzn−1n+α (
R(z) ) Γ(n)= Φ̃p (0)
Γ(n+α) Γ( ) pn n0 z+R(z) Γ(n+α)
In conclusion we can truncate the Taylor series of the threshold functions at a certain order
in the expansion provided that w is sufficiently small. For |w| < 1 the Taylor series converge,
which is particular interesting for the semiclassical approximation.
Finally, the special case of n= 0 follows from (8.4) in the limit n→ 0+ [122]:
p p
Φ0 (w) = Φ0 (w) = (1+w)
−p (8.18)
This concludes our general discussion on threshold functions. If not stated otherwise we later
evaluate Φpn (w) and Φ̃
p
n (w) for the optimized cutoff shape function, for which the integrals can
be explicitly computed.
8.2 Beta-functions for the coefficients
In this section we list the beta-functions for the coefficients that appear in the truncation ansatz
of eq. (5.15) for general spacetime dimension d and arbitrary cutoff shape function.
8.2.1 Comparison of coefficients
As in any (finite) dimensional vector space, a full set of linearly independent basis elements
is the basic ingredient to understand and study the whole space. While theory space in its
complete form is surely not finite dimensional, the considered truncated subspace (5.15) is
definitely finite, in fact a vector space of dimension 6. Therefore, the standard tools of algebra
are directly applicable. In particular any vector of this truncated subspace is uniquely expressed
in terms of a set of basis elements and as such, we can compare coefficients in any equality.
To this end, we have to project the set of traces given in eq. (7.69) onto the basis invariants
of the truncation ansatz using the heat kernel expansion and the results of the previous section.
While the actual expansion of the RHS is not illustrative, we focus here on some remarks only
and present the results in the following subsections.
A. From the eight contributions in RHS|C=0 only the first three terms affect the level-(0)
coefficients, since all others are linear in Ω.
B. For eq. (7.69) we observe again that there are far more terms linear in Ω which thus only
trigger the RG evolution of the dynamical invariants.
C. Our truncation ansatz contains two basis invariants of orderC2, the level-(0) and level-(1)
Einstein-Hilbert functionals, and another one of order C for the boundary curvature term,
thus
LHS ≡ LHS| +( d LHS) ·C+ 1( d2C=0 d 2 2 LHS) ·C
2
C dC
On the other hand, the operators under the functional trace are given by
RHS≡ STr[ · · · O(C0)]+STr[ · · · O(C2)]+O(C4)
8.2 Beta-functions for the coefficients 255
In the heat kernel expansion these terms are multiplied by additional factors ofC0,C, and
C2. Hence, the heat kernel as∫ym√ptotics of the C-quadratic terms in eq. (7.69) reduces
to the bulk volume invariant M ḡ only. All other field monomials that appear in the
expansion would combine to terms of at least order C3 which is beyond the scope of the
present truncation.
D. The last remark in particular implies that all boundary effects stem from the terms con-
tained in RHS|C=0, while the Einstein-Hilbert coefficients obtains corrections also from
1 d2the 2( 2RHS) ·C2. This asymmetry between boundary and bulk contributions makes adC
matching of ūDk and ū
∂D
k unlikely. However, for the present truncation we are unable to
resolve ū∂Dk . If the conformal projection technique is replaced by its x-dependent analog
Ω(x) we will have additional terms that may compensate this imbalance.
E. For the ghost contributions, later marked with the parameter ρgh = 1, the last argument of
(∆;ℓ; ghWp ūk ≡ ūgh) is in fact k-independent and thus its associated anomalous dimension
vanishes, ηghk ≡−∂ ght ln ū = 0. Hence, all
p
qn that occur contain the anomalous dimension
of the dynamical Newton coupling ηD ≡−∂t ln ūDk = ∂t lnGDk :
qpn (w,η
D)≡ Φpn (w)− 1ηD Φ̃p2 k n (w) (8.19)
In the sequel we list the beta-functions for the dimensionful coefficients that are obtained by
comparison of coefficients for the six basis invariants. Therefore, we rewrite eq. (7.39) into the
level-language, which leads to
( ) ∫ √ ( )
Πall ∂tΓk[Φ,Φ̄] = +2 · dd ∂ ( (0) λ̄ (0)x ḡ t ūk k )+dΩ ·∂ (
(1) (1)
t ūk λ̄k )
∫ M √ ( )
− dd 2 ∂ (0) +( −2)Ω ·∂ (1)x ḡ C t ūk d t ūk
M∫
− √+2 · dd 1 ∂ ( ∂ (0) λ̄ ∂ (0)x H̄ t ūk k )
∫∂M √ √
−2 · dd−1 (d−1)x H̄ C ∂ ∂ (0)t ūk +O(Ω2) (8.20)d
∂M
Each of the six basis invariants can be uniquely identified by its power in Ω and C as well as
the integration range, which is either M or ∂M. While level-(1) contributions are linear in Ω,
we can project onto level-(0) invariants by setting Ω = 0 on both sides. Along similar lines the
beta-functions for all coefficients then follow from the truncated FRGE:
[ ]
1 ( [ ] )−1
Πall (∂tΓk[g, ḡ]) = ΠallSTr ΠallĤessϕ Γk[Φ,Φ̄] +Rk[Φ̄] ∂tRk[Φ̄]
2
8.2.2 The Einstein-Hilbert invariants
Let us start wit∫h the E√instein-Hilbert invariants of the dynamical and background metric. They
correspond to M d
dx ḡ C2 which are either linear in or independent of Ω. The coefficient for
this invariant on the LHS reads
Π =−∂ (0) − ( −2)Ω ·∂ (1)EHLHS t ūk d t ūk
The corresponding terms on the RHS can be easily projected using ΠEH(•) = ∂C2(•)|C=0, since
it is the only invariant of order O(C2). The different levels, level-(0) and level-(1), are then
obtained using Π(0)(•) = (•)|Ω=0 or Π(1)(•) = ∂Ω(•)|Ω=0, respectively. This results in two beta-
functions
∂ (0) ≡ β̄ (0)( (1), λ̄ (1)t ūk ū ūk k ;k) and ∂
(1)
t ūk ≡ β̄
(1)( (1) (1)ū ūk , λ̄k ;k)
256 Chapter 8. The beta-functions
Since level-(0) contribution do not enter the Hessian operator and thus the RHS of the flow
equation, there is no back-reaction of (0)ūk in the beta-function of
(1)
ūk . For level-(0) the explicit
form of the beta-function is given by
kd−2 d {(0) (1) (1) − (d+1)
( )
β̄ū (ūk , λ̄k ;k) = q
1 (1) D
d 6 (d−2)/2
−2λ ,η
2(4π) 2
( )}
− (d−1)ρ 2 (1) DPqd/2 −2λ ,η
kd−2 {d }
+ ρgh Φ
1 2
d 3 (d−2)/2
(0)+2ρP Φd/2 (0) (8.21)
(4π) 2
Here, we have introduced the dynamical dimensionless cosmological constant k2λ̄ (1) =.. λDk k ≡
λ (1)k = λ
(2)
k = · · · and the dynamical anomalous dimension associated to Newton’s coupling
ηD ..=−∂t ln (1)ūDk ≡−∂t ln ūk . Notice that the beta-functions are indeed independent on the level-
(0) coefficients. Again, we emphasize that ρP = 1 = ρgh are only bookkeeping parameters that
ultimately are set to unity. They are particularly convenient to understand how the different
contributions affect the RG flow. We will make use of this technique in part III of this thesis.
The RG evolution for the higher level Newton-type coefficients is described by the follow-
ing beta-function
d−2 { ( )
β̄ (1)( (1)
k d
, λ̄ (1); ) = (d−6)(d+1)ū 2 (1) Dū k k k d 6 qd/2 −2λ ,η (8.22)
4(4π) 2
− 2 (d−4)(d+1)
( )
(1) 2
( −2) λ q(d−2)/2 −2λ
(1),ηD
3 d
−2 (d−6)(d−1)
[ ( ) ( )]
3 (1) D 2 (1) D
( −2) ρP d q(d+2)/2 −2λ ,η −qd/2 −2λ ,ηd
}
+8 (d−4)(d−1)
( )
ρ λ (1) q3 −2λ (1),ηD(d−2) P d/2
kd−2 {( ) }− ρ d 4gh 3 − ( −2)ρP Φ2d/2 (0)+ 4d ρ Φ3d d (d−2) P (d+2)/2 (0)
(4π) 2
( )
Notice that the entire dependence of β̄ (1)ū (
(1)
ūk , λ̄
(1)
k ;k) on
p
ūD D Dk arises from qn −2λk ,η -functions.
Thus, one could equivalently write β̄ (1)ū (η
D, λ̄ (1)k ;k). This in turn gives rise to an implicit equation
for the anomalous dimension
ηD ≡−∂ ln D = β̄ (1)(ηD, λ̄ (1)t ūk ū k ;k)
that we are going to study later on in this chapter, after introducing the full set of dimensionless
couplings.
In order to obtain the corresponding beta-functions in the D/B-language, one simply has to
employ the following relations
D ≡ (1) (0) (1)ūk ūk and ūBk ≡ ūk − ūk
In the case of beta-functions for couplings we mostly make use of the D/B formulation.
8.2.3 The bulk volume invariant
∫ √
In the next step, we consider the volume invariant on the bulk dM d x ḡ. It is associated to the
running of the dynamical and background cosmological constant. In contrast to the previous
8.2 Beta-functions for the coefficients 257
case of the Einstein-Hilbert functional, there is another basis invariant of the same degree in
C, namely the volume counterpart of the boundary. Hence, the projection should look like
Π .vol(M)(•) .= (•)|C=0;∂M=0/ , for which the LHS assumes the form
Π (0) (0) (1) (1)vol(M)LHS= 2∂t(ūk λ̄k )+2dΩ ·∂t(ūk λ̄k )
We then disentangle the different levels by means of Π(0) and Π(1), yielding the following two
beta-functions for the coefficients:
∂ ( (0) λ̄ (0))≡ β̄ (0)( (1), λ̄ (1); ) and ∂ ( (1) λ̄ (1))≡ β̄ (1)( (1) (1)t ūk k λ ūū k k k t ūk k λ ūk , λ̄k ;k)ū
Applying Π(0) and Πvol(M) to the RHS of the flow equation results in
d ( ) d
β̄ (0)( (1), λ̄ (1)
k d(d+1) 1 − k dλ ūk k ;k) = qd/2 2λ
(1),ηD − ρ 1
ū d d gh
Φd/2 (0) (8.23)
4(4π) 2 (4π) 2
The level-(1) and therefore D-counterpart has a quite similar structure with an additional term
proportional to λ (1) which vanishes in d = 4:
kd(d+1) { ( ) ( )}
β̄ (1)( (1), λ̄ (1)ū ;k) = 2(d−4)λ (1) q2 −2λ (1),ηD − (d−6)d q2 −2λ (1)λ k k d/2 2 (d+2)/2 ,η
D
ū d
4(4π) 2
kd dρgh
+ Φ2
d (d+2)/2 (0) (8.24)
(4π) 2
By construction, both beta-functions are purely diamagnetic in nature, since ρP multiplies terms
of order O(C2) and higher. Notice that eq. (8.22) and (8.24) give rise to a closed system of two
coupled differential equations which can be solved independently of the other equations. Once
the level-(1) or D solutions are obtained, one can insert them into the remaining beta-functions
and solve for the level-(0) coefficients.
The transition to the D/B formulation of the beta-functions is straightforward. The conver-
sion rules are given in eq. (5.18) and (5.17). They read:
(0) (0) (0)
ū λ̄ = ūD λ̄Dk k k k + ū
B
k λ̄
B
k ūk = ū
D
k + ū
B
k
(p)λ̄ (p)ū = ūD D (p) Dk k k λ̄k ūk = ūk ∀ p≥ 1
Along the same lines, one obtains the relations for the boundary coefficients we consider next.
8.2.4 The Gibbons-Hawking-York invariant
The Gibbons-Hawking-York basis invariant is the only one of order O(C). It is furthermore only
of level-(0) since the next order vanishes under conformal projection. The corresponding LHS
is given by
√
ΠGHYLHS=−2 (d−1) ∂ ∂ (0)d t ūk
On the RHS only the first three terms of eq. (7.68) contribute, for they are of order O(Ω0,C0).
Applying ΠGHY to the full flow equation, a comparison of coefficients leads to
∂ ∂ (0) ∂ (0) (1) (1)t ūk ≡ β̄ū (ūk , λ̄k ;k)
Hereby, the beta-functions of the level-(0) Gibbons-Hawking-York invariant reads
d−2 { ( ) }
β̄ ∂ (0)( (1)
−k d
ū ūk , λ̄
(1)
k ;k) = (d+1)q
1
d/2−1 −2λ (1),ηD −4ρ Φ112(4π)d/2 gh d/2−1 (0) (8.25)
Notice that it contains only diamagnetic contribution.
258 Chapter 8. The beta-functions
8.2.5 The boundary volume invariant
Finally, we consider the last basis invariant, the volume element on the boundary ∂M. In order
to project the flow equation to this invariant, we have to set C to zero and ignore the bulk con-
tribution. Under the corresponding projection Π .vol(∂M)(•) .= (•)|C=0;Int(M)=0/ the LHS assumes
the following form:
Π ∂ (0) ∂ (0)vol(∂M)LHS = 2∂t(ūk λ̄k )
This time, Dirichlet boundary conditions for the metric fluctuations ensure that there are in fact
no higher order terms. The k-dependence of the coefficient is then governed by the following
equation:
∂t(
∂ (0)
ūk λ̄
∂ (0)
k )≡ β̄
∂ (0) (1)
λ (ūk , λ̄
(1)
k ;k)ū
Except for the global pre-factor, the explicit form of the beta-functions resembles eq. (8.25):
d−1 { }
β̄ ∂ (0)
−k d
λ (
(1)
ūk , λ̄
(1)
k ;k) = − (d+1)q
1 D 1
ū (d 1) (d−1)/2 (−2λ ,η )−4ρgh Φ(d−1)/2 (0)
16(4π) 2
(8.26)
This however, is not by chance but rather due to the fact that both beta-functions result from the
same three traces and are evaluated only on the basis of the Seeley-DeWitt coefficient a1( f ;∆)
in eq. (7.73).
8.3 Beta-functions for dimensionful couplings
In physics language, the coefficients discussed in section 8.2 are usually replaced by the dimen-
sionful couplings. For the considered truncation ansatz, this only affects the coefficients asso-
ciated to the curvature invariants, which can be rewritten in terms of Newton-type couplings
Gk:
I ≡ 1ūk for all I ∈ {(p), ∂ (p), B, ∂B, D, ∂D} (8.27)16πGIk
Thus, the corresponding beta-functions of the Newton-type couplings can be read off from the
results of section 8.2 using the following identity:
∂ GI =− 1 (ūI )−2∂ ūI ≡−16π(GI 2 It k 16π k t k k) ∂t ūk ≡ G
I
kη
I
k
In the final step we inserted the anomalous dimension η Ik ≡−∂t ln ūI of ūIk k. For the dimensionful
GIk it assumes the form:
η Ik ≡ ∂t lnGIk
The beta-functions β̄ IG({GJk, λ̄Dk };k) of the dimensionful Newton couplings GIk are defined via
∂ I ItGk ≡ β̄G({GJk, λ̄Dk };k) and thus – except for a factor of GIk – their k-dependence is fully deter-
mined by the anomalous dimension:
β̄ IG({GJ D I I J Dk, λ̄k };k) = Gk ηk({Gk, λ̄k })
8.3 Beta-functions for dimensionful couplings 259
From these rules of conversion, we can deduce the anomalous dimensions of the three involved
Newton-type couplings:
2kd−2 d {(d+1) ( )
η (0)( (1) (1)k ū
1 (1) D
k , λ̄k ) = q(d−2)/2 −2λ ,ηd
(4π) 2−1 6
( )}
− ( −1)ρ 2 −2λ (1),ηD (0)d P qd/2 Gk
− 4k
d−2 {d }
ρ Φ1gh (d−2)/2 (0)+2ρ Φ
2 (0)
d P d/2 (0) Gk (8.28a)
(4π) 2−1 3
kd−2 d { (d−6)(d+1) ( )η (1) (1) (1)k (ūk , λ̄k ) = − 2 (1) Dd−1 6 qd/2 −2λ ,η (8.28b)(4π) 2
2
+ (d−4)(d+1)
( )
(1) 2 (1) D
3 (d−2)
λ q(d−2)/2 −2λ ,η
+2 (d−6)(d−1)
[ ( ) ( )]
( −2) ρP d q
3 (1)
(d+2)/2 −2λ ,ηD −q2 −2λ (1)d/2 ,ηDd
}
−8 (d−4)(d−1)
( )
( −2) ρ λ
(1)
P q
3
d/2 −2λ (1),ηD
(1)
G
d k
4kd−2 {( ) }
+ ρ d 4 2gh 3 − ( −2)ρP Φd/2 (0)+
4d
( −2)ρ Φ
3 (1)
d− P1 d d (d+2)/2
(0) Gk
(4π) 2
∂ (0) d−2 { ( ) }
η ∂ (0)( (1)
G · k d
k ūk , λ̄
(1)
k ) =
k (d+1)q1 −2λ (1),ηD −4ρ Φ1gh (0) (8.28c)
3(4π)d/2−1 d/2−1 d/2−1
Notice that for the D/B-formulation we have the following identities
ηB/ B ≡ η (0)/ (0) −η (1)/ (1) (1) (1) (1)k Gk k Gk k Gk and ηD(ūD, λ̄Dk k k )≡ ηk (ūk , λ̄k ) (8.29)
These equations will be extensively used in the following sections.
The remaining couplings are of the cosmological constant type. They multiply the basis
invariants associated to volume elements. As coefficients they only occur in the combination
ūIkλ̄
I
k . In order to isolate their RG behavior, we use the following relation
∂ λ̄ I = η I λ̄ I +(ūI )−1∂ (ūI λ̄ I)≡ η It k k k k t k k k λ̄ Ik +16πGIk ∂t(ūIkλ̄ Ik ) (8.30)
When applied to the beta-functions of section 8.2 we obtain
d ( ) d
∂ λ̄ (0) = η (0)λ̄ (0)
k d(d+1)
+ 1 −2λ (1),ηD (0) − 4k d ρ Φ1 (0) (0)t k k k qd/2 Gk gh d/2 Gk (8.31a)d−1 d(4π) 2 (4π) 2−1
kd(d+1) { ( ) ( )}
∂ λ̄ (1) = 2(d−4)λ (1) q2t k d/2 −2λ (1),ηD −
(d−6)d
q2 (1) D
(1)
d−1 2 (d+2)/2
−2λ ,η Gk
(4π) 2
4kd dρgh
+ Φ2(d+2)/2 (0)
(1)+η (1) (1)G
d−1 k k
λ̄k (8.31b)
(4π) 2
260 Chapter 8. The beta-functions
d−1 { }
∂ λ̄ ∂ (0)
k d
t k = η
∂ (0)λ̄ ∂ (0)k k − − (d+1)q1(d−1)/2 (−2λ ,ηD)−4ρ Φ1
∂ (0)
(d 3) gh (d−1)/2 (0) Gk
4(4π) 2
(8.31c)
In the sequel of this chapter, we will consider dimensionless couplings, where the canonical
scaling is subtracted and for which the topological features of theory can be studied, as for
instance fixed points.
8.4 Beta-functions for dimensionless couplings
In this section, we present the beta-functions of all dimensionless couplings for general space-
time dimension d. Therefore, it is necessary to subtract the canonical scaling dimension of
the couplings and retain only their anomalous k-dependence. The conversion rules relating
dimensionless couplings to dimensionful differ for boundary and bulk couplings only in case
of the cosmological constant, where λ Ik = λ̄
I
k k
−2 holds for all I ∈ {D, B, (0), (1), (2), · · ·} and
λ ∂ (0)k = λ̄
∂ (0) −1
k k for the boundary counterpart. On the other hand, the Newton-type couplings all
share the same canonical dimension and thus can be converted to dimensionless quantities via
gIk =G
I
k k
d−2 for all I ∈ {D, B, ∂ (0), (0), (1), (2), · · · }. This amounts to the following relations
between their scale derivatives:
( )
∂ gI = (d−2)k(d−2) I d−2 I I I It k Gk+ k ∂tGk = d−2+ηk gk ≡ βg ∀ I (8.32a)
Likewise for the cosmological constant type couplings, we obtain
∂ λ I = k−2∂ I −2 I −2 I I It k t λ̄k −2k λ̄k = k ∂t λ̄k −2λk ≡ βλ ∀ I 6= ∂ (0) (8.32b)
∂ λ ∂ (0) = −1∂ λ̄ ∂ (0) − −1λ̄ ∂ (0) = −1∂ λ̄ ∂ (0)t k k t k k k k t k −λ
∂ (0)
k ≡ β
∂ (0)
λ (8.32c)
In the following we will list the dimensionless beta-functions in their explicit, d-dependent form
first in the {D,B}, then in the level-description. The threshold functions pΦn (w) and pΦ̃n (w)
which they contain are defined in section 8.1 and are the same as in ref. [122]. Keep in mind
that in order to identify the paramagnetic ghost and graviton contributions, respectively, the
corresponding terms are multiplied by the factors ρP and ρgh which should be put to unity if this
information is not needed.
In the sequel of this section we first study the beta-functions for arbitrary cutoff shape
functions, while in the afterwards we specialize for the ‘optimized’ one. We have published the
results for the bulk sector in ref. [173] and part of the following sections is carried over from
this publication.
8.4.1 Arbitrary cutoff shape function
For practical applications, it is more convenient to use a specific choice of cutoff shape function
that is consistent with the requirements we impose on the cutoff operator and at the same time
simplifies the expression in a suitable way. However, in order to prove the cutoff scheme inde-
pendence of certain results one either makes use of the general properties of pΦn (w) and
p
Φ̃n (w)
discussed in section 8.1 or one tests the stability of the results for different shape functions.
In this subsection, we present the beta-functions of the dimensionless couplings for arbitrary
cutoff shape functions and in general spacetime dimensions d.
The anomalous dimension related to gD
The additional term in βDg that adds to the canonical k-dependence in eq. (8.32a) is the anoma-
lous dimension ηD that encapsulates all the non-trivial contributions generated by the RG flow.
8.4 Beta-functions for dimensionless couplings 261
In case of the anomalous dimension for ‘D’-Newton constant, we have to solve an implicit equa-
tion to obtain the corresponding ηD, since it also appears on the RHS of the flow equation. It
yields
BD(λD;d)gD
ηD(gD,λD;d) = 1
1− (8.33)BD(λD2 ;d)gD
Here, BD1(λ
D;d) and BD2(λ
D;d) are functions depending on the dynamical cosmological constant
λD only, as well as parametrically on the spacetime dimension d. Their explicit forms are, for
the function in the numerator,
d {
D(λD; ) = − (d−6)(d+1)B 2 D1 d d−1 6 Φ(d/2) (−2λ )(4π) 2
2
+ (d−4)(d+1) λD Φ2 D( −2) (d−2)/2 (−2λ )3 d
+2 (d−6)(d−1)
[ ]
ρ dΦ3 D 2 D(d−2) P (d+2)/2 (−2λ )−Φ(d/2) (−2λ )
}
−8 (d−4)(d−1)ρ D 3( −2) P λ Φ(d/2) (−2λ
D)
d
4 {( ) }
+ ρ d 4 2 4d 3
d− gh1 3
− (d−2)ρP Φ(d/2) (0)+ ( ρ Φ (0) (8.34)d−2) P (d+2)/2
(4π) 2
Likewise the function in the denominator reads:
d {
D(λD; ) = (d−6)(d+1)B2 d
2 D
d−1 6
Φ̃(d/2) (−2λ )
2(4π) 2
− 2 (d−4)(d+1) D 2( −2) λ Φ̃(d−2)/2 (−2λ
D)
3 d
−2 (d−6)(d−1)
[ ]
3 D 2 D
(d−2) ρP d Φ̃(d+2)/2 (−2λ )− Φ̃(d/2) (−2λ )}
+8 (d−4)(d−1) D 3 D( −2) ρP λ Φ̃(d/2) (−2λ ) (8.35)d
Notice that in the present truncation the dimensionalities d = 4 and d = 6 play a special role: in
these cases the graviton contributions to BD1 and B
D
2 are either all given by terms with an extra
factor of λD multiplying the threshold functions, in d = 6, or precisely those terms are all absent,
in d = 4. This pattern is not found in the ghost contributions (which can be identified by their
ρgh factor).
The beta-function of λD
The running of the dynamical cosmological constant is governed by the beta-function
4dρ
βD(gD,λD;d) = (ηDλ −
gh
2)λD+gD Φ2 (0) (8.36)
d−1 (d+2)/2(4π) 2
(d+1) { }
+gD 2(d−4)λD 2 D D (d−6)dq 2 D D
d−1 (d/2)
(−2λ ,η )− 2 q(d+2)/2 (−2λ ,η )
(4π) 2
Note that the non-canonical graviton contributions on the RHS of (8.36) show the same property
as discussed above: they are proportional to either (d−4)λD or (d−6).
Together with eq. (8.35), eq. (8.36) gives rise to a closed system of two coupled differential
equations which can be solved independently of the other equations.
262 Chapter 8. The beta-functions
The anomalous dimension related to gB
The non-canonical k-dependence of the background Newton constant is given by the anomalous
dimension
ηB(gD,λD,gB;d) = (8.37)
d {(d+1)
= q1(d−2)/2 (−2λD,ηD)−
2 (d−4)(d+1) λD q2 D D
d−1 3 3 (d−2) (d−2)/2
(−2λ ,η )
(4π) 2
(
+ (d−6)(d+1) −8 (d−1)
)
2
6 ( −2)ρP q(d/2) (−2λ
D,ηD)
d
+8 (d−4)(d−1) D 3 D D(d−2) ρP λ q(d/2) (−2λ ,η ) }
−2 (d−6)(d−1)d 3 D( −2) ρPq(d+2)/2 (−2λ ,η
D) gB
d
4 {d ( }− ρ 1 d 2(d−4)
)
2 4d 3 B
d− gh
Φ
1 3 (d−2)/2
(0)+ 3 + (d−2) ρP Φ(d/2) (0)+ (d−2)ρPΦ(d+2)/2 (0) g
(4π) 2
The property found of ηD concerning its λD-dependence in d = 4 and d = 6 is only partially
shared by ηB; there are contributions from the graviton sector that neither drop out for d = 4
nor for d = 6.
The beta-function of λB
The last member in the hierarchy of the differential system of the bulk sector is the cosmological
constant λ B, which does not influence - but in turn shows a dependence on - all the other 3 bulk
couplings. Its beta-function is given by
4d { }
β Bλ (g
D,λD,gB,λ B;d) = (ηB−2)λ B− ρgh Φ1 2(d/2) (0)+ Φ(d+2)/2 (0) gBd
(4π) 2−1
(d+1) {
+ d q1 D(d/2) (−2λ ,ηD)−2(d−4)λD q2(d/2) (−2λD,ηD)d
(4π) 2−1
d }
+(d−6) q2(d+2)/2 (−2λD,ηD) gB (8.38)2
At this point all 4 RG equations of the D-B system are fully specified for the bulk sector.
What remains are the boundary couplings that we are going to discuss in the level language.
In the level description, the ‘D’ beta-functions provide the beta-functions at all higher levels
p= 1,2,3, · · ·
η (p) = ηD and β (p)λ = β
D
λ for all p≥ 1.
What is special are the level-(0) couplings. They are governed by combinations of the ‘B’ and
‘D’ beta-functions.
The anomalous dimension related to g(0)
The anomalous dimension of the level-(0) Newton coupling fulfills
η (0) ηD ηB
= + ,
g(0) gD gB
from which, using eqs. (8.35) and (8.37), it follows that
2d {(d+1) }
η (0) =+ q1 (−2λD,ηD)− (d−1)ρ q2 (−2λD,ηD) g(0)
d
(4π) 2−
P
1 6 (d−2)/2 (d/2)
4 {d }− ρ 1 2 (0)
d− gh
Φ
1 3 (d−2)/2
(0)+2ρP Φ(d/2) (0) g (8.39)
(4π) 2
8.4 Beta-functions for dimensionless couplings 263
For notational consistency, one should identify λD = λ (1) = λ (2) = · · · on the RHS of this equa-
tion.
As a check on our calculation, we mention that if instead we identify ηD =η (0) and λD = λ (0)
in the pqn (−2λD,ηD) functions we re-obtain precisely the anomalous dimension of the single-
metric truncation found in [122], η (0) = η sm, as it should be. (See also Section 10.3)
The beta-function of λ (0)
For the cosmological constant at level-(0) one finds
β (0) = (η (0)−2)λ (0) +g(0) d(d+1) 1 − − ( ) 4dq Dλ (d/2) ( 2λ ,η
D) g 0 ρ Φ1 (0) (8.40)
d−1 d gh(4π) 2 (4π) 2−1
(d/2)
where λD = λ (1) = λ (2) = · · · in the present truncation. Again, it can be checked that β (0)λ gives
rise to precisely the single-metric beta-function β smλ , obtained in [122] if instead we make the
identification ηD = η (0) and λD = λ (0).
The anomalous dimension related to g∂ (0)
For the present truncation, it is difficult to speak of D/B couplings in case of the boundary
invariants. The reason is that level-(1) is not resolved in our final ansatz and thus it is purely
speculation which contribution can be associated to the dynamical or background RG evolution.
While for the cosmological constant, there is in fact no level-(1) counterpart, the direction in
theory space associated to the Newton-type coupling ∂ (1)gk vanishes under conformal projection.
Thus, consider the anomalous dimension of ∂ (0)Gk given in eq. (8.28c) and express it in terms
of the dimensionless couplings.
g∂ (0) d { ( ) }
η ∂ (0)(gD,λD,g∂ (0)) = − (d+1)q
1 (1) D 1
3(4π)d/2 1 d/2−1
−2λ ,η −4ρgh Φd/2−1 (0)
(8.41)
For this anomalous dimension there is no obvious reduction for any spacetime dimension d,
as observed for ηD in particular. However, for λD = 0 and ηD = 0, which corresponds to the
semiclassical regime, there is a global factor of (d−3). Notice that due to the general properties
of the threshold functions the bracket is positive for d ≥ 3. Thus non-trivial fixed point values
for g∂ (0) will be negative.
The beta-function of λ ∂ (0)
The last ingredient describes the RG evolution of the boundary volume element.
β ∂ (0) = (η ∂ (0) −1)λ ∂ (0)λ
d { }− (d+1)q1 D 1 ∂ (0)− (d−1)/2 (−2λ ,η )−4ρgh Φ(d−1)/2 (0) g (8.42)(d 3)
4(4π) 2
The single-metric results obtained in [166] are structurally the same however with ηD and λD
replaced by their single-metric analogs.
Notice that similar to the hierarchy ( (1) (1) (0) (0)gk ,λk )→ gk → λk on the bulk sector, we obtain a
chain of dependencies for the bulk-boundary couplings:
( (1),λ (1)gk k )→
∂ (0) → λ ∂ (0)gk k
This has some very practical consequences. For instance in deriving solutions for the full 6-
dimensional truncated theory space, one first solves for the level-(1) bulk couplings and then
insert the results into the remaining four differential equations to obtain a full RG trajectory.
264 Chapter 8. The beta-functions
8.4.2 Optimized cutoff shape function
The ‘optimized’ shape function [188] is given by R(0)(z) = (1− z)θ(1− z) and allows for an
explicit evaluation of the threshold-functions:
( ) ( )
Φp
1 −p 1 −p
n (w) = 1+w , and Φ̃
p (w) = 1+w (8.43)
Γ(n+1) n Γ(n+2)
In the following we list the beta-functions in general spacetime dimensions d within this opti-
mized scheme.
The anomalous dimension related to gD
The key ingredients for the anomalous dimension ηD in the D-sector, eq. (8.33), are the func-
tions BD D1(λ ;d) and B
D D
2(λ ;d) given by (8.34) and (8.35), respectively. In the optimized scheme
they are obtained upon inserting (8.43) into (8.34):
(1−2λD)−2 {
D(λD; ) = − (d+1)
(
( −6)−2 (d−4)d
)
B1 d d λ
D
d 3 d−2
(4π) 2−1Γ(d/2) ( )
(d−1) (6−d)+2(d+2)λD }
+4 ρP
( ) (d+2) (1−2λ
D)
8 13 +
4
d(d+2)ρP
+ ρ (8.44a)
d gh
(4π) 2−1Γ(d/2)
d(1−2λD)−2 {
D(λD; ) = (d−6)(d+1) − 1 (d−4)(d+1) λD +2 (d−6)(d−1)B2 d d−1 6(d+2) 3 (d−2) ( ρd−2)(d+2) P(4π) 2 Γ(d/2+1)
( )
4(d− (d−6)d1) ( +4) −2(d−4)λD }− d− − ρP (8.44b)(d 2)(d+2) (1 2λD)
If we are not interested in distinguishing the paramagnetic from diamagnetic terms, and ghost
from metric contributions, we may set ρgh = 1 = ρP. This yields the following simplified ex-
pressions:
(1−2λD)−3 { (d−7)(d−6)(d−2) 4(d3−11d2+18d−6)
BD1(λ
D;d) = − + λD
d
3(4π) 2−1Γ(d/2) (d+2) (d−2)( )
1 4
− 4d(d−4)(d+1)
} 8
2 3
+
d(d+2)
λD− + (8.45a)(d 2) d(4π) 2−1Γ(d/2)
d(1−2λD)−3 {(d−6)(d3−9d2+54d−56)
BD D2(λ ;d) = (8.45b)d
3 2(4π) 2−1Γ(d/2+1) 2(d −4)(d+4)
2(d3−10d2− +15d−10) 2(d−4)(d+1)
}
λD+ λD2
(d2−4) (d−2)
The beta-function of λD
The beta-function of eq. (8.37) for the cosmological constant, in the optimized scheme, looks
as follows:
4dρ gD
βD(gD,λD;d) = (ηD ghλ −2)λD+ (8.46)d
(4π) 2−1Γ(d/2+2)
2(d+1)gD { ( D ) ( )}D
+ 2(d−4)λD 1− η − (d−6)d 1− η
d
(4π) 2−1 − 2 (d+2) (d+2) (d+4)dΓ(d/2)(1 2λD)
8.4 Beta-functions for dimensionless couplings 265
The reduction to ρgh = 1 = ρP is omitted here, since this expression contains no factor of ρP,
and ρgh occurs only trivially in the second term on the RHS. So no further simplification can
be achieved.
The anomalous dimension related to gB
Eq. (8.38) for the choice (8.43) results in
ηB(gD,λD,gB;d) = (8.47)
(1−2λD)−1 { (d+1) − 2
D
= ( ηD)− (d−4)(d+1) (d−η )d λD
d−1 3 3 (d−2)4 D( π) 2 Γ(d/2) (1−2λ )
( D
+ (d−6)(d+1) −16 (d−1)
) (1−η /(d+2))
3 (d−2)ρP (1−2λD)
D
(d−4)(d−1) (1−η /(d+2))+16 ( −2) ρP λ
D
d (1−2λD)2
− (1−η
D/(d+4))}
8 (d−6)(d−1)d B
(d2− ρ4) P (1− g2λD)2
{ }
− 4 (d+2) 4(d+4)ρ + ρ gB
d− gh P(4π) 2 1Γ(d/2) 3 d(d+2)
This lengthy expression reduces slightly when we switch off the separation given by the ρ-
parameters. For ρgh = 1= ρP we obtain
ηB(gD,λD,gB;d) = (8.48)
(1−2λD)−1 { (d+1) − − 2 (d−4)(d+1) (d−η
D)
= (d ηD) D
d−1 3 3 (d−2)
λ
4 D( π) 2 Γ(d/2) (1−2λ )
( D
+ (d−6)(d+1)(d−2)−16(d−1)
) (1−η /(d+2))
3(d−2) (1−2λD)
(d− D4)(d−1) (1−η /(d+2)) D }+16 λD −8 (d−6)(d−1)d (1−η /(d+4)) B(d−2) (1−2λD)2 (d2− g4) (1−2λD)2( )
− 4 d
3+4d2+16d+48
gB
d
(4π) 2−1Γ(d/2) 3d(d+2)
The last term of (8.48) contains the ghost contributions.
The beta-function of λB
For the cosmological constant in the B-sector we only write down the result prior to setting
ρgh = 1:
8(d+4)ρ
β B
gh
λ (g
D,λD,gB,λ B;d) = (ηB−2)λ B− gB (8.49)
d
(4π) 2−1(d+2)Γ(d/2)
(d+1)(1−2λD)−2{ ( (d−1) )+ 2d (d−2)− ηD
d (d+2) (d+4)
(4π) 2−1Γ(d/2+1)
}
−4(d−2)(1−ηD/(d+2))λD gB
This completes the list of the bulk beta-functions in the {D,B}-description. As above, the
boundary expressions are described in the level-language to which we turn next.
266 Chapter 8. The beta-functions
The anomalous dimension related to g(0)
For the above made choice of the shape function the anomalous dimension for g(0) assumes the
form
2(1−2λD)−1 {(d+1) (d−1) ((d+2)−ηD }
η ( )
)
0 =+ (d−ηD)−2 ρ (0)
d− − D P g(4π) 2 1Γ(d/2) 6 (d+2) (1 2λ )
{ }
− 4 d 4ρ + ρ g(0) (8.50)
d
(4π) 2−
gh P
1Γ(d/2) 3 d
Setting ρgh = 1= ρP this further simplifies to
{
( ) 4 (d+1) (d−ηD) − (d−1) ((d+2)−η
D) (d2+12)}
η 0 = − g(0)
d
4 −1 2 12 (1−2λD) (d+2) (1−2λD 2( π) 2 Γ(d/ ) ) 3d
(8.51)
which can also be derived using the identity η (0)/g(0) = ηB/gB +ηD/gD directly.
The beta-function of λ (0)
For the scale-dependence of the cosmological constant λ (0) we have to consult eq. (8.40) which
yields
( )
d
( ) − (d+1) (d+2)−η
D
0 (0) (0) ( ) 2 − ( ) 4dρ0 0 ghβλ = (η 2)λ +g − gd−1 (1 2λD) d(4π) 2 Γ(d/2+2) (4π) 2−1Γ(d/2+1)
(8.52)
Omitting the separation given by the ρ-parameters we find
( )
d (d−3+8λD)(d+2)− (d+1)ηD
β (0) = (η (0)−2)λ (0)λ +g
(0) 2 (8.53)
d
(4π) 2−1Γ(d/2+2)(1−2λD)
The higher level beta-functions are described by the D couplings given in eqs. (8.44a),
(8.44b) and (8.46).
The anomalous dimension related to g∂ (0)
Substituting the explicit expressions for the threshold functions in case of the optimized scheme,
eq. (8.41) assumes the following form:
g∂ (0) { (d−ηD }
η ∂ (
)
0) = (d+1) − −4dρ (8.54)d3 4 −1 2 (1 2λD
gh
( π) 2 Γ(d/ ) )
This anomalous dimension defines the non-canonical scaling behavior of the boundary Newton-
type coupling multiplying the Gibbons-Hawking-York functional.
The beta-function of λ ∂ (0)
Finally, we consider eq. (8.42) and insert the optimized shape functions. The differential equa-
tion for λ ∂ (0)k is then given by
∂ (0) { D }
β ∂ (0) ∂ (0)λ = (η −
g d (d+1−η )
1)λ ∂ (0) + − +4ρgh (8.55)(d 3)
4(4π) 2 Γ(d+1 ) (1−2λD)2
Identifying ρgh = 1 does not lead to any significant reduction of the formula, so we omit it here.
8.5 The semiclassical approximation 267
8.5 The semiclassical approximation
In this section we present an explicit solution to the RG equations which is valid approximately
provided gIk ≪ 1 and λDk ≪ 1. (The magnitude of λ Bk plays no role for the validity.) We refer to
it as the semiclassical approximation since in an (k/mPlanck)-expansion it describes the leading
deviations from the strictly classical behavior with exactly constant dimensionful couplings
GI ≡GIk 0 and λ̄ I ≡ λ̄ Ik 0 , respectively.
Newton constants
The k-dependence of the Newton constants is covered by the corresponding anomalous dimen-
sion: k∂ IkGk = η
IGI I Ik for I ∈ {D, B, (p), ∂ (p)}. If gk ≪ 1 we may expand in gk and retain
the linear term only. Returning to dimensionful variables all anomalous dimensions have the
following form then:
η I = BI (λ̄D/k21 k )G
I
k (8.56)
The solution to the ensuing RG equation with initial conditions posed at k = 0 is given by
∫
1 1 k
= − dk′BI1(λ̄D/k′2)k′(d−1)k (8.57)
GI Ik G0 0
The next step in the approximation consists in expanding the integrand of (8.57) in small values
of λD ≡ λ̄Dk k /k2. In the lowest order we have BI (λ̄D1 k /k2) = BI (0) +O(λ̄D1 k /k2) for which the
integration can be performed easily. Thus to leading order for gIk ≪ 1 and λDk ≪ 1:
I G
I [ ]
G = 0k − = G
I I I d−2
0 1−ωdG0 k (8.58)1+ω I I d 2dG0 k
Here we have defined the coefficients ω Id ≡ −BI1(0)/(d − 2) which are analogous to those in
[122] and its later generalizations.
For the D Newton coupling the coefficient ωDd follows from (8.34):
( )
1 { }
D (d
4−7d3+4d2+12d)−8d(d−2)ρ
ω = gh Φ2 − 2d (d−6)(d−1)d+8ρgh 3d (0) ρ Φ (0) ,d−1 6(d−2)2 (d/2) (d−2)2 P (d+2)/2(4π) 2
1 { }3 2 2
ωD = (d +7d −74d−48) d(2d −10d−8)d 6( −2) Φ
2 (0)− Φ3
d−1 d (d/2) (d−2) (d+2)/2
(0) (8.59)
(4π) 2
The first equation contains the separation between contributions of different origin, the second
relaxes this division and sets ρgh = 1= ρP. For the B Newton constant, we find from eq. (8.37),
with and without the ρ-factors, respectively:
1 {
B − (d(d+1)−4dρgh) 1 2d((d−6)(d−1)d+8ρgh)ρω = Φ (0)+ P Φ3d 3( −2) (d−2)/2 − (0)d−1 d (d 2)2 (d+2)/2(4π) 2
− (d−
}
2)d((d−5)d−6−8ρgh)−48((d−1)+(d−4)ρgh)ρP Φ2 (0) (8.60)
6(d−2)2 (d/2)
1 { d(d−3) 1 (d3−7d2ωB = − +74d+48) 2d d−1 3(d−2) Φ(d−2)/2 (0)− 6(d−2) Φ(d/2) (0)(4π) 2
}
+ d(2d
2−10d−8) 3
(d−2) Φ(d+2)/2 (0)
268 Chapter 8. The beta-functions
In the level description the ω-coefficient is determined by eq. (8.39):
{ }
ω (0) = 2 − d(d+1−4ρgh) 1 2(d(d−1)+4ρ )d d − − Φ(d−2)/2 (0)+
gh 2
1 3(d 2) (d−2)
ρP Φ(d/2) (0)
(4π) 2
{ }
ω (0) 2 − d(d−3)= Φ1 (0)+ (d2−d+4) 2d d − − (d−2)/2 ( Φ (0) (8.61)1 6(d 2) d−2) (d/2)(4π) 2
In the last equation we set again ρgh = 1= ρP to simplify the result.
Finally, the ω-coefficient for the boundary Newton-type coupling can be read off from eq.
(8.41):
∂ (0) − d
{(d+1) }
ωd =
1
− − Φd/2−1 (0)−
4ρgh
Φ1− d/2−1 (0) (8.62)3(4π)d/2 1 (d 2) (d 2)
Omitting the separation of ghost and gravitational contributions by setting ρgh = 1, eq. (8.62)
simplifies to
d(d−3) { }
ω ∂ (0)d =− 1 13(4π)d/2− ( −2) Φ1 d d/2−1 (0) (8.63)
As already mentioned ω ∂ (0)d vanishes in d = 3 and is negative (positive) for d > 3 (d < 3).
Cosmological constants
The RG equations for the bulk cosmological constants λ Ik occurring in the present truncation
have the structure
( )
∂ λ I = η I It k −2 λk +AI(λD D Ik ,gk )gk for all I 6= ∂ (0) (8.64)
and for the boundary counterpart
( )
∂ λ ∂ (0) = η ∂ (0) −1 λ ∂ (0) +A∂ (0)(λDt k k k ,
∂ (0)
gDk )gk (8.65)
∣
whereby AI(λDk ,g
D
k ) ≡ AI1(λ I 1 D I I I I D ∣k )− 2η A2(λk ). Here we define A1 ≡ A (λk ) to be the con-ηD=0
tribution that remains after omitting ηD on the RHS of the flow equations. The dimensionful
analog of eq. (8.64) and (8.65) can be solved formally to yield
I I ∫λ̄ λ̄ k ∂ (0) ∫k 0 ′ I λ̄ λ̄
∂ (0) k
= + dk A (λD Dk′ ,gk′)k
′d−1 or k = 0 + dk′A∂ (0)(λD,gD )k′d−2
GI GI ∂ ( ) ∂ ( )
k′0 0 k′
k 0 0 Gk G0 0
(8.66)
For λDk ≪ 1 we expand AI D 2 I1(λ̄k /k ) = A1(0)+O(λ̄D 2k /k ) and solve the integral to leading order.
Furthermore, we also insert the approximate result for Newtons coupling (8.58) into eq. (8.66).
We finally obtain for the semiclassical approximation on the bulk
GI ( )
λ̄ I = k λ̄ Ik 0 +ν
I I
dG0 k
d +O(λ̄D0 /k
2) = λ̄ I +ν I GI kd0 d 0 +O(λ̄
D 2 I 2
0 /k , G0 k
d−2) (8.67)
GI0
whereby we defined the coefficients ν Id ≡ AI1(0)/d, again following the conventions used in
earlier publications [189, 190]. Likewise on the boundary we have the following semiclassical
expansion
∂ (0)( )
λ̄ ∂ (0)
G
= k λ̄ ∂ (0) +ν ∂ (0) ∂ (0) d−1 D 2k ∂ (0) 0 d G0 k +O(λ̄0 /k )
G0
= λ̄ ∂ (0) +ν ∂ (0) ∂ (0)G kd−10 d 0 +O(λ̄
D
0 /k
2, GI 20 k
d−2) (8.68)
8.6 Beta-functions in d = 4 269
this time however with ν ∂ (0) ≡ ∂ (0)d A1 (0)/(d−1)
For the D cosmological constant ν Id can be deduced from eq. (8.36):
1 { }
νD =− (d−6)d 2 (d+1)−4ρ Φ
2
d− gh1 (d+2)/2
(0) ,
(4π) 2
−(d−7)(d+2)νD = Φ2d d−1 (d+2)/2 (0) (8.69)2(4π) 2
For the B cosmological coupling we find from eq. (8.38)
1 {( ) ( ) }
νBd = (d+1)−4ρ 1gh Φ(d/2) (0)−
(d−6) 2
d− 2
(d+1)−4ρgh Φ
1 (d+2)/2
(0) ,
(4π) 2
1 { }
νBd = ( −3) 1 (0)− (d−7)(d+2)d Φ Φ2d−1 (d/2) 2 (d+2)/2 (0) (8.70)(4π) 2
and in the level-description we obtain from eq. (8.40):
{ }
ν (0)
1 1
d = (d+1)−4ρ Φd− gh1 (d/2) (0) ,(4π) 2
ν (0)
(d−3)
d = Φ
1
d−1 (d/2)
(0) (8.71)
(4π) 2
As always, the first equation of the above pairs retains arbitrary ρ’s, while the second is the true
final result with ρgh = 1= ρP.
Finally, we use eq. (8.68) and the result obtained in eq. (8.42) to derive the semiclassical
expansion for λ̄ ∂ (0)k :
{
∂ (0) − d (d+1)−4ρ
}
gh
ν 1d = Φ(d−3) (d−1) (d−1)/2 (0) ,4(4π) 2
d(d−3)
ν ∂ (0)d =− − Φ1(d 3) − (d/2)
(0) (8.72)
4(4π) 2 (d 1)
Notice that ν ∂ (0)d has the opposite sign w.r.t. its bulk counterpart in eq. (8.71).
8.6 Beta-functions in d = 4
In this section we present the beta-functions for the dimensionless couplings derived in the last
section for the particular interesting case of four spacetime dimensions, i.e. d = 4.
8.6.1 The D sector: beta-functions for gD, λ D
The RG equation for the dynamical Newton constant gDk is governed by the anomalous dimen-
sion ηD. The FRGE provides the following relation for it:
1 { [ ]}
ηD(gD, λD) = + 5 q2 (−2λD,ηD)−6ρ 4q3 (−2λD3 2 P 3 ,η
D)−q22 (−2λD,ηD) gDπ
2 {( ) }
+ ρ 2 −ρ Φ2gh 3 P 2 (0)+4ρPΦ
3
3 (0) g
D (8.73)
π
It has a global factor of gDk and depends on λ
D
k through the threshold functions. In addition,
p
q Dn (w,η ) contains another term proportional to ηD. As a consequence, eq. (8.73) is an implicit
equation for the anomalous dimension. It can be solved to yield the final result:
BD D1(λ ;4)g
D
ηD(gD, λD) = − (8.74)1 BD2(λD;4)gD
270 Chapter 8. The beta-functions
We obtain a non-polynomial dependence of ηD on gD, and BD1(λ
D;4) and BD2(λ
D;4) are functions
of the cosmological constant, here specialized for d = 4. The first one contains graviton as well
as ghost contributions,
1 { [ ]}
BD(λD;4) = 5 Φ2 (−2λD)−6ρ 4Φ3 (−2λD)−Φ2 (−2λD1 3 2 P 3 2 )π
2 {( ) }
+ ρ 2gh 3 −ρP Φ
2
2 (0)+4ρ
3
PΦ3 (0) (8.75)π
whereas the second one stems entirely from the graviton sector:
{ [ ]}
BD2(λ
D;4) =− 1 5 Φ̃23 2 (−2λ
D)−6ρP 4Φ̃33 (−2λD)− Φ̃2 D2π 2 (−2λ ) (8.76)
Furthermore the running of the dynamical cosmological constant λDk is described by
1 [ ]
∂ λD = (ηD−2)λDt k k +gD 5q23 (−2λDk ,ηD)+4ρgh Φ23 (0) (8.77)π
Remarks:
The beta-functions for the ‘D’ couplings exhibit certain properties that are reminiscent of the
single-metric truncation [122, 191]: First, notice that the threshold functions Φpn (−2λD) and
Φ̃pn (−2λD) have singularities when the argument approaches −1. This leads to a boundary of
theory space at λD = 1/2 in both types of truncations. Second, in the single- and the bi-metric
truncation we find a further divergence in ηD that can restrict the physically relevant part of
theory space even stronger, namely a boundary caused by the singular curve 1−Bsmgsm2 = 0 or
1−BD2gD = 0, respectively. A certain class of RG trajectories, later referred to as type (IIIa)D
trajectories, terminate on these lines. We find that even though quantitatively the singularity
properties change when moving from the single- to the bi-metric beta-functions, the qualitative
picture remains the same.
8.6.2 The B-sector: beta-functions for BgB, λ
In the B-sector, the essential RG running is inherited from gDk and λ
D
k . The B-couplings them-
selves appear on the RHS of their differential equations only in the trivial canonical term since
Tr[· · · ] did not depend on them. In particular, Newton’s constant gB enters its own anomalous
dimension ηB only as a global factor. Besides that it depends on the dynamical couplings only:
ηB(gD, λD, gB) = BB(gD1 ,λ
D)gB (8.78a)
1 { ( ) }
BB(gD,λD) = 5q1 (−2λD,ηD)− 51 3 1 3 +12ρ
2
P q2 (−2λD,ηD)+24ρ 3 D DP q3 (−2λ ,η )π
− 4
{ }
ρ 1gh 3Φ
1
1 (0)+
1Φ23 2 (0)+2ρPΦ
3
3 (0) (8.78b)π
Notice that eq. (8.78a) has no denominator analogous to its D counterpart (8.74). Note also that
ηB, via the pqn -functions (8.19), indirectly depends on ηD = ηD(gD,λD).
The running of the cosmological constants in the B-sector is described by the differential
equation
∂ λ B = (ηB−2)λ B+AB(λD,gDt k k k k )gBk (8.79a)
Here the RG-effects that are not already covered by ηB are encoded in the function
1{ ( ) ( )}
AB(λD D 1 D D 2 D D 1 2k ,gk )≡ 5 q2 (−2λk ,η )−q3 (−2λk ,η ) −4ρgh Φ2 (0)+ Φ3 (0) (8.79b)π
8.6 Beta-functions in d = 4 271
Remarks:
The beta-functions of the ‘B’ (and also of the level-(0)) couplings are free from any additional
singularities that would further reduce the physical part of theory space. In particular, as it is
apparent from (8.78), the anomalous dimension ηB is well-defined for any value of gB on which
it depends linearly. Notice also that the threshold functions are evaluated at −2λDk , and never at
−2λ Bk , so that there is no restriction in the λ B direction of theory space that would be analogous
to the λD = 1/2 boundary. The same holds true for the level-(0) plane.
We thus have all four beta-functions of the bulk sector at hand and can, at least in principle,
solve the bulk system of differential equations. This will be carried out in part III of this thesis.
Next, let us consider the level-language and the boundary differential equations.
8.6.3 The level-description: beta-functions for g(p), λ (p)
The {B, D}-language is equivalent to the one based upon the level number where the a priori
different higher levels p = 1,2, · · · happens to be identical within the present truncation. The
higher level Newton constants (p)gk have the same running for all p = 1,2,3, · · · , for instance,
and their beta-functions coincide in turn with that of gDk .
D B
(0) g g (p)
gk =
k k and g = gDk k for all p≥ 1 ,gDk +gBk
λD gB+λ B gD(0) (p)
λ k k k kk = and λ = λ
D for all p≥ 1 . (8.80)
gD+gB k kk k
Hence, for the anomalous dimensions, we obtain
η (1) = η (2) = · · ·= ηD (8.81)
with ηD given in equation (8.74). For the level-(0) anom√alous dimension, i.e. the anomalous
dimension corresponding to the running pre-factor of the -term 1/ (0)gR G = 1/GB+1/GDk k k , we
obtain instead:
{ (
η (0)(gD,λD,g(0)
2 5
) = q1 (−2λD,ηD)−3ρ q2 (−2λD,ηD)−ρ 21 P 2 gh 3Φ
1
1 (0)π 6
)}
+ρ 2 (0)P Φ2 (0) g (8.82)
Note that η (0)/g(0) is a function of λD and gDk k only.
For the cosmological constant at level-(0) we find
{ }
∂ λ (0) = (η (0) −2)λ (0) + (0) 1g 5q1 (−2λD Dt k k k 2 k ,η )−4ρ Φ1gh 2 (0) , (8.83)π
while for the higher cosmological constants
λ (1) (2) Dk = λk = · · ·= λk . (8.84)
At all levels p = 1,2,3, · · · their running is locked to that of λDk , which in turn is governed by
equation (8.77).
8.6.4 The boundary sector: beta-functions for ∂ (0), λ ∂ (0)g
Not surprisingly, the structure of the anomalous scaling of the boundary couplings almost agrees
with the one for λ (0)k in eq. (8.83). That is, the anomalous dimension of g
∂ (0) assumes the
following form in d = 4:
{ ( ) }
η ∂ (0)(gD,λD,g∂ (
1
0)) = 5q11 −2λ (1),ηD −4ρ Φ1 ∂ (0)gh3π 1 (0) g (8.85)
272 Chapter 8. The beta-functions
The term in curly brackets is reminiscent of the one in eq. (8.83). Similarly, for the differential
equation of the boundary cosmological constant we obtain in d = 4:
{ }
∂ λ ∂ (0) = (η ∂ (0)−1)λ ∂ (0) 1t − √k k 5q13/2 (−2λk,ηD)−4ρgh Φ1
∂ (0)
2 π 3/2
(0) gk (8.86)
The reason for this structural similarity is that all three invariants obtain RG corrections to the
canonical scaling from the same traces of eq. (7.69). All are of zeroth order in Ω and C and
thus only the first three terms of eq. (7.68) contribute to the beta-functions of the associated
couplings.
8.A Expanded beta-functions in d = 2+ ε
Besides the most relevant spacetime dimension d = 4, it is instructive to consider the case
of d = 2+ ε . Here, we can study the dynamical effects that add to the trivial topological 2-
dimensional theory by ‘turning on’ ε . The results we obtain give rise to certain universal terms
that can be ascribed to the universal value Φn+1n (0) = 1/Γ(n+ 1) which is obtained for any
shape function R(0). In the sequel we display the beta-functions obtained by inserting d = 2+ ε
into the general results and expanding in ε , thereby discarding contributions of order ε and
higher. If the limit ε → 0 is finite, the result is a single term of order ε0, if not, there will be
additional pole terms in 1/ε .
The anomalous dimension related to gD:
Instead of giving the full expression of the anomalous dimension ηD, it is more useful for further
investigations to present the numerator and denominator functions BD1,2 expanded in ε separately.
We obtain for the numerator in eq. (8.33):
−8
{ ( )}
BD(λD) = 2Φ3 (−2λD1 2 )−2ρP(Φ2 D1 (−2λ )+λD Φ2 (−2λD)−2Φ30 1 (−2λD)ε
− 4
{
−2ρgh(1+3ρP)−λD(−2+ log(64π3))Φ20 (−2λD) (8.87)3 ( )
+3 −1+ρP(−5+ log(16π2)) Φ2 D[ 1
(−2λ )
+3 4λDρP(−2+ log(4π))Φ31 (−2λD)+λDΦ2 (0,1,0) D0 (−2λ )
(
−2ρP (−7+ log(16π2))Φ3 (−2λD)−ρ Φ2 (0,1,0)2 gh 1 (0)
+2λDΦ3 (0,1,0) (− D)+2ρ Φ3 (0,1,0)1 2λ gh 2 (]0)
2 ) }+Φ (0,1,0)1 (−2λD)−2Φ
3 (0,1,0)
2 (−2λ
D) +O(ε)
We see that BD1 gives rise to a Laurent series with a singular part ∝ 1/ε . The function that
appears in the denominator of (8.33) assumes the form
4{ ( )}
BD2(λ
D) = 2Φ32 (−2λD)−2ρ (Φ2P 1 (−2λD)+λD Φ20 (−2λD)−2Φ3 (−2λD)ε 1
2{
+ −λD(−2+ log(64π3))Φ20 (−2λD) (8.88)3 ( )
+3 −1+ρP(−5+ log(16π2)) Φ21 (−2λD)[
+3 4λDρP(−2+ log(4π))Φ3 (−2λD)+λDΦ2 (0,1,0) (−2λD1 0 )
(
−2ρP (−7+ log(16π2))Φ3 D2 (−2λ )+2λDΦ3 (0,1,0) (−2λD]}1
)
+Φ2
)
(0,1,0)
1 (−
D)−2Φ3 (0,1,0)2λ 2 (−2λD) +O(ε)
8.A Expanded beta-functions in d = 2+ ε 273
BD(λD;2+ε)gD
Obviously BD2 , too, is again divergent in ε . However, due to the relation η
D = 1 ,
1−BD2 (λD;2+ε)gD
the anomalous dimension is finite in the limit ε → 0:
ηD(gD,λD) (8.89)
( )
ρ Φ2 (−2λD)−2Φ3 (−2λD)+2λDΦ3 (−2λD) −2λDΦ2P 1 2 1 0 (−2λD)= 2 ( ) +O(ε)
ρ Φ̃2 (−2λD)−2Φ̃3 (−2λD)+2λDΦ̃3 D D 2 DP 1 2 1 (−2λ ) −2λ Φ̃0 (−2λ )
The beta-function of λD:
The running of the D cosmological constant expanded in terms of ε is given by
βDλ (g
D,λD) = (ηD−2)λD+8ρgh Φ2 D2 (0) g (8.90)( )
+3 −4λD q21 (−2λD,ηD)+4q22 (−2λD,ηD) gD +O(ε)
Notice that the anomalous dimension in the beta-function has to be understood as the expanded
version of eq. (8.89).
The anomalous dimension related to gB:
ηB(gD,λD,gB) =
8{ ( )}
= 4q3 D D2 (−2λ ,η )−2ρ Φ2 (−2λD D 2P )+λ q (−2λD,ηD)−4q3 (−2λD,ηD) gBε 1 0 1
− 1
{
4λD(−2+ log(64π3))q2 (−2λD0 ,ηD)−6q1 D D3 0 (−2λ ,η )( )
+12 1−2ρP(−3+ log(4π)) q21 (−2λD,ηD) (8.91)[
−12 4λDρ (−2+ log(4π)) 3 (−2λD,ηD)+λDΦ2 (0,1,0)q (−2λDP 1 0 )
(
−2ρP (−7+ log(16π2))q3 D D2 (−2λ ,η )+2λD 3 (0,1,0)q (−2λD1 ,ηD)]
+ 2
)
(0,1,0)
q (−2λD1 ,ηD)−2
3 (0,1,0)
q D2 (−2λ ,ηD)
( ( )) }
+8 2+ρ 6−3 2 (0,1,0)P q1 (0,ηD)+6
3 (0,1,0)
q D2 (0,η ) ρgh +O(ε)
This series contains a divergent part ∝ 1/ε for the anomalous dimension ηB itself. It happens
to vanish however for λD = 0.
The beta-function of λB:
( )
β B(gD,λD,gB,λ B) = (ηB−2)λ B−8ρ Φ1λ gh 1 (0)+ Φ22 (0) gB (8.92)( )
+3 2q11 (−2λD,ηD)+4λD q2 D D1 (−2λ ,η )−4q22 (−2λD,ηD) gB +O(ε)
Notice that while β Bλ contains no explicit 1/ε-poles the anomalous dimension η
B to be used in
(8.92) does have such poles.
The anomalous dimension related to g(0):
η (0)(g(p),λ (p)) (8.93)
( ) ( )
= 2 q10 (−2λD,ηD)−2ρ q2 (−2λD,ηD) g(0)P 1 −4ρ 2gh 3 +2ρP g
(0) +O(ε)
274 Chapter 8. The beta-functions
The beta-function of λ (0):
β (0)λ (g
(p),λ (p)) = (η (0) −2)λ (0) +6q11 (−2λD,ηD)g(0) −8ρ Φ1gh 1 (0)g(0) +O(ε) (8.94)
Note that in the level-description neither η (0) nor β (0)λ has explicit 1/ε-poles.
The anomalous dimension related to g∂ (0):
For the boundary Newton-type coupling the expansion in d = 2+ ε yields
{ ( ) }
η ∂ (0)(gD,λD,g∂ (0)) = g∂ (0) 2q1 −2λ (1),ηD − 8 ρ Φ10 3 gh 0 (0) +O(ε)
{
∂ ( ) (2−ηD) }= g 0 8− − 3 ρgh (8.95)(1 2λ (1))
In the second step we used the property pΦ0 (w) =
p
Φ̃0 (w)≡ (1+w)−p. Since ηD is also free of
1/ε-poles, the anomalous dimension η ∂ (0) is well defined in the limit ε → 0+.
The beta-function of λ ∂ (0):
Finally, the beta-function for the cosmological constant λ ∂ (0) is also found to be free of 1/ε-
poles and its ε → 0+-limit is given by
√ { }
β ∂ (0) = (η ∂ (0) −1)λ ∂ (0) − π 3q1 (−2λ ,ηD)−4ρ Φ1 (0) g∂ (0)λ 1/2 gh 1/2 +O(ε) (8.96)
8.B Beta-functions in d = 3
Next, we apply the results of section 8.4 to the special case of d = 3. We proceed in the usual
manner, starting with the D couplings, then consider the B-sector, and finally take a look at the
boundary couplings employing the language description of the beta-functions.
The anomalous dimension related to gD:
The structure of the anomalous dimension follows eq. (8.33), with the numerator function
3 {
BD(λD1 ;3) = √ (2+12ρP)Φ23/2 (−2λD)+16ρ D 3 DP λ Φ3/2 (−2λ )2 π }
− 8 λD Φ2 (−2λD3 1/2 )−36ρ Φ
3
P 5/2 (−2λD)
√2
{( ) }
+ ρ 2gh 1−4ρP Φ3/2 (0)+12ρ Φ3Pπ 5/2 (0) (8.97)
We see that BD1 receives contributions of both gravitons and ghosts, of both dia- and para-
magnetic nature. The denominator function BD2 is unaffected by the ghosts:
√3
{
BD(λD;3) = (2+12ρ 2 D D 3 D2 P)Φ̃3/2 (−2λ )+16ρP λ Φ̃3/2 (−2λ )4 π }
− 8 λD Φ̃2 (−2λD)−36ρ Φ̃3 D3 1/2 P 5/2 (−2λ ) (8.98)
The beta-function of λD:
For the cosmological constant λD the beta-function reduces into the following form for d = 3:
6
βDλ (g
D,λD;3) = (ηD−2)λD +gD√ ρ 2gh Φ5/2 (0) (8.99)π
√1
{ }
+gD −4λD q23/2 (−2λD,ηD)+9q2 D Dπ 5/2 (−2λ ,η )
8.B Beta-functions in d = 3 275
The anomalous dimension related to gB:
Next, consider the anomalous dimension for the B-Newton coupling.
3 {4 ( )
ηB(gD,λD,gB;3) = √ q1 D1/2 (−2λ ,ηD)+ 83 λ
D q21/2 (−2λD,ηD)− 2+16ρ 2 D DP q3/2 (−2λ ,η )2 π 3
}
−16ρ λD q3 D D 3 D D BP 3/2 (−2λ ,η )+36ρP q5/2 (−2λ ,η ) g
2 { ( ) }− √ ρ Φ1 2gh 1/2 (0)+ 1−2ρP Φ3/2 (0)+12ρPΦ35/2 (0) gBπ
(8.100)
The beta-function of λB:
Finally, we consider the beta-function of λ B which represents the last ingredient in the hierarchy
of the bulk couplings. In d = 3 it is given by
6 { }
β B(gD,λD,gBλ ,λ
B;3) = (ηB−2)λ B− √ ρgh Φ1 2 B3/2 (0)+ Φ5/2 (0) g (8.101)π
√1
{ }
+ 6q1 (−2λD,ηD)+4λD q2 D D3/2 3/2 (−2λ ,η )−9q2 (−2λD,ηD) gBπ 5/2
The anomalous dimension related to g(0):
Instead of usingD/B it is sometimes more convenient to employ the level-language. The anoma-
lous dimension associated to g(0) is given by
( ) (p) (p) √3
{ }
η 0 (g ,λ ;3) = + 2q1 (−2λD,ηD)−2ρ q2 (−2λD,ηD3 1/2 P 3/2 ) g
(0)
π
− √2
{ }
ρ Φ1 2 (0)gh 1/2 (0)+2ρP Φ3/2 (0) g (8.102)π
The beta-function of λ (0):
Combining the D and B results, one can deduce the beta-function of λ (0), which reads in d = 3:
β (0)(g(p),λ (p);3) = (η (0)−2)λ (0) +g(0) √6 q13/2 (−2λD,ηD)−g(0)√
6 1
λ ρghΦ3/2 (0)π π
(8.103)
In the level-description, the RG equations for the higher couplings (p ≥ 1) involve η (p) = ηD
(p)
and β Dλ = βλ .
The anomalous dimension related to g∂ (0):
The boundary Newton-type coupling has the following anomalous dimension in d = 3:
∂ (0) { ( ) }
η ∂ (0)(gD,λD,g∂ (0)
g
) = √ 4q1 −2λ (1)1/2 ,ηD −4ρ Φ1gh 1/2 (0) (8.104)2 π
The beta-function of λ ∂ (0):
Finally, in d = 3, the beta-function for the cosmological constant describing the RG evolution
of the boundary volume element is given by
β ∂ (0) ∂ (0)λ = (η −1)λ
∂ (0)
3 { }− 4q1 D1 (−2λ ,η )−4ρ Φ1 (0) g∂ (0)gh 1 (8.105)4

Results
III
9 Investigating Asymptotic Safety . . . . . . . . . . . . . . . 281
9.1 Structure
9.2 Fixed points
9.3 Predictivity
9.4 Conclusion
9.A Critical exponents of the background-sector
10 Background Independence . . . . . . . . . . . . . . . . . . . 305
10.1 Introduction
10.2 Can split-symmetry coexists with Asymptotic Safety?
10.3 Single-metric vs. bi-metric truncation: a confrontation
10.4 An application: the running spectral dimension
10.5 A brief look at d = 2+ ε and d = 3
10.6 Summary and conclusion
11 A generalized C-theorem . . . . . . . . . . . . . . . . . . . . . 353
11.1 Introduction
11.2 The effective average action as a ‘C-function’
11.3 Asymptotically safe quantum gravity
11.4 Discussion and outlook
11.A The special status of Faddeev-Popov ghosts
11.B The trajectory types Ia and IIa
12 Propagating modes in Quantum Gravity . . . . . . . 397
12.1 Introduction
12.2 Anomalous dimension in single- and bi-metric truncations
12.3 Interpretation and Applications
12.4 Summary
13 The Gibbons-Hawking-York boundary term . . . . 421
13.1 Introduction
13.2 Global properties of the boundary sector
13.3 Variational problems
13.4 Counting field modes
13.5 Conclusion
14 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
14.1 Summary
14.2 Outlook
One of the central questions we have to answer in this final part of the thesis, is whether or not
the employed bi-metric-bulk – pure-background-boundary truncation studied in part II of this
thesis supports the Asymptotic Safety conjecture for Quantum Gravity (QG). In the Effective
Average Action (EAA) approach this corresponds to the existence of a non-trivial fixed point
of the Renormalization Group (RG) flow with a finite dimensional ultraviolet (UV)-critical hy-
persurface, dimSUV < ∞, rendering the underlying theory renormalizable in a non-perturbative
sense. Today, there is strong evidence that theories of QG are indeed asymptotically safe [99,
100, 121, 123, 126, 130–135, 138, 139, 141, 142, 146–171]. There are also first observations
that describe a reduction of infinite dimensional theory spaces to a finite set of relevant basis
invariants [129]. While both aspects, a non-trivial fixed point and dimSUV < ∞, are necessary
conditions in order to obtain a fully predictive fundamental theory, most of the employed trun-
cations are for technical reasons finite dimensional and thus extensions including an infinite
number of basis invariants is very interesting to confirm the Asymptotic Safety conjecture. In-
deed, recent studies of f (R)-theories provide first promising results that find dimSUV < ∞ even
for infinite-dimensional truncations.
Nevertheless, a suitable theory of Quantum Gravity has to fulfill an even more fundamental
requirement than renormalizability, i.e. Background Independence (BI). Since the geometry of
spacetime is the dynamical object, there is a priori no background over which one can construct
a Quantum Field theory (QFT). In the EAA approach one introduces an arbitrary background
metric on intermediate steps in order to apply the standard techniques for the path integral while
preserving the Ward Identities for split-symmetry (WISS) at least on-shell. The emergence
from a suitable non-trivial fixed point in the UV is then just one criterion that a suitable theory
of QG described by an RG trajectory of the corresponding Functional Renormalization Group
Equation (FRGE) has to satisfy. In addition, in order to make the underlying theory manifestly
background independent, one has to ensure that the same RG trajectory enters a regime of intact
split-symmetry in the infrared (IR). One of our main objectives in this thesis, is to investigate
the status of this global property by searching for RG solutions that are simultaneously com-
patible with the requirements of Background Independence and Asymptotic Safety. Thus, our
truncation ansatz proceeds into a different direction then adding more and more basis invariants
to existing calculations. We explore an up to now almost ‘dark’ region of the theory space by
throw some light on the bi-metric nature of T. While previous works almost exclusively relied
on a severe assumption of fully intact split-symmetry in the gravitational sector along the entire
RG evolution by invoking the so-called single-metric approximation, only recently [172] a first
fully fledged bi-metric calculation was performed. Extrapolating the split-symmetry require-
ment from the limit k→ 0 up to the UV at k→ ∞ is a very strong assumption, since the gauge
fixing functional and the cutoff action explicitly break split-symmetry. The technical difficulties
which arise in studying this part of theory space prevented further progress in this direction and
thus most of our today’s understanding is based on the single-metric results only.
In this thesis, we focus on exactly this almost undiscovered region of theory space by em-
ploying a new technique that significantly reduces the effort in evaluating bi-metric truncations.
Furthermore, we allow spacetime to have a non-vanishing boundary, ∂M 6= 0/ , and thus intro-
duce another very important generalization to previous truncations. These geometries are in
particular for cosmological applications very interesting.
This part of the thesis is structured as follows. In chapter 9 we study the UV properties
of the RG flow by searching for suitable non-trivial fixed point in order to support the Asymp-
totic Safety conjecture also in case of ∂M 6= 0/ and for bi-metric calculations. Then, in chapter
10 we consider the IR regime of RG trajectories in order to understand whether or not Back-
ground Independence and Asymptotic Safety can coexist. Furthermore, we compare the results
279
derived in part II with the bi-metric calculation done in the ref. [172] which is based on a
TT-decomposition. Both are used to study the reliability of their single-metric counterpart and
define regions in theory space where the latter seems to be a good approximation to the general
bi-metric RG flow.
After the UV and IR analysis, we discuss global aspects of the RG evolution, which is
thus based on global solutions interconnecting the UV with the IR. In chapter 11 we pro-
pose a C-function like feature of exact solutions that reflects the ‘mode-counting’ property of
Zamolodchikov’s C-function. We study its monotonicity for the two bi-metric truncations and
the single-metric calculation and demonstrate its value in judging the reliability of truncations.
Chapter 12 confronts the propagation of gravitons with the global results. Basically we distin-
guish three branches of the RG evolution, the UV, the semiclassical regime, and the IR. We
present a possible interpretation of the observed results in terms of ‘dark matter’.
Except for chapter 9, the entire discussion so far is based on the beta-functions of the
bulk-sector only. This is for the reason that the considered truncation has little to say about
split-symmetry violation on the boundary. Thus, in chapter 13 we present some results of the
boundary sector and apply their RG evolution to study black hole thermodynamics.
We conclude with a short summary of this part and present an outlook on future work along
the direction presented in this thesis.
C This part of the thesis relies mostly on already published work. Large parts of the fol-
lowing chapters follow closely ref. [166, 173, 192–194]. We reference the corresponding
publication at the beginning of each chapter.

9.1 Structure
General remarks
The hierarchical structure of the truncated
RG equations
No anti-screening in the semiclassical regime
9.2 Fixed points
The dynamical sector
The level-(0) or B-sector
The boundary level-∂ (0)-sector
Recovering the mechanism of paramag-
netic dominance
Impact of the ghost contributions
Summary
9.3 Predictivity
The classification of trajectories in the lin-
earized region
Critical exponents and scaling fields
9.4 Conclusion
9.A Critical exponents of the background-
sector
9. INVESTIGATING ASYMPTOTIC SAFETY
C This chapter is is based on [173]. It includes new results for the boundary couplings not
presented in [173].
In this chapter we study the Asymptotic Safety conjecture for Quantum Gravity employing the
beta-functions of the bi-metric-bulk – pure-background-boundary truncation derived in part II
of this thesis.
The FRGE endows theory space, T, with a vector field whose integral curves describe the
RG evolution of theories. Each point in theory space, corresponding to an action functional
respecting the symmetry constraints, thus generates a trajectory on T and thus interconnects
a UV theory with its IR limit. From the general perspective all solutions of the FRGE are
equally valid, even though they exhibit different properties. The Asymptotic Safety program
sets two conditions for an RG trajectory to describe a well-behaved QFT. First, the underlying
RG flow should emanate from a fixed point, thus its RG evolution should converge to a certain
‘stable’ theory which is regular in a sense and gives rise to a fundamental ‘bare’ prescription of
the generating functional. Second, in order to have a full predictive formalism, the number of
free parameters fixed by experiments should be finite dimensional. This translates into a finite
dimensional UV-critical hypersurface, SUV, which is spanned by all relevant basis invariants
associated to the considered fixed point. The family of RG trajectories which are asymptotically
safe are parametrized by dim(SUV) < ∞ free parameters and can be though of as emanating
from the corresponding fixed point in the UV and then evolve within the hypersurface SUV.
The vicinity of critical points of the flow equation can be described by linearizing SUV. The
obtained structure defines the universality class of the underlying RG trajectories and is associ-
ated to the conformal (or in general scale-invariant) theory related to the fixed point functional.
The characteristics of the universality class are best described by the critical exponents. Thus, it
might be that conceptually very different theories reflecting possibly unrelated physical models
exhibit the same critical behavior. While their description converge towards the fixed point their
global evolution differs in general.
We start this chapter with section 9.1 making some general remarks on properties of the
282 Chapter 9. Investigating Asymptotic Safety
RG flow for general spacetime dimensions d and for the special case of d = 4. In the follow-
ing we then focus exclusively on the latter case of a four dimensional spacetime and search for
fixed points of the RG flow, see section 9.2. The results are then further discussed by disen-
tangling the contributions from the ghost and graviton sector as well as from paramagnetic and
diamagnetic terms to understand their respective impact.
In section 9.3 we study the second condition for asymptotically safe theories, namely the
UV critical hypersurface of the respective fixed points. With section 9.4 we conclude this
chapter.
9.1 Structure
In this section we discuss some very general properties of the beta-functions derived in part II
of this thesis and listed in chapter 8. Later on, we will study the RG equations only for the
special case of d = 4. However, before we will make various remarks concerning their general
structure which is valid for any d > 2. In particular, we investigate the hierarchy among the six
beta-functions and take a first look at the semiclassical regime.
9.1.1 General remarks
Our truncation ansatz comprises a total of six independent couplings, three of Newton-type and
three associated to the cosmological constant. Their respective impact on the RG evolution
however is quite different. We will consider their general structure and certain topological
properties in what follows.
Structure of the beta-functions
One can summarize the structure of the corresponding beta-functions by means of the anoma-
lous dimensions η I(gIk,λ
D,ηD) and a function AI(λD,ηD). For all Newton-type couplings we
have the following differential equation:
( )
∂tg
I
k = d−2+η I(gI D D Ik,λ ,η ) gk ∀ I ∈ {(p), ∂ (p), B, ∂B, D, ∂D} (9.1)
While this structure is the same in the boundary and bulk sector, for the cosmological type
couplings there is a difference related to their canonical scaling dimension. Thus, we have the
two sets of differential equations:
( )
∂ λ I = η I(gIt k k,λ
D,ηD)−2 λ Ik +AI(λD,ηD)gIk ∀ I ∈ {(p), B, D} (9.2a)( )
∂ λ ∂ (0) = η ∂ (0)( ∂ (0) D Dt k gk ,λ ,η )−1 λ
∂ (0) +A∂ (0)k (λ
D,ηD) ∂ (0)gk (9.2b)
On closer examination, it appears that the dynamical (or level-(p) for p ≥ 1) couplings λD and
ηD play a central role in defining the right-hand-side (RHS) of the differential equations. Except
for this additional dependence on λD and gD, the pair (λ I ,gI) decouples from the remaining
equations for each I. This gives rise to a hierarchy among the couplings, which we discuss in
section 9.1.2. The reason for this ‘imbalance’ between the dynamical sector and the remaining
couplings is due to the structure of the FRGE,
1 [( [ ] ) ]
∂tΓk[ϕ ;Φ̄] = STr Ĥessϕ Γk[ϕ ;Φ̄] +Rk[Φ̄] ∂tRk[Φ̄]
2
Formally expanding the EAA in powers of the fluctuation fields ϕ , i.e. the level expansion
Γk[ϕ ;Φ̄] = ∑
∞ 1 Γ(n)[Φ̄]ϕnn=0 ! whereby the couplings of level-(p) multiply terms of order O(ϕ
p),
n
the flow equation reads
[( ) ]
∞ 1 ∞
∑ 1 ∂ Γ(n)! t [Φ̄]ϕ
n = STr ∑ 1 Γ(m)( −2)! [Φ̄]+Rk[Φ̄] ∂tRn m k[Φ̄]2
n=0 m=2
9.1 Structure 283
Since EAA only appears in its Hessian form on the RHS of the FRGE, all invariants of level-(0)
and level-(1), along with their coefficients, disappear under the trace. Thus, for the truncation
ansatz studied in this thesis, the level-(2) coefficients are trigger the non-trivial flow of the RG
evolution; since here we identified D ≡ (1) (2)gk gk = gk = · · · we can also substitute the level-(1)
on the RHS of the flow equation. In fact, for a truncation ansatz that distinguishes between
the various levels, the beta-functions of level-(p) can only dependent on other couplings of
level-(p+2) and higher.
For the beta-functions of chapter 8 this considerations imply that neither the bulk level-(0)
nor the boundary level-∂ (0) couplings affect the RHS of the flow equation. Their appearance
in the beta-functions of the dimensionless couplings is purely due to the non-linear conversion
rules relating them to the dimensionful coefficients. If we consider the (non-trivial) dependence
of the beta-functions on λD and ηD for general spacetime dimensions d, as given in chapter
8, one observes that, quite remarkably, all terms proportional to λD are also proportional to
(d− 4). Thus, for the special case of d = 4 these contributions drop out, and the remaining
λD-dependence is via the threshold functions, Φp pn (w) and Φ̃n (w), only, see section 8.1 for their
definitions and properties. The threshold functions on the other hand only appear in two com-
binations. Either they are evaluated for zero argument, i.e. pΦn (0), rendering the corresponding
terms independent on λD and ηD, or they combine to expressions of type pqn (−2λD,ηD) which
are defined as
qp
1
n (w,η
D)≡ Φpn (w)− ηD Φ̃pn (w) (9.3)2
These latter contributions are associated to the graviton traces, while the former can be attributed
to the ghost sector. In fact, keeping track of the bookkeeping parameter ρgh reveals this feature.
Note that it is always the anomalous dimension ηD that enters pqn (−2λD,ηD) and there is
no extra dependence of gD on the RHS of the flow equation.
Separating para- and diamagnetic contributions
Similar to the pre-factor introduced for the traces of the ghost sector, ρgh, all formulae contain
also a parameter ρP ∈ {0,1} which we also implemented as a book keeping device. Contri-
butions proportional (not proportional) to ρP originate from the paramagnetic (diamagnetic)
interaction terms1 [165]. We refer to all terms that are not proportional to ρgh as ‘graviton’ con-
tributions. In the nomenclature of ref. [165] those are either of ‘paramagnetic’ or ‘diamagnetic’
origin and correspondingly carry a factor of ρP or are missing a factor of ρP, respectively. In
view of the physical picture of Asymptotic Safety developed in ref. [165] it will be instructive
to keep the two types of contributions separately. Later on, when we are not interested in this
distinction we ‘turn on’ all terms, putting ρP = ρgh = 1, unless stated otherwise.
Topological properties of the truncated RG flow
The general structure of the beta-functions for the six invariants, given in eq. (9.1) and (9.1),
already reveals some topological features of the associated RG evolution. These are described
by critical regions in the truncated theory space, where the beta-functions vanish. Values of the
couplings where the entire set of equations becomes trivial define a critical point or in other
terms a fixed point of the RG flow.
Focusing our attention on the Newton-type couplings their beta-functions vanish if the RHS
of the differential equation (9.1) turns zero, i.e.
( )
d−2+η I(gI !,λD,ηD) gI = 0 (9.4)
1By definition [165, 195], and in accordance with the identical nomenclature in Yang-Mills theory, derivative
(non-derivative) interaction terms coupling metric fluctuations hµν to the background ḡµν are referred to as diamag-
netic (paramagnetic), typical structures being hD̄2h (hR̄h).
284 Chapter 9. Investigating Asymptotic Safety
Assuming that η I is regular at gI = 0, there is a trivial and possibly a set of non-trivial solutions
for this equation, namely
gI = 0 and η I(gI ,λD∗ ∗ ,η
D) = (2−d) (9.5)
Notice that the first, trivial solution is independent on any value of λD or ηD and thus the ‘line’
(gI = 0,λ I ,λD,gD) divides theory space into two portions, gI < 0 and gI > 0, for any I. No RG
trajectory can cross this line and thus an initial positive (negative) Newton coupling will remain
positive (negative) during for the entire RG flow.
R In fact, the anomalous dimension related to g
D is plagued by a singularity at BD(λD;d)gD2 =
1, in case of Einstein-Hilbert truncations. This affects the only type IIIa trajectories in the
limit k→ 0.
On the other hand, solutions of η I(gI∗,λ
D,ηD) = (2− d) will depend on the explicit form
of η I and the point (λD,gD) and thus defines running critical points of gI . They give rise to
a non-Gaußian fixed point (NGFP) and will be derived numerically in section 9.2 and further
discusses in chapter 10.
The k-dependence of the cosmological-type couplings follows equation (9.2) and reads
( )
∂ λ I = η I(gI ,λD,ηD)− cI λ I +AI(λD,ηD)gIt k k M/∂M k k (9.6)
Here, for simplicity we introduced cI I/∂ ≡ 1 for I = ∂ (p) and c /∂ = 2 otherwise. TogetherM M M M
with its the dynamical sector and its associated Newton-type coupling it forms a complete dif-
ferential system. The critical regions in the λ I direction is less general, however in combination
with eq. (9.5) we can spot the joined critical values (λ I∗ ,g
I
∗).
First, we insert the trivial solution gI∗ = 0 which yields the critical values (g
I
∗ = 0,λ
I
∗) with
λ I∗ solving
( )
η I(0,λD,ηD)− cI λ I !M/∂M ∗ = 0 (9.7)
Hence, (gI∗ = 0,λ
I
∗ = 0) is a valid solution for any I independent on the values of λ
D and gD.
If all three pairs I = (0), I = (1), and I = ∂ (0) assume this trivial solution, the trajectory has
entered the Gaußian fixed point (GFP). The second option of η I(0,λD,ηD) ≡ cI /∂ does notM M
constraint the λ I∗ in general, but can be only satisfied for particular values of λ
D and gD. This
set of solutions, is only interesting for I ≡ D.
Second, the non-trivial solutions gI satisfying η I(gI ,λD∗ ∗ ,η
D) = (2− d) lead to an implicit
equation for λ I I∗ and g∗ which also depends on the value of the D-couplings. Their properties
can only be discussed making use of the explicit forms of the beta-functions.
In figure 9.1 we depict the phase diagram of the (λD,gD)-plane for the bi-metric-bulk –
pure-background-boundary truncation. In this plot the separation of RG trajectories at gD = 0
becomes apparent. In the upper half plane there seems to be a fixed point with imaginary
critical exponents, which can be inferred by the spirals in the vicinity of arctan(gD) ≈ 0.5 and
arctan(λD)≈ 0.2.
9.1.2 The hierarchical structure of the truncated RG equations
As we have mentioned above the size of all physical RG-effects is controlled by theD-couplings
only, in fact by couplings of at least level-(2) which, in the present case, are equal to GDk or λ̄
D,
respectively. The level-(0) and level-(1) couplings, and correspondingly the B-couplings, as
9.1 Structure 285
arctan(gD)
1.5
1.0
0.5
0.0 arctan(λD)
-0.5
-1.0
-1.5
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Figure 9.1: This plot shows the phase diagram in the (λD,gD)-plane for the beta-functions of
part II, using arctan(· · · ) to cover the full range gD, λD ∈ {−∞,∞}. Apparently, the solution
gD = 0 separates the RG flow of negative and positive Newton coupling. Notice the signal of at
least two fixed points, one situated at the origin, the other at positive gD and λD.
well as their boundary analogs, cannot enter the RHS of the FRGE, for they are at most linear
in the dynamical fields and hence drop out in the calculation of the Hessian operator.
No matter whether we employ the {B,D,∂ (0)} or the {∂ (0),(0),(1),(2), · · · } language,
the truncated FRGE always comprises a coupled system of ordinary differential equation in six
variables, {gDk ,
∂ (0) ∂ (0)
gBk ,gk ,λ
D
k ,λ
B
k ,λk } or {
∂ (0), (0), (p) ≡ D,λ ∂ (0),λ (0),λ (p)gk gk gk gk k k k ≡ λDk , p≥ 1}, respec-
tively. In either case the dynamical couplings gDk and λ
D
k influence the beta-functions of the level-
∂ (0) ∂ (0)
(0)- (or B-) and the boundary-couplings, but there is no back-reaction of {gB Bk , λk , gk ,λk }
(0) (0) ∂ (0) ∂ (0) B/(0)/∂ (0)
or {gk , λk , gk ,λk } on the ‘D’ variables; furthermore λk does not back-react on
B/(0)/∂ (0)
gk . The system of equations decomposes in the following hierarchical way:
{ D,λD} −→ B/(0)/∂ (0) −→ λ B/(0)/∂ (0)gk k gk k . (9.8)
Notice that there is no mutual influence of (0)/B and the boundary counterparts ∂ (0). An
illustration of the hierarchy, which is formally written in terms of (9.8), is given as follows
λD gD
g(0) g∂ (0)
λ (0) λ ∂ (0)
(9.9)
Hereby, the level-(0) and ∂ (0)-couplings are seen to be mutually independent and the depen-
dence propagates from top to bottom.
This observation fixes the strategy for the (mostly numerical) solution of the RG equations
on which we embark in the following sections: First we solve the non-trivially coupled gDk -
λDk system, then we insert its solution into the single differential equation of the level-(0) (or
alternatively, level-∂ (0) ) Newton coupling, solve for their k-dependence, and finally use all
286 Chapter 9. Investigating Asymptotic Safety
three solution functions obtained already to get the running of the corresponding cosmological
constants of level-(0) (or its counterpart on the boundary level-∂ (0)).
The hierarchy (9.8) raises the following question: Is it really necessary for the couplings
in the background sector (a) to assume fixed point values for k → ∞, and (b) to restore split-
symmetry at k= 0? After all, the B-couplings could even diverge in the UV without causing∣any
problems for the D-couplings and the hµν -correlation functions given by (∣δ/δh)
n Γk[h; ḡ]∣ .h=0
Or equivalently, should we insist that the level-(0) part of Γk, that is, Γk[h; ḡ]∣ , is well defined,h=0
too?
The answer to these questions is ‘yes’. In fact, the EAA at level zero is related to the
partition function Zk[ḡ] whose ḡ-dependence is indeed of physical interest, for instance, when
it is used as a tool for ‘counting’ the field modes which are already integrated out at a given
scale [166, 173]. This quantity is similar to a state sum and makes its appearance when one
studies quantum gravity effects in black hole thermodynamics, for instance. (See ref. [166]
or chapter 13 for further details.) Therefore we want also the level-(0) part of the EAA to be
asymptotically safe, and to be linked to the higher levels in a split-symmetric way.
Moreover, this third part of the thesis is supposed to pave the way for future work on more
complicated truncations which also allow lifting the degeneracy among all higher levels p =
1,2, · · · which is still assumed here. In this sense our analysis of the level-(0) /level-(1) interplay
is supposed to have the character of a model for the general case.
9.1.3 No anti-screening in the semiclassical regime
In the next chapter we shall present a comprehensive analysis of the RG equations using the
strategy of the previous subsection. Here, as a warm up, we present a simple analytic solution
to these equation which is valid in the semiclassical regime, i.e. at scales in, and slightly above
those of the classical regime where all dimensionful Newton and cosmological constants are
strictly k-independent.
The technical details related to this approximation, over and above the approximation due
to the truncation can be found in section 8.5. There we find that if gIk ≪ 1 and λDk ≪ 1 all of the
dimensionful quantities GI and λ̄ Ik k , I ∈ {D,B,(p), ∂ (p)}, behave as
[ ]
GI I Ik ≈ G0 1−ωdGI kd−20 (9.10a)
λ̄ I ≈ λ̄ I +ν I I d or λ̄ ∂ (0) ≈ λ̄ ∂ (0) +ν ∂ (0) ∂ (0)k 0 dG0 k d−1k 0 d G0 k (9.10b)
The dimension-dependent constants ω I and ν Id d are tabulated in section 8.5. The general struc-
ture of the solution (9.10) is quite familiar; it also obtains in the single-metric (‘sm’) Einstein-
Hilbert truncation. There it was found that the crucial ω-coefficient which governs the running
of Gk ≡ Gsmk is positive in the most interesting case d = 4. With ω sm4 > 0 Newton’s constant
decreases for increasing k, and this was interpreted as a kind of gravitational anti-screening
[122].
Within the approximation (9.10), the anomalous dimension is given by
η I =−(d−2)ω Id gI (9.11)
so that ω Id > 0, i.e. anti-screening, corresponds to a negative anomalous dimension η
I < 0.
Therefore, the positive (negative) sign of ω sm smd (η ) was highly welcome fro(m the Asym) ptotic
Safety point of view since at a non-trivial fixed point of the equation ∂ I I Itgk = d−2+ηk gk the
anomalous dimension is negative, too:2 η I∗ =−(d−2)< 0 whenever gI > 0.
It is therefore somewhat discomforting to discover that in the semi-classical regime the anti-
screening sign is not re-obtained within the bi-metric truncation. In fact, in the bi-metric setting
2Here and in the following we always assume d > 2.
9.2 Fixed points 287
it is the dynamical Newton constant GDk that should be compared toG
sm
k . Therefore the hallmark
of anti-screening is now ηD < 0, that is, ωDd > 0 in the semiclassical approximation. However,
using the explicit equations of section 8.5 for d = 4, we find that, with any cutoff, ωD4 < 0.
Hence the dynamical Newton co∣nstant shows a screening rather than anti-screening behavior
in the semiclassical regime: D∣η ∣ > 0. Instead, the background Newton constant G
B
k runssemiclass
in the opposite direction, ηB∣ < 0, but this has no direct physical meaning.
semiclass
Fortunately later on we shall also discover that in other parts of theory space ηD does be-
come negative actually, and that even non-trivial fixed points form. Nevertheless, this simple
example is a severe warning showing the limitations of the single-metric truncations and the
range of validity of the semiclassical approximation.
For the anomalous dimension related to g∂ (0) the semiclassical approximation yields a neg-
ative ω ∂ (0)4 < 0 for any cutoff shape function. In contrast to the dynamical bulk coupling, there
is however no problem in having a negative ω ∂ (p)4 . Therefore, notice that a negative ω
I
4 < 0
that remains negative beyond the semiclassical regime for k → ∞, is still compatible with the
existence of a non-trivial fixed point, defined by η I∗ = −(d−2)< 0. However, in this case the
fixed point value of the associated Newton coupling gI has to be negative, too. While this is
perfectly fine for the boundary couplings, there is experimental evidence that the pre-factor of
the Einstein-Hilbert action should be positive in the classical regime, when the theory resembles
General Relativity. Since under the RG flow the sign of gIk is preserved, this in turn restricts
gDk > 0 for all k, in order to recover the classical limit. On the other hand, except for case of
enforcing the classical matching condition for the coefficients of Einstein-Hilbert- and Gibbons-
Hawking-York-functional even on the quantum level, there is no preferred choice for the sign
of g∂ (p).
9.2 Fixed points
Based on the strategy outlined in subsection 9.1.2 we are going to solve the truncated flow
equations in a stepwise manner, following the hierarchical order of dependencies by starting
with the D-couplings and then proceed to the level-(0) counterparts. In this section, we are not
relying on any approximation beyond the employed truncation techniques of part II and study
the existence of special topological points, fixed points, of the RG flow.
Whenever numerical calculations are involved we employ the ‘optimized’ shape function
[188] R(0)(z) = (1− z)Θ(1− z) for which the threshold functions p pΦn (w) and Φ̃n (w) [122] are
given by a simple closed form, see eq. (8.43),
p 1 ( )−p 1 ( )−pΦn (w) = 1+w , and Φ̃p (w) = 1+wΓ(n+1) n Γ(n+2)
However, while the explicit numerical values may differ for the various cutoff choices, the
qualitative statements about the fixed points and their properties remain valid.
9.2.1 The dynamical sector
λD gD
g(0) g∂ (0)
λ (0) λ ∂ (0)
According to the hierarchy, we start with the dynamical couplings gD, λD and try to find common
zeros of their beta-functions: βDg = 0= β
D
λ . In the following analysis we set ρP = 1, ρgh = 1, i.e.
288 Chapter 9. Investigating Asymptotic Safety
we include all (dia- and paramagnetic, as well as ghost) contributions. Using the explicit form
of the beta-functions given in chapter 8 we find three fixed points (gD∗ ,λ
D
∗ ):
◮ A Gaußian fixed point at λD∗ = g
D
∗ = 0, henceforth denoted G
D-FP
◮ Two non-Gaußian fixed point at which both gD and λD are positive and negative, respec-
tively; they will be denoted NGD+-FP and NG
D
−-FP in the following
For the optimized shape function the fixed point coordinates are given by
u NGD-FP NGD∗ + −-FP G
D-FP
gD∗ = 0.703 −3.54 0 (9.12)
λD∗ = 0.207 −0.302 0
We recall that for the same shape function the well-known fixed point of the single-metric
truncation [135, 191] is located at (gsm∗ = 0.707,λ
sm
∗ = 0.193). These coordinates are remarkably
close to those of the NGD+-FP.
An RG trajectory that initially starts at gD = 0 will remain so for all k ∈ [0,∞). These class
of trajectories are subject to the simple D-equations:
gD = 0 , λD = λDk k k (k0/k)
2 (0≤ k < ∞) (9.13)
0
Here λDk denotes the value of the cosmological constant at k = k0. This trajectory separates the0
positive gD > 0 from the negative gD < 0 regime and thus there is no RG trajectory which ever
crosses the gD = 0 plane.
As mentioned already, this is a very general feature of any Newton-type coupling. Thus, for
any I there exist trajectories that separate the positive from the negative gB/(0)/∂ (0) regime:
B/(0)/∂ (0) = 0 , λ B/(0) = λ B/(0) ( / )2 or λ ∂ (0) ∂ (0)gk k k k0 k k = λk (k0/k) (9.14)0 0
The uniqueness of the integral curves ensures that there is no trajectory ever passes through
gB = 0, g(0) = 0, or g∂ (0) = 0. This, in particular implies that there exists no crossover trajectory
connecting NGD+-FP to NG
D
−-FP. More generally, there is no trajectory along which any of
the various Newton constants gI would cross zero. Since, in the classical regime we observe
a positive dynamical Newton coupling, we expect NGD+-FP to be more appropriate to describe
real physics.
9.2.2 The level-(0) or B-sector
λD gD
g(0) g∂ (0)
λ (0) λ ∂ (0)
So far, we have seen that there are two non-Gaußian and one Gaußian fixed point solutions in the
dynamical sector. Next, we insert their coordinates into the beta-functions of the background
or similarly of the level-(0) quantities (gB,λ B) and (g(0),λ (0)), respectively. We then search for
zeros which would generalize the ‘D’ fixed points to the full four dimensional bulk part of
theory space.
R Notice that the boundary couplings are unaffected by the level-(0) sector of the bulk and
vice versa. Thus the fixed points can be computed independently.
9.2 Fixed points 289
The bulk fixed points in the D/B language
It turns out that for each of the fixed points in the D-sector there exists precisely one non-
trivial solution of (gB B∗, λ∗ ), i.e. which satisfies (g
B
∗,λ
B
∗ ) 6= (0,0). Those related to the three
non-Gaußian D-FPs are located at
u NGD∗ +-FP NG
D
−-FP G
D-FP
gB∗ = 8.18 1.531 1.396 (9.15)
λ B∗ = −0.008 −0.12 −0.111
Furthermore, for gB = 0 = λ B the beta-functions for the B-sector vanish as well, for any value
of the D-couplings. Thus, also a Gaußian fixed point (gB B∗,λ∗ ) = (0,0) exists in the B-sector,
and it can be combined with each one of the D fixed point in (9.12).
In total we have six fixed points therefore: one which is purely Gaußian, having both
(gD D B∗ ,λ∗ ) = 0 and (g∗,λ
B
∗ ) = (0,0), three mixed ones , and two are purely non-Gaußian ones.
The table (9.16) gives a summary of all six combined fixed points and introduces the notation
we shall use for them.
d=4 NGD+-FP NG
D
−-FP G
D-FP
(gD∗ =0.7,λ
D
∗ =0.2) (g
D
∗ =−3.5,λD∗ =−0.3) (gD∗ =0,λD∗ =0)
NGB -FP NGB⊕NGD-FP NGB⊕NGD-FP NGB D+ + + + − +⊕G -FP
(9.16)
(gB∗=8.2,λ
B
∗ =−0.01) (gB∗=1.5,λB∗ =−0.1) (gB∗=1.4,λB∗ =−0.1)
GB-FP GB⊕NGD-FP GB+ ⊕NGD−-FP GB⊕GD-FP
(gB=0,λB=0) (gB∗ ∗ ∗=0,λ
B
∗ =0) (g
B
∗=0,λ
B
∗ =0)
An analogous table in the level language will be given below.
The bulk fixed points in the level-description
In the level-language the dynamical couplings, i.e. those at the higher levels p = 1,2,3, · · · ,
influence the beta-functions at level-(0), but not vice versa. The fixed points found in eq. (9.12)
for the ‘D’ couplings entail ( (p)g∗ ,λ
(p)
∗ ) = (gD∗ ,λ
D
∗ ) for the levels p≥ 1.
Regarding p = 0, the fixed points listed in (9.16) give rise to three non-trivial ones for the
(0) (0)
level-(0) couplings. Two of them, namely NG D D+ ⊕NG+-FP and NG+ ⊕NG−-FP are easily
computed using the conversion rules for the dimensionless couplings, i.e.
D B
(0) gk gg = k
(p)
k and g
D
k = gk for all p≥ 1 ,gD+gBk k
D B
(0) λk gk +λ
B gD
λ = k k
(p)
k and λ
D
k = λk for all p≥ 1 . (9.17)gD Bk +gk
(0)
To obtain the third one, NG+ ⊕GD-FP, related to the Gaußian fixed points of the ‘D’ sector,
one must be careful: When directly using relation (9.17) any of the remaining 4 fixed points
would seem to be located at (0) = 0 = λ (0)g∗ ∗ . Because of a division by zero this is incorrect,
however. From the coupled system (8.82)-(8.84) derived in part II, it turns out that the correct
(0)
coordinates of NG D+ ⊕G -FP are actually non-zero:
(0)
NG+ ⊕GD-FP : g(0) = 1.713 , λ (0)∗ ∗ = 0.068 (9.18)
290 Chapter 9. Investigating Asymptotic Safety
They are obtained if we directly solve for zeros in the full { (0)gk , D,λ
(0)
gk k ,λ
D} system.
In total we thus have found the following three non-trivial fixed point values for the level-(0)
couplings:
u NGD∗ -FP NG
D-FP GD+ − -FP
(0)
g∗ = 0.647 2.697 1.713 (9.19)
λ (0)∗ = 0.190 0.0168 0.068
As a result, the list of all six combined fixed points looks as follows:
d=4 NGD D D+-FP NG−-FP G -FP
(gD∗ =0.7,λ
D
∗ =0.2) (g
D
∗ =−3.5,λD=−0.3) (gD=0,λD∗ ∗ ∗ =0)
(0) (0) D (0) D (0)NG D+ -FP NG+ ⊕NG+-FP NG+ ⊕NG−-FP NG+ ⊕G -FP
(0) (0) (0) (0) (0) (0) (9.20)
(g∗ =0.65,λ∗ =0.2) (g∗ =2.7,λ∗ =0.02) (g∗ =1.7,λ∗ =0.1)
G(0)-FP G(0)⊕NGD+-FP G(0)⊕NGD−-FP G(0)⊕GD-FP
(0) (0) (0) (0) (0) (0)
(g∗ =0,λ∗ =0)) (g∗ =0,λ∗ =0)) (g∗ =0,λ∗ =0))
Summary
Each fixed point in the level description has an analog in the B - D system. Regardless of
whether we consider the {B,D} or the {(0),(1),(2), · · · } language, we find a total of six fixed
points on the bulk sector. Two of them lie in the negative gD half-plane and will be not rele-
vant for our further investigation. The remaining four are a doubly non-Gaußian fixed point
NGB+⊕NGD+-FP, a purely Gaußian one, and two mixed fixed points at which either the dynam-
ical or the background couplings vanish. Their potential relevance to the Asymptotic Safety
construction will be explored in the following chapters.
9.2.3 The boundary level-∂ (0)-sector
Before we study to which extend the dia- and paramagnetic, as well as ghost and graviton
contribution affect the presented results, let us consider the topological structure of the boundary
sector, first.
λD gD
g(0) g∂ (0)
λ (0) λ ∂ (0)
This relates to the other branch of the hierarchy (9.9) and is completely independent of the fixed
point structure of the level-(0) bulk couplings.
Different to the previous fixed point values of the level-bulk couplings, the qualitative pic-
ture of the semiclassical regime remains true for the non-Gaußian fixed points, leading to a
negative Newton coupling as pre-factor of the Gibbons-Hawking-York term. Explicitly, we ob-
tain again precisely one non-trivial solution ( ∂ (0),λ ∂ (0)g∗ ∗ ) for each NGFP of the D-sector, similar
to the case of their bulk counterparts. For the optimized shape function, we find
u∗ NG
D
+-FP NG
D
−-FP G
D-FP
∂ (0)
g∗ = −2.14 −27.9 −18.8 (9.21)
λ ∂ (0)∗ = 1.204 0.72 1.33
9.2 Fixed points 291
∂ (p)
If this picture of g∗ < 0 prevails for higher levels, the matching of the pre-factors of the
Einstein-Hilbert and the Gibbons-Hawking-York functional would be inconsistent with the
Asymptotic Safety requirement. Future work that disentangles the various level-contributions
will be able to answer this question.
Apart from the non-trivial solutions for ( ∂ (0) ∂ (0)g∗ ,λ∗ ), there exists a Gaußian type fixed point,
independent on the values of the D-sector. Thus, we have again a total of six combined fixed
points for the dynamical bulk level-(0) boundary sector. The complete list along with the nam-
ing conventions reads as follows:
d=4 NGD-FP NGD+ −-FP G
D-FP
(gD∗ =0.7,λ
D D
∗ =0.2) (g∗ =−3.5,λD∗ =−0.3) (gD∗ =0,λD∗ =0)
∂ (0) ∂ (0)
NG− -FP NG− ⊕
∂ (0) ∂ (0)
NGD+-FP NG− ⊕NGD−-FP NG− ⊕GD-FP
∂ (0) − ∂ (0) ∂ (0) ∂ (0) ∂ (0) ∂ (0) (9.22)(g∗ = 2.14,λ∗ =1.2) (g∗ =−27.9,λ∗ =0.72) (g∗ =−18.8,λ∗ =1.3)
G∂ (0)-FP G∂ (0)⊕NGD ∂ (0)+-FP G ⊕NGD-FP G∂ (0)⊕GD− -FP
∂ (0) ∂ (0) ∂ (0) ∂ (0) ∂ (0) ∂ (0)
(g∗ =0,λ∗ =0)) (g∗ =0,λ∗ =0)) (g∗ =0,λ∗ =0))
Summary
In the second branch in the hierarchy structure of the beta-functions, the boundary couplings
appear. Similar to the previous results on the bulk sector we find a total of six fixed points
for the D-∂ (0) combinations. The important difference resides in the sign of the fixed point
values of the boundary Newton-type coupling g∂ (0) which is negative for all non-trivial solutions.
However, there is no physical argument why this coupling should non-negative and thus it is
perfectly compatible with the Asymptotic Safety conjecture and experimental constraints.
9.2.4 Recovering the mechanism of paramagnetic dominance
In the single-metric analysis of [165] it was found that in d = 4 the NGFP owes its existence
entirely to the paramagnetic interactions of the hµν fluctuations with the background metric;
the diamagnetic interactions disfavor the formation of a fixed point instead. (Below d = 3
the situation changes, showing that the respective physical mechanism behind the NGFPs in
d = 2+ ε and d = 4 are quite different [165].) In this subsection we are study this picture
in the bi-metric case making use of the parameter ρP to disentangle dia- and paramagnetic
contributions.
In the gD-λD-regime of interest, all qualitative properties of ηD in eq. (8.74) are well de-
scribed by its linear approximation ηD ≈ BD1(λD)gD. The contribution of the denominator in
(8.74), involving BD D2(λ ), influences the resulting anomalous dimension only weakly. So let
us focus on BD1(λ
D). Assuming, as always, a positive dynamical Newton constant, the nega-
tive anomalous dimension which is indicative of anti-screening and is necessary for a NGFP,
requires a negative BD1 .
Now, what we find is that the paramagnetic interactions indeed drive BD1 negative, but the
diamagnetic ones have an antagonistic effect trying to make BD1 and η
D positive. In fact, switch-
ing off the paramagnetic contributions yields, in the bi-metric setting,
∣
BD(λD)∣1 > 0 for all values of λD . (9.23)ρP=0
Instead including the paramagnetic parts we find that there is a crossover from negative to
292 Chapter 9. Investigating Asymptotic Safety
positive values of the anomalous dimension at a certain critical value λDcrit, that is
{
∣ D D
BD(λD)∣ > 0 ∀λ < λcrit1 (9.24)ρP=1 < 0 ∀λD > λDcrit
The precise value of the critical cosmological constant, λDcrit, is cutoff scheme dependent, but in
any scheme we find 0< λD Dcrit < λ∗ .
The behavior (9.24) makes it explicit that, first, the absence of anti-screening in the semi-
classical regime (BD1(0)> 0) can be reconciled with a non-trivial fixed point existing simultane-
ously (BD D1(λ∗ )< 0) and, second, the NGFP can form only because in some part of theory space
the paramagnetic interactions are stronger than the diamagnetic ones.
Thus we essentially recover the single-metric picture according to which ‘paramagnetic
dominance’ is the physical mechanism responsible for Asymptotic Safety. However, the bi-
metric analysis suggests that this mechanism can occur only for a non-zero, positive cosmolog-
ical constant.3 The implications of this result will be further discussed in chapter 12.
A similar effect of the diamagnetic terms rendering BI1 > 0 can be observed for the anoma-
lous dimension of the boundary Newton coupling g∂ (0). The associated b∫eta-√functions only
contain diamagnetic contributions, since the corresponding basis invariant ∂M H̄K̄ does not
feel the additional R̄-factors in the heat kernel expansion. In this case, the diamagnetic con-
tributions are not compensate by counteracting paramagnetic terms and ∂ (0)B1 > 0 stays always
positive requiring a negative fixed point value for g∂ (0).
9.2.5 Impact of the ghost contributions
It is instructive to look at the dependence of the fixed point data on the ghost contributions to the
beta-functions. Since we multiplied each trace associated to the ghost functional with a factor
of ρgh = 1, it is easy to assess the importance of ghosts relative to the gravitons by varying this
parameter away from ρgh = 1, the value used in the previous subsections. This will give us an
idea of the numerical accuracy one may expect within the present approximation which neglects
the running of the ghosts’ wave function renormalization.
The fixed point coordinates in the dynamical sector depend on the ρgh non-trivially as shown
in Fig. 9.2. While the trivial fixed point GD-FP is independent of ρgh, the non-Gaußian ones
gD∗ λD∗
ρ
1 2 3 4 gh
0.2
0.1
-2
ρ
1 2 3 4 gh
-4 GD-FP
NGD−-FP -0.1
NGD-FP G
D-FP
+
6 NG
D
- −-FP
-0.2 NGD+-FP
-0.3
-8
(a) The dependence of gD∗ on ρgh. (b) The dependence of λ
D
∗ on ρgh.
Figure 9.2: The dependence of gD∗ and λ
D
∗ on ρgh for the three types of fixed points, G
D-FP,
NGD−-FP, and NG
D
+-FP. While the Gaußian one is insensitive to independent on ρgh the λ
D
∗
values of the others only slightly change when increasing ρgh. As for the Newton constant g∗D,
the fixed point in the lower half-plane (gD < 0) is by far more sensitive to ρgh than the one in
the upper, NGD+-FP.
3The same conclusion can be drawn from the beta-functions found in [172] with a bi-metric truncation too.
9.2 Fixed points 293
NGD+-FP and NG
D
−-FP are not. However, it turns out that g
D
∗ of NG
D
−-FP is very sensitive to a
change in ρgh, especially going from ρgh = 0 to ρgh = 1. This suggests that NG
D
−-FP is likely
to be a truncation artifact. On the other hand, the fixed point values of NGD+-FP are quite stable
when changing the strength of the ghost-contributions.
From the point of view of the Asymptotic Safety scenario both of these facts are encour-
aging: First, the fixed point at positive dynamical Newton constant, NGD+-FP, seems well de-
scribed within the present approximation and hardly could be a truncation artifact. If we define
a k→ ∞ limit there, since the running gDk never changes its sign, the resulting theory has a posi-
tive GD0 , as it should be. Second, the fixed point at negative g
D, NGD−-FP, is much less stable and
more likely to be a truncation artifact. If so, this would actually be welcome since then no UV
limit could be taken there, and no asymptotically safe theory with GD0 < 0 could be constructed,
and the result of positive GD0 implied by NG
D
+-FP is a true prediction.
For the background couplings the picture is as follows. The Gaußian solution gB∗ = 0= λ
B
∗ ,
leading to the three fixed points GB⊕GD-FP, GB⊕NGD−-FP, and GB⊕NGD+-FP, is independent
of ρgh, and so the same is true for their (gB,λ B)-coordinates. The fixed point that is situated
gB B∗ λ∗
10 1
5
0 ρ
1 2 3 4 gh
0 ρ
1 2 3 4 gh
-1
GB⊕ (· · ·)D-FP
5 GB- B⊕ D ⊕ (· · ·)
D-FP
NG+ G -FP B D
NGB⊕NGD NG ⊕G -FP+ −-FP +
B⊕ D 2 NG
B⊕NGD-FP
-
10 NG+ NG+-FP
+ −
NGB⊕NGD- + +-FP
-15 -3
(a) The dependence of gB∗ on ρgh. (b) The dependence of λ
B
∗ on ρgh.
Figure 9.3: The dependence of λ B∗ and g
B
∗ on ρgh for the six fixed points. There are three Gaußian
and three non-Gaußian fixed points in the B-sector, one for each D-fixed point. The Gaußian
ones are independent on ρgh. In contrast to the D-part of these fixed points, the B fixed point
values vary most strongly for NGB+⊕NGD+-FP. Especially for small ρgh it starts out negative,
diverges and then turns positive for ρgh < 1.
in the negative gD half-plane, NGB D+⊕NG−-FP, is again very sensitive to the influence of the
ghost sector. The background fixed point values exhibit a strong dependence on ρgh even for
NGB+⊕NGD+-FP which has the positive gD∗ . The B-fixed point values actually change the sign
and turn negative below ρgh < 0.6, see Fig. 9.3.
We do not think that this apparent instability of (gB,λ B∗ ∗ ) sheds any doubt on the viability of
the Asymptotic Safety construction at the doubly non-Gaußian NGB+⊕NGD+-FP. In fact, the B-
couplings are entirely unphysical and owe their existence only to the extra ḡ-dependence of Γk,
hence to the split-symmetry violation in unobservable quantities. In fact, considering the level-
(0) couplings in fig. 9.4 the instability of the B-sector is not visible for the ‘true’ coefficients
multiplying the level-(0) invariants. The picture we obtain in this description is thus in full
agreement with the D-results.
Finally, let us consider the ρgh-dependence of the boundary couplings, depicted in fig. 9.5.
From this perspective it seems that the very large fixed point values associated to ∂ (0)NG− ⊕NGD−-
∂ (0)
FP and NG D− ⊕G -FP are caused by a ρgh-pole in the vicinity of the true value ρgh = 1. This
further strengthen the assumption that those fixed points are artifacts of the truncation rather
than stable solutions of the full RG flow. On the other hand, the D-fixed point of most interest,
∂ (0)
NG ⊕NGD− +-FP, turns out to be almost insensitive on the ρgh parameter, especially in the
294 Chapter 9. Investigating Asymptotic Safety
(0)
g∗
4 λ (0)∗
0.6
3
G(0)⊕ (· · ·)D-FP (0)
(0)⊕ D G ⊕ (· · ·)
D-FP
NG (0)
(+
G -FP D
0)⊕ D- 0.4 NG ⊕G -FPNG+ NG− FP (+0)(0)⊕ NG ⊕NG
D-FP
2 NG+ NG
D
+-FP
+(0) −
NG ⊕NGD+ +-FP
0.2
1 ρ
1 2 3 4 gh
-0.2
ρ
1 2 3 4 gh
(a) The dependence of (0)g∗ on ρgh. (b) The dependence of λ
(0)
∗ on ρgh.
Figure 9.4: The dependence of λ (0)∗ and
(0)
g∗ on ρgh for the six fixed points. The general structure
is quite similar to the D-sector, with a ρgh-insensitive Gaußian fixed point and the non-Gaußian
one of the upper half-plane g(0) > 0 being more stable than its negative counterpart. The diver-
gence of the B-couplings does not affect the level-(0) couplings.
∂ (0)
g∗ λ ∂ (0)∗
30
G∂ (0)⊕ (· · ·)D-FP
∂ (0) 3 ∂ (0) D
NG− ⊕GD-FP G ⊕ (· · ·) -FP∂ (0)20 ∂ (0) DNG− ⊕NGD−-FP NG− ⊕G -FP∂ (0)∂ (0) D
NG− ⊕NGD-FP NG2 − ⊕NG -FP+ ∂ (0) −NG− ⊕NGD+-FP10
ρ 1
0.5 1.0 1.5 2.0 2.5 3.0 3.5 gh
ρ
-10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 gh
20 -1-
-30 -2
(a) The dependence of ∂ (0)g∗ on ρgh. (b) The dependence of λ
∂ (0)
∗ on ρgh.
Figure 9.5: The dependence of λ ∂ (0)∗ and
∂ (0)
g∗ on ρgh for the six fixed points. From very general
arguments it follows that the Gaußian fixed point is independent on ρgh. In contrast, especially
∂ (0)
the non-Gaußian fixed point NG− ⊕NGD−-FP is very sensitive to a change in ρgh. Indeed, in
the vicinity of ρgh = 1 this fixed point value approaches a singularity and changes sign. On the
other hand ∂ (0)NG− ⊕GD-FP is almost constant in the ρgh-range of interest.
vicinity of ρgh = 1 which corresponds to including all ghost contributions in the way they appear
on RHS of the flow equation.
9.2.6 Summary
In this section, we investigated the fixed point structure of the beta-functions in the bi-metric-
bulk – pure-background-boundary truncation. Therefore, we made use on a hierarchical order
in the differential system, by first solving the fixed point condition for the D-couplings, yielding
one Gaußian and two non-Gaußian fixed points. The latter two are separated by the trajectories
satisfying gDk = 0 and lay in the upper (lower) half-plane of Newton’s coupling g
D and are
denoted NGD+-FP andNG
D
−-FP, respectively. For asymptotically safe RG trajectories describing
a QFT that enters the familiar classical regime of General Relativity there is the additional
constraint gD∗ > 0. Thus NG
D
+-FP seems to be favored by experimental observations. In fact, it
turns out that NGD−-FP is very sensitive to changes of the ghost sector indicating that this non-
trivial solution might be only an artifact of the truncation, while the dependence of NGD+-FP on
ρgh is almost negligible. If this picture confirms for extended truncations, then the requirement
of Asymptotic Safety yields a unique fundamental theory in the D-sector. Inserting the solution
separately into both branches of the hierarchy model, we found for each of the three fixed
9.3 Predictivity 295
points (λD∗ ,g
D
∗) a trivial and a non-trivial solution for the zeros of the bulk and boundary level-
(0) beta-functions, respectively. This gives rise to a total of 3D × 2(0) + 3D × 2∂ (0) = 12 fixed
points in the six dimensional truncated theory space of the bi-metric-bulk – pure-background-
boundary truncation. From the physical and – as the stabilization checks indicate – also from the
mathematical point of view the most interesting fixed points are combinations as (· · · )⊕NGD+-
(0) ∂ (0)
FP, in particular the ‘double’-NGFPsNG+ ⊕NGD+-FP and NG− ⊕NGD+-FP. Notice that bulk
and boundary level-(0) couplings are independent of each other and thus the complete form of,
(0) (0) ∂ (0) (0)
say NG D+ ⊕NG+-FP, is given by G∂ (0)⊕NG ⊕NGD+ +-FP or NG− ⊕NG+ ⊕NGD+-FP. In the
next section, we study the UV critical hypersurfaces associated to the twelve fixed points.
9.3 Predictivity
In investigating AS conjecture for QG in the bi-metric-bulk – pure-background-boundary trun-
cation, we identifying the topological landscape of theory space by searching for trivial points
of the RG flow. The obtained structure characterizes the underlying differential system and is
of uttermost importance to define non-perturbatively renormalizable theories in the AS sense.
In section 9.2, we have found a total of twelve fixed points within the six dimensional truncated
theory space. In the sequel we use their linearized critical hypersurfaces SUV to get a first un-
derstanding of their properties and the associated scale invariant theory. As mentioned above,
the corresponding critical exponents classify the universality class of the fixed point theory and
can in fact be the same for fundamentally very different physical models. Thus, while the global
properties is hidden in the non-linear parts of SUV, its UV structure is fully described by its uni-
versality class. For the AS conjecture the most important information is the dimensionality of
SUV, which in order to describe fully predictive theories has to be finite dimensional. However,
due to the complexity of the FRGE on usually projects the RG flow of the full infinite dimen-
sional theory space T to a subspace by means of truncations. For finite dimensional truncations
the UV-critical hypersurface is always finite and thus there is little to say about this aspect of
Asymptotic Safety. Nevertheless, by increasing the number of basis invariants one observes a
promising trend, suggesting that in the UV asymptotically safe trajectory can be fixed by only
a finite set of experiments, which is also confirmed by first infinite dimensional truncations of
f (R)-type, where the UV-critical hypersurface satisfies dim(SUV)< ∞ [129].
We start this section by recapitulating the classification of RG trajectories in the vicinity of
critical points and introduce the notion of stability matrix, relevant and irrelevant directions, and
critical exponents. Afterwards, we follow the lines of the previous section and study the two
orthogonal subspaces (gD,λD,g(0),λ (0)) and (gD,λD,g∂ (0),λ ∂ (0)), separately, which is justified by
the hierarchical structure of the differential system (9.9).
9.3.1 The classification of trajectories in the linearized region
In this subsection, we go back to the general formulation, denoting u(n)(k) to be a generic
dimensionful coupling and β (n)({u(m)(k)}) its associated beta-function. Furthermore, let Nu
be the number of couplings of the possibly truncated theory space. If the beta-functions are
sufficiently smooth, there is a neighborhood of every point in theory space ≡ (m)u0 (u0 ) which is
well described by the linearized RG flow, obtained by expanding the beta-functions up to linear
order in the couplings, i.e.
∣
Nu
(n) { (m) } (n) { (m)} ∂β
(n)({ (m)}) ∣u ( )
β ( u (k) ) = β ( u0 )+∑ ∣
∣ (r) (r)
∣ u (k)−u∂u(r) 0 + . . .r u(m)≡ (m)u0
296 Chapter 9. Investigating Asymptotic Safety
The coefficient of the linear term defines the stability matrix B at u0 and contains all essential
information of the RG flow in the vicinity of u0. Its components are thus given by
∣∣∂β (n)({u(m)}) ∣
Bnr(u0) = ∣ (9.25)
∂u(r) ∣
u(m)≡ (m)u0
We thus arrive at a linear differential system of Nu variables, that can be solved independently
if the stability matrix B is diagonalizable and thus the system can be decoupled. In the fol-
lowing we assume that there is a basis of (right) eigenvectors of the stability matrix, satisfying
∑Nun B
(n) (r) 4
rn(u0)v = λr v , with
Nu Nu
v(r)(k)≡ ∑S−1u(m)(k) , u(m)rm (k) = ∑S (r)mrv (k) (9.26)
m r
Here S denotes an invertible linear map such that S−1BS is diagonal. Notice that the eigenvalues
λr are in general complex for B usually fails to be symmetric.
If we substitute (9.26) into the linearized differential equations, the system indeed decou-
ples, yielding Nu isolated ordinary differential equations of the following form
[ ]
Nu Nu
k∂ v(r)
(m)
k (k) = λ · v(r)(k)+∑S−1 β (n)r rn (u0)−∑Bnm(u0)u0 (9.27)
n m
The second term is a constant, independent of k and v(r), which we abbreviate in the sequel by
b(r)(u ) ≡ ∑Nu S−1[β (n)(u0)−∑Nu (m)0 n rn m Bnm(u0)u0 ]. In case of λr = 0, the RHS of eq. (9.27) is
constant and the the k-dependence of the associated eigenvectors becomes logarithmic:
( )
v(r)(k) = b(r)(u0) ·
k
ln + v(r)(k0) (9.28)
k0
For the more interesting case of non-vanishing eigenvalues, λr 6= 0, eq. (9.27) is linear in
t = ln(k/k0). Using an exponential ansatz. the eigenvectors of the stability matrix associated to
λr 6= 0 shows the following k-dependence
( ) ( )−θr (r)
(r) k b (u0)v (k) = v(r)(k0)− 1 b(r)θ (u0) · + for λr 6= 0 (9.29)r k0 θr
In the last two equations we have introduced v(r)(k0) as the value of the eigenvector at k0 and
θ .r .=−λr, the negative of the associated eigenvalue, which is referred to as critical exponent if
u0 ≡ u∗ describes a fixed point. The sign of its real part, i.e. of ℜ(θr), indicates if a direction
that is generated by a local eigenvector diverges from the chosen coordinate u0 or converges
when lowering the scale k. We thus classify all eigenvectors in the linear regime of an arbitrary
point u0 in theory space as relevant, marginal or irrelevant direction depending on the real part
of the negative eigenvalue θr that might be either positive, zero or negative, respectively, see
table 9.1. In the marginal case, in which the eigenvalue vanishes, the linear approximation is
insufficient to deduce a converging or diverging behavior of the solutions along this direction.
Therefore higher order contributions have to be considered.
Now, we apply this machinery to very special points in theory space, fixed points. The
previous discussion simplifies for critical points u∗, since the beta-function vanishes there and
thus the zeroth order in the Taylor expansion of β (n) drops out, implying in particular
Nu
(r) ≡− −1 ∗ (m) (m) (r)b (u∗) ∑ Srn Bnmu∗ ≡ θ ∑S−1r rm u∗ ≡ θr v∗ (9.30)
n,m m
4 In this context, the eigenvalues of B denoted λr should not be confused with the cosmological constant.
9.3 Predictivity 297
Classification of trajectories
Critical exponent (=−λr) Behavior Classification
ℜ(θ )> 0 k−|θr|r relevant
ℜ(θr) = 0 not considered marginal
ℜ(θr)< 0 k+|θr| irrelevant
Table 9.1: Classification of trajectories in the linearized region of a point u0
Here we wrote B∗nm ≡ Bnm(u∗) and noticed that B∗nm diagonalizes w.r.t. S. We then used eq.
(9.26) to identify the remaining sum as the fixed point coordinate in the basis of the eigenvec-
(r)
tors v(m), i.e. v∗ ≡ −1 (m)∑m Srm u∗ . The resulting expression for the k-dependence of the (right)
eigenvectors (9.29) assumes the following very illuminating form:
( ) ( )−θr
(r) (r) − (r) · k (r)v (k) = v (k0) v∗ + v∗ for θr 6= 0 (9.31)
k0
Discount marginal direction for a moment, let us consider the RG evolution in the vicinity of
v∗ for a generic trajectory k 7→ v(k) with initial condition v(k0) ≡ v0 6= v∗. The first term in eq.
(9.31) describes the deviation from the fixed point coordinate. To be explicit, we focus on the
UV flow of the trajectory, hence assuming k> k0 and thus k/k0 > 1. For positive (negative) real
(r) (r)
part of θr the initial separation of v0 from the fixed point coordinate v∗ shrinks (blows up)
while increasing k. This has the consequence that if the RG trajectory is asymptotically safe,
i.e. it converges to v∗ for k→ ∞, the coordinate in all ‘irrelevant’ directions, ℜ(θr)< 0, has to
coincide with the fixed point value, i.e.
7→ ⇒ (r) ≡ (r)k v(k) asymptotically safe = v0 v∗ ∀r ∈ Nu with ℜ(θr)< 0
Otherwise, any initial discrepancy potentiates pulling the trajectory away from the fixed point.
On the other hand, if ℜ(θr)> 0 the trajectory is pulled into the fixed point for k→ ∞.
Thus, the critical exponent indeed characterize the asymptotic behavior of the correspond-
ing couplings. Depending on their sign small disturbances from the fixed point value in the
respective direction of theory space either leads to an attractive or repulsive evolution. In com-
Classification of fixed point trajectories
Critical exponent (=−λr) Classification Attractive Repulsive
ℜ(θr)< 0 Attractive IR (k→ 0) UV (k→ ∞)
ℜ(θr)> 0 Repulsive UV (k→ ∞) IR (k→ 0)
Table 9.2: Classification of trajectories in the linearized region of a fixed point u∗.
bination with the AS requirement, all trajectories which are attracted towards the fixed point
for increasing k, are suitable candidates for a renormalizable quantum theory of gravity. On the
other hand, if we are in search for a safe IR theory the arguments have to be reverted. Hence,
any repulsive direction for increasing k, say, describes an attractive direction for decreasing k.
In the Wilsonian renormalization group approach, the flow evolves from the UV towards the IR
and therefore we may classify the trajectories that enter into an IR fixed point as attractive and
298 Chapter 9. Investigating Asymptotic Safety
those which depart as repulsive. Table 9.2 presents an incomplete overview of the classifica-
tion scheme, where we omitted the marginal directions since the vanishing of their eigenvalues
require a separate analysis.
T
SUV
Figure 9.6: This plot illustrates the vicinity of a non-Gaußian fixed point embedded in theory
space T. Its UV-critical hypersurface, SUV, is highlighted in blue along with a set of asymp-
totically safe RG trajectories (marked in white and red). Here, as usually, arrows point from
k→ ∞ to k = 0. The case of RG solutions which do not respect the AS condition is presented
by the (orange) line that separates from SUV under the inverse flow. Corresponding theories
are plagued by UV divergences and are therefore not renormalizable.
In the sequel, we focus on the UV interpretation of the fixed point regimes, since we are in
search for a suitable UV completion of Quantum Gravity. In this case, the relevant eigenvectors
span the subspace of UV attractive trajectories which defines the linear part of the UV-critical
hypersurface SUV. While each irrelevant direction, ℜ(θr)< 0, is constraint by the Asymptotic
Safety program, the relevant counterparts are unspecified. Hence, the dimensionality of SUV,
which equals the number of relevant directions, is related to the predictivity of the system.
Any trajectory within SUV gives rise to the same fundamental UV theory, for it converges to
the fixed point for k → ∞, see fig. 9.6. Thus, we need s .UV .= dim(SUV) experiments to fix
the remaining freedom of the differential system in order to obtain a unique renormalizable
theory of Quantum Gravity. From this perspective it is very natural to impose the condition of
dim(SUV)< ∞ being finite to define a generalized criterion of renormalizability [168].
R Remember that any initial parameter of a theory has to be fixed by some additional con-
straint, either experimentally or by theoretical consistency. Asymptotic Safety is a very
strong constraint of the latter case, where usually infinitely many free parameters are allo-
cated values specified by the mathematical formalism.
9.3.2 Critical exponents and scaling fields
Following the analysis of the previous subsection, we are now going to study the critical be-
havior of the twelve fixed points for the bi-metric-bulk – pure-background-boundary truncation.
This discussion presents the level-language, first for the bulk and then independently for the
boundary couplings. The corresponding results for the B-sector are found in appendix 9.A of
this chapter.
The bulk-sector
In the previous section we found a total of twelve fixed points that represents trivial, though very
interesting parts of the RG flow. We now extend the list of properties, which so far consists of
their coordinates, by describing their associated universality classes characterized by the critical
exponents θr.
9.3 Predictivity 299
Making again use of the hierarchy among the beta-functions, we start with the bulk sector
and consider all bulk projections of fixed points associated to the dynamical (or from level-(1)
upwards) Gaußian fixed point for which (gD D∗ ,λ∗ )≡ (0,0) holds true. Besides a full Gaußian one,
there is mixed combination with non-canonical scaling properties of the level-(0) couplings, i.e.
G(0)⊕ (0)GD-FP (0) (0) (0) (0)(g∗ ,λ∗ )=(0,0) NG D+ ⊕G -FP (g∗ ,λ∗ )=(1.7,0.07)
θ eigenvectors v(r) θ eigenvectors v(r)r r
−2 8π3 êD+ êDg λ −2 0.97êDg +0.12êDλ −0.23
(0)
êg +0.03
(0)
êλ
× (9.32)2 +êDλ 2 5 10−17êD
(0) (0)
λ +1.00êg +0.38êλ
−2 8π (0) (0)êg + êλ 2 1.00
(0)
êg +0.04
(0)
êλ
2 (0) 4 (0)êλ êλ
In both tables, we list the critical exponents along with the ‘scaling fields’, the eigenvectors v(r)
of the associated stability matrix. Here the ê’s are Cartesian unit vectors; D ≡ (1) (2)êg êg = êg = · · ·
points in the direction of the dynamical (or level-(p), p ≥ 1) Newton constant, etc. For the
directions emanating from the Gaußian part of the fixed points we observe the canonical scaling
behavior with critical exponent −2 and 2 for Newton’s and cosmological coupling, respec-
tively. Notice that the Gaußian fixed point is situated on the separating line gI = 0 that prevents
zero-crossing of RG trajectories. It is UV repulsive (attractive) in the direction of the cos-
mological (Newton) coupling and thus there is a unique RG solution for gD > 0 entering the
point (0,0). This is also reflected in the dimensionality of its UV critical manifold, which is
s .UV .= dim(SUV) = 2, as given by the canonical scaling behavior of the couplings. For the
non-trivial fixed point combination of D- and level-(0) sector, (0)NG ⊕GD+ -FP, there appears an
additional relevant direction along λ (0) yielding a three dimensional subspace of asymptotically
safe trajectories, i.e. sUV = 3.
Next, we consider the fixed points associated to NGD−-FP. Its sensitivity on the ghost sector
contributions has raised the question, if its existence is rather an artifact of the truncation than
a true property of the exact flow. Nevertheless, it is instructive to discuss its properties listed in
the following table
G(0)⊕ (0)NGD−-FP (0) (0)(g∗ ,λ∗ )=(0,0) NG ⊕NGD-FP (0) (0)+ − (g∗ ,λ∗ )=(2.7,0.02)
θ eigenvectors v(r) θ eigenvectors v(r)r r
2.21 1.0 êDg +0.1 ê
D
λ 2.21 −0.50êDg −0.03 D −0.88
(0) −0.01 (0)êλ êg êλ
5.12 −0.6 êDg +0.8 êDλ 5.12 0.2êD Dg −0.3êλ −
(0) (0) (9.33)0.5êg +0.8êλ
−2 1.00 (0)êg +0.01 (0)êλ 2 1.00
(0) +0.01 (0)êg êλ
2 (0) 4 (0)êλ êλ
The same structure as discussed previously appears for the mixed combination of non-trivial
and Gaußian fixed points, whereby in the present case the level-(0) part obeys the canonical
scaling relation, yielding G(0)⊕NGD−-FP. This fixed point has again a three dimensional UV-
critical manifold, s (0) DUV = 3, with qualitatively interchanged roles of g and g . On the other
hand, the doubly non-Gaußian fixed point (0)NG D+ ⊕NG−-FP is UV-attractive in all four bulk
directions, and thus each coefficient of the truncation ansatz has to be experimentally fixed by
an independent measurement. Notice that the hierarchy structure is also visible on the level of
critical exponents, which remain the same in the D-sector for different values of the level-(0)
couplings.
We now turn to the most promising candidate for an asymptotically safe quantum theory of
gravity, NGD+-FP. This fixed point seems to be stable under severe deformations and extensions
300 Chapter 9. Investigating Asymptotic Safety
of theory space and exhibits also very appealing properties concerning its dependence on the
ghost contributions, for instance. Due to the existence of the separating line at gD = 0, it is so
to say also situated in the half-plane gD > 0 consistent with a classical IR limit reproducing the
positive sign of Newton’s coupling in GR. In combination with the level-(0) results for the fixed
point condition, we obtain two further critical points, characterized as follows:
G(0)⊕NGD-FP (0) (0)+ (g∗ ,λ∗ )=(0,0)
θr eigenvectors V ( j)
3.6+4.3ı −0.99êDg − (0.04+0.15ı)êDλ
3.6−4.3ı −0.99êDg − (0.04−0.15ı)êDλ
−2 0.96 (0)êg +0.28 (0)êλ
2 (0)êλ
(0)
NG ⊕NGD-FP (0) (0)+ + (g∗ ,λ∗ )=(0.65,0.19)
θr eigenvectors v(r)
3.6+4.3ı 0.86êDg +(0.04+0.13ı)ê
D
λ +(0.40−0.03ı)
(0)
êg − (0.2−0.2ı) (0)êλ
3.6− (9.34)4.3ı 0.86êDg +(0.04−0.13ı)êDλ +(0.40+0.03
(0)
ı)êg − (0.2+0.2 (0)ı)êλ
2 0.96 (0) +0.28 (0)êg êλ
4 (0)êλ
According to eq. 9.34, the fixed points which stem from NGD+-FP, located in the physically
relevant gD > 0 half-space, are special in that some of their critical exponents are complex. This
leads to an oscillating part in the solution of eq. (9.31) which further gives rise to the charac-
teristic spirals of the RG trajectories approaching the FP for k → ∞. This feature was already
observed in the single-metric Einstein-Hilbert truncation [135, 191] and in fact, the complex
conjugate pair θ1/2 of eq. 9.34 is reminiscent of the single-metric results. This close numerical
similarity lends further credit to the conjecture that the single-metric fixed point should corre-
spond to one of the two fixed points of the full system which are related to the NGD+-FP. in
chapter 10 we investigate the cause of this coincidence. In contrast to the complex, non-integer
values of the critical exponents in the D-sector their level-(0) counterparts are found to be al-
ways of integer type. In fact, from all previous results we see that they either appear with the
canonical scaling −2 and 2 for the Gaußian solution related to G(0)-FP, or with 2 and 4 in case
(0)
of NG+ -FP. For the dimensionality of the UV critical hypersurface we read off sUV = 3 for
the mixed fixed point G(0)⊕NGD+-FP and sUV = 4 for the ‘doubly non-Gaussian’ fixed points,
(0)
NG+ ⊕NGD−-FP.
For the present truncation, we observe that each Gaußian part of the fixed point is associated
with one relevant and one irrelevant direction, while a non-Gaußian component introduces two
relevant dimensions. In particular for the ‘doubly’ NGFPs we thus have a four dimensional
UV-critical hypersurface, implying that four independent experimental results are needed in
order to impose sufficient initial condition to obtain the unique trajectory realized in Nature.
Furthermore, whereas the critical exponents related to the level-(0) couplings are always real
integers, the D-sector is of non-integer type usually and in the most interesting case of NGD+-FP
we find a pair of complex numbers for θ1/2, resulting in a spiral behavior of the RG trajectories.
The UV-critical manifold for the boundary sector
We conclude this section by presenting the critical behavior of the fixed points in the boundary
sector. We are not going to repeat the discussion for the D-sector, since the results remain the
9.3 Predictivity 301
same and are unaffected by g∂ (0) and λ ∂ (0). In fact, we will only focus on the difference of the
level-(0) bulk and boundary results and keep the discussion short on those aspects, which turn
out to be the same in both cases.
However, the first distinction already appears for the Gaußian fixed point in the boundary
sector. Since it described the canonical scaling of the free theory, the critical exponents differ
for λ (0) and λ ∂ (0), since for the latter the dimensionful coupling scales as k1 rather than k2 in
the canonical case. Hence, for GD-FP, we have the following characterization of the critical
regimes when combined with the Gaußian and non-trivial fixed point solutions of the boundary
sector:5
G∂ (0)⊕GD-FP ∂ (0) ∂ (0) ∂ (0) ∂ (0) ∂ (0)(g∗ ,λ∗ )=(0,0) NG− ⊕GD-FP (g∗ ,λ∗ )=(−18.8,4/3)
θr eigenvectors v(r) θr eigenvectors v(r)
−2 8π êD D D D ∂ (0)3 g + êλ −2 0.12êg +0.01êλ +0.99êg +0.02
∂ (0)
êλ
∂ (0) ∂ (0) (9.35)2 +êDλ 2 0.0ê
D
g +0.0ê
D
λ +0.0êg +0.0êλ
−2 8π ∂ (0)êg − ∂ (0)êλ 2 1.00
∂ (0)
êg −0.07 ∂ (0)êλ
1 ∂ (0)ê 3 g∂ (0)λ
We obtain the same picture as for the bulk sector: each Gaußian part of the fixed points comes
along with one irrelevant direction, while for the non-Gaußian component both directions turn
out to be relevant and thus have to be fixed by additional constraints. The only apparent differ-
ences compared to the bulk contribution are the negative sign multiplying the λ ∂ (0)-part of the
eigenvector and the decremented critical exponent associated to g∂ (0).
Next, consider the vicinity of the NGD−-FP fixed point and its combination with the zeros of
the beta-functions for the boundary couplings. The resulting critical exponents and the scaling
fields read:
G∂ (0)⊕ ∂ (0)NGD-FP ∂ (0) ∂ (0) D ∂ (0) ∂ (0)− (g∗ ,λ∗ )=(0,0) NG− ⊕NG−-FP (g∗ ,λ∗ )=(−27.9,0.72)
θ eigenvectors v(r) θ eigenvectors v(r)r r
2.21 1.0 D +0.1 D 2.21 0.02 D +0.001 D +1.0 ∂ (0) −0.03 ∂ (0)êg êλ êg êλ êg êλ
5.12 0.6 êDg −0.8 êDλ 5.12 0.04êDg −0.05 D +9.97
∂ (0) +0.7 ∂ (0)êλ êg êλ
−2 1.00 ∂ (0) −0.03 ∂ (0) 2 1.00 ∂ (0) −0.03 ∂ (0)êg êλ êg êλ
1 ∂ (0) ∂ (0)êλ 3 êλ
(9.36)
Except for the differences that already appeared on for the Gaußian fixed point, there is no
further peculiarity. In fact, the results resembles the properties of the bulk sector exhibiting
integer eigenvalues in the descending boundary part of the differential system.
Finally, let us consider the most interesting fixed point NGD+-FP. In combination with the
trivial solution for the boundary couplings, we obtain the familiar picture of a three dimensional
UV-critical hypersurface with real, integer critical exponents on the boundary and a complex
5 ∂ (0)For technical reasons, we could not resolve the eigenvector associated to θ2 of
D
NG− ⊕G -FP. Whether this
is a numerical problem or if it is related to the large negative or the fractional value of g∂ (0) and λ ∂ (0), respectively,
remains unclear.
302 Chapter 9. Investigating Asymptotic Safety
pair in the D-sector. The details of G∂ (0)⊕NGD+-FP read:
G∂ (0)⊕NGD-FP ∂ (0) ∂ (0)+ (g∗ ,λ∗ )=(0,0)
θr eigenvectors V ( j)
3.6+4.3ı −0.99êDg − (0.04+0.15ı)êDλ
− (9.37)3.6 4.3ı −0.99êDg − (0.04−0.15ı)êDλ
−2 0.87 ∂ (0)êg −0.49 ∂ (0)êλ
1 ∂ (0)êλ
The second combination results in the doubly non-Gaußian fixed point. As any non-trivial zero
of the beta-functions of the boundary part, the corresponding fixed point value of the Newton
coupling is negative, ∂ (0)g∗ . Thus, asymptotically safe trajectories evolve in the lower half-plane
g∂ (0) < 0 which is opposite to the bulk level-(0) as well as D-solutions. Apart from this subtlety,
the general structure of SUV is very similar compared to (9.34):
∂ (0)
NG D− ⊕NG+-FP ∂ (0) ∂ (0)(g∗ ,λ∗ )=(−2.14,1.20)
θr eigenvectors v(r)
3.6+4.3ı (0.3+0.1 ∂ (0) ∂ (0)ı)êDg +(0.003+0.01ı)ê
D
λ −0.9êg − (0.2−0.2ı)êλ
3.6−4.3ı (0.3−0.1ı)êDg +(0.003−0.01 ) D −0.9 ∂ (0)ı êλ êg −
(9.38)
(0.2+0.2 ∂ (0)ı)êλ
2 0.87 ∂ (0) −0.49 ∂ (0)êg êλ
3 ∂ (0)êλ
While the D-sector forces the trajectories to spiral in the vicinity of NGD+-FP, the real exponents
in the boundary part implies a linear translation along ∂ (0) and ∂ (0)êλ êg , respectively. This, as well as
sUV =, resembles the findings of the previous subsection where the level-(0) boundary couplings
are replaced by their bulk counterparts.
Hence, besides the canonical difference that also affects the non-Gaußian solutions, the
critical behavior of bulk and boundary sector on level-(0) are quite similar. Nevertheless, we
already discovered a very fundamental distinction between both cases, namely the half-spaces
of the Newton-couplings asymptotically safe RG trajectories live in. Furthermore, there are
subtle differences in the eigenvectors, where certain directions are combined with a relative
opposite sign compared to its bulk analog.
9.4 Conclusion
So far, the Asymptotic Safety conjecture for Quantum Gravity has undergone several, all suc-
cessful tests each revealing a non-trivial fixed point compatible with the classical theory of
General Relativity. Besides increasing the number of basis invariants to cover more and more
regions of theory space, one also studies the stability of the previous findings by deforming the
underlying assumptions or by employing conceptual extensions to the standard program. In this
thesis, we study two such conceptual different directions in theory space, one which modifies
the topological class of the underlying spacetime, allowing for manifolds with boundary, the
other addressing the intrinsically bi-metric character of the Effective Average Action. While
for technical reasons, most of our today’s understanding is based on so-called single-metric cal-
culations, which impose an a priori unjustified split-symmetry along the entire RG evolution,
recently there has been first evidence for the continued existence of the NGFP in a bi-metric
setting [172].
9.A Critical exponents of the background-sector 303
Remarkably, employing a new technique that should encourage further work beyond the
single-metric approximation, we recover the familiar NGFP reminiscent of the single-metric
Einstein-Hilbert results [122]. Searching for extensions including the boundary invariants, each
fixed point in the dynamical sector gives rise to a trivial and non-trivial solution for the boundary
level-(0) couplings. In combination with the level-(0) bulk couplings we obtain twelve critical
points of the RG flow, out of which those associated to NGD+-FP are the most interesting ones.
In order to understand the topological properties of asymptotically safe trajectories, we first
studied the general properties of the differential equations, revealing a separating line at gI = 0
which implies that for each RG solution the sign of Newton’s coupling is the same for all k. We
then utilized the hierarchical structure of the beta-functions obtained the fixed point values for
the optimized shape function and studied the possible combination of individual solutions to a
full fledged critical point in the six-dimensional truncated theory space.
For the boundary Newton coupling we observed that only negative non-trivial solutions can
occur, which is in contrast to the bulk sector where in fact gD is bounded to be positive by
experimental data. While numerically and also qualitatively different situated, the respective
directions of the UV-critical hypersurfaces are very similar for level-(0) on bulk and boundary.
In both directions only real integer critical exponents occur, while in the D-sector the especially
important NGD+-FP fixed point is related to complex scaling exponents resulting in the spiral
effect observed in fig. 9.1. Concerning the predictivity of the underlying theory, within this
truncation we revealed a correspondence between the number of Gaußian components in the
fixed point and the number of its irrelevant directions. While each non-trivial solution for a pair
(gI ,λ I) also requires two initial conditions, thus gives rise to positive real parts of θ only, a
trivial contribution (gI ,λ I) = (0,0) is UV-attractive (-repulsive) along êIg (or ê
I
λ ). For the very
∂ (0) (0)
important case of the threefold non-Gaußian fixed point NG− -NG+ -NG
D
+-FP this results in a
six-dimensional UV-critical manifold, being restricted only to the respective half-planes of the
Newton couplings. Hence, we may conclude that we found further evidence for the Asymptotic
Safety conjecture by severely deforming the standard truncations, exploring different topolog-
ical spacetimes and the study the effect of split-symmetry breaking on the existence of the
non-Gaußian fixed point. Its stability seems quite promising indicating that the so far observed
topological property might be in fact a feature of the exact flow equation. For the bi-metric-bulk
– pure-background-boundary truncation the Asymptotic Safety condition imposes the following
requirement on the solutions of FRGE to be asymptotically safe:
EH-GHY
SUV ≡ {k→ Ak[ϕ ;Φ̄;{gI ,λ I}] ∈ TEH-GHY |gD,g(0) > 0 ∧ g∂ (0) < 0}
Therefore, we indeed obtain dim(S EH-GHYUV ) = 6 and the present truncation includes four more
relevant invariants, compared to the single-metric Einstein-Hilbert truncation. In the next chap-
ter, we will combine the requirements of Asymptotic Safety and Background Independence to
form a reduction of S EH-GHYUV to the physical relevant part, including only those theories which
respect the fundamental requirement of Background Independence.
9.A Critical exponents of the background-sector
This appendix list the characterizing features of the UV-critical hypersurface for the six fixed
points in the background sector. It includes the critical exponents as well as the corresponding
304 Chapter 9. Investigating Asymptotic Safety
scaling fields.
GB⊕GD-FP (gB,λB∗ ∗ )=(0,0) NGB+⊕GD-FP (gB∗ ,λB∗ )=(1.40,−0.11)
θ (r) (r)r eigenvectors v θr eigenvectors v
−2 8π êD+ êD −2 8π D D3 g λ 3 êg + êλ (9.39)
2 +êD 2 +êDλ λ
−2 −4π êB+ êBg λ −2 −4π êBg + êBλ
2 êB 2 êBλ λ
GB⊕NGD−-FP B D(gB,λB∗ ∗ )=(0,0) NG+⊕NG−-FP (gB∗ ,λB∗ )=(1.53,−0.12)
θr eigenvectors v(r) θ eigenvectors v(r)r
2.21 1.0 êD +0.1 êD 2.21 −0.93êD −0.06êDg λ g λ −0.35êBg +0.04êBλ
5.12 − (9.40)0.6 êD +0.8 êD 5.12 0.4êD −0.6êDg λ g λ −0.4êB Bg +0.5êλ
−2 1.0 êBg −0.1 êBλ 2 1.00êB −0.08êBg λ
2 êB Bλ 4 êλ
NGB+⊕NGD+-FP (gB∗ ,λB∗ )=(8.18,−0.01)
θr eigenvectors v(r)
3.6+4.3ı −(1.6−0.2ı)10−2êD − (0.9+2.4ı)10−2êDg λ +1.0êBg+(1.7−2.3ı)10−2êBλ
3.6−4.3ı −(1.6+0.2ı)10−2êDg − (0.9−2.4ı)10−2êD +1.0êBλ g+(1.7+2.3ı)10−2êBλ
2 êBg−9×10−4êBλ
4 êBλ
GB⊕NGD+-FP (gB,λB∗ ∗ )=(0,0)
θr eigenvectors v(r)
3.6+4.3ı −0.99êDg − (0.04+0.15ı)êDλ (9.41)
3.6−4.3ı −0.99êDg − (0.04−0.15ı)êDλ
2 êBg−9×10−4êBλ
2 êBλ
10.1 Introduction
10.2 Can split-symmetry coexists with Asymp-
totic Safety?
10.3 Single-metric vs. bi-metric truncation: a
confrontation
10.4 An application: the running spectral di-
mension
10.5 A brief look at d = 2+ ε and d = 3
10.6 Summary and conclusion
10. BACKGROUND INDEPENDENCE
C This chapter follows closely section 1 and 5-9 of [173].
The most momentous requirement a quantum theory of gravity must satisfy is Background In-
dependence, necessitating in particular an ab initio derivation of the arena all non-gravitational
physics takes place in, namely spacetime. Using the background field technique, this require-
ment translates into the condition of an unbroken split-symmetry connecting the (quantized)
metric fluctuations to the (classical) background metric. If the regularization scheme used vio-
lates split-symmetry during the quantization process it is mandatory to restore it in the end at the
level of observable physics. In this chapter we present a detailed investigation of split-symmetry
breaking and restoration within the EAA approach to Quantum Einstein Gravity (QEG) with
a special emphasis on the Asymptotic Safety conjecture, based on the results of part II for the
bulk sector. In ref. [173] we demonstrate for the first time in a non-trivial setting that the two
key requirements of Background Independence and Asymptotic Safety can be satisfied simulta-
neously.This important result is shown by the existence of a subset of RG trajectories which are
both asymptotically safe and split-symmetry restoring: In the ultraviolet they emanate from a
non-Gaussian fixed point, and in the infrared they loose all symmetry violating contributions in-
flicted on them by the non-invariant functional RG equation. As an application, we compute the
scale dependent spectral dimension which governs the fractal properties of the effective QEG
spacetimes at the bi-metric level. Earlier tests of the Asymptotic Safety conjecture almost ex-
clusively employed ‘single-metric truncations’ which are blind towards the difference between
quantum and background fields. We explore in detail under which conditions they can be re-
liable, and we discuss how the single-metric based picture of Asymptotic Safety needs to be
revised in the light of the new results. We shall conclude that the next generation of truncations
for quantitatively precise predictions (of critical exponents, for instance) is bound to be of the
bi-metric type.
306 Chapter 10. Background Independence
10.1 Introduction
One of the key requirements every candidate for a quantum theory of the gravitational interac-
tion and spacetime geometry should satisfy is Background Independence. The theory’s basic
kinematical rules and dynamical laws should be formulated without reference to any distin-
guished spacetime such as Minkowski space, for instance. Rather, the possible states of a
‘quantum spacetime’ should be a prediction of the theory. In addition, it must provide us with
a set of special observables which, by means of their expectation values in a given state, ‘in-
terpret’ this state in terms of classical geometry, or a generalized notion thereof. Among them
the expectation value of the metric would play a significant role. If non-degenerate, smooth
and approximately flat, on large length scales at least, the underlying state might appear like a
classical spacetime macroscopically, possibly similar to the real Universe we live in. We can
then try to match the predictions against concrete measurements and observations.[72, 73, 75,
76, 93, 196–198]
However, in general one would also expect states without any interpretation in terms of
concepts from classical General Relativity, Riemannian geometry in particular. A simple exam-
ple are situations in which the metric has an expectation value which is degenerate, identically
vanishing, for instance. While the gravitational physics implied by such states is certainly very
different from the one we know, they might realize a ‘symmetric phase’ of gravity which ar-
guably is easier to understand than the broken phase we live in. In the latter, diffeomorphism
symmetry is broken down to the stability group of the metric expectation value, the Poincaré
group in the flat case.
There exist two fundamentally different approaches to deal with the requirement of Back-
ground Independence. They differ in particular in the way they deal with the rather severe
conceptual and technical difficulties which are caused by this requirement and are of a kind
never encountered in conventional matter field theories on Minkowski space:
A. The most obvious strategy is to literally employ no background geometry at all in setting
up the foundational structures of the theory, then work out its quantum dynamical con-
sequences, and try to find states on which appropriate geometric operators (metric etc.)
signal the existence of almost classical spacetimes at the expectation value level. Ex-
amples of such literally background independent settings include statistical field theory
models like the Causal Dynamical Triangulations (CDT) [94–98], and Loop Quantum
Gravity (LQG) [72, 75, 93]. Following this route, there are (at least) two characteristic
difficulties one has to cope with. First, since most of the traditional tools of quantum
(field) theory presuppose a rigid background metric, considerable conceptual problems
must be overcome. The proposed solutions are usually outside the realm of quantum field
theory. Often they encode the fundamental degrees of freedom in new types of variables
which replace the quantum fields, triangulated spaces or spin-foams being typical exam-
ples. Second, at the computational level the difficulty arises that an ab initio explanation
of a (non-degenerate, smooth, approximately flat) macroscopic metric has to bridge an
enormous gap of scales if one starts from a microscopic theory governing the ‘atoms of
geometry’. For Planck sized building blocks of spacetime, say, even the typical scales of
particle physics are about 20 orders of magnitude away. This calls for an application of
Wilson’s renormalization group but, again, most of the existing tools are inapplicable in
the background-free context [199, 200].
B. Rather than using literally no background at all, ‘Background Independence’ can also
be achieved in the diametrically opposite way, namely by actually taking advantage of
background structures in formulating and evaluating the theory, but making sure that all
possible backgrounds are treated on a completely equal footing. This latter requirement
implies in particular that the background geometry may appear only in intermediate steps
10.1 Introduction 307
of the calculations while predictions for observables may never depend on it [27].
This second approach is realized for instance in the present case employing the Effective
Average Action to study theories of quantum gravity [122]. In the simplest case of metric
gravity we use the background-quantum field split of the dynamical metric ĝµν , the integration
variable in the underlying functional integral, in the form ĝµν = ḡµν + ĥµν . Here ḡµν is the
arbitrary classical background metric, and ĥµν a nonlinear fluctuation which plays the role
of the quantum field, i.e. it is functionally integrated over (or promoted to an operator, in
the canonical formulation). Thanks to the presence of the arbitrary but, by assumption, non-
degenerate background metric ḡµν the original problem appears in a new, somewhat different
guise now: it consists in the conceptually easier task of quantizing the, now matter-like field
ĥµν ‘living’ on the classical background geometry given by ḡµν .
At this point the technical challenge resides in the fact that this background is completely
generic and enjoys no special symmetries in particular. However, the availability of ḡµν opens
the door for the application of a considerable arsenal of quantum field theory techniques, namely
basically all those that have been developed for matter systems on Minkowski space or on
curved classical backgrounds. The price we pay for this enormous advantage is that in the
ĥµν -theory we must have complete control over the ḡµν -dependence of all expectation values,
the n-point functions of ĥµν in particular. Their 1PI version, with an IR cutoff at the scale k, is
generated by the Effective Average Action Γk[hµν ; ḡµν ]. As usual this functional depends on the
expectation value field hµν ≡ 〈ĥµν〉, but also the background metric, ḡµν , which here acquires
the status of an indispensable second argument of Γk. The expectation value of the full metric
is
gµν ≡ 〈ĝµν〉= ḡµν + 〈ĥµν〉 ≡ ḡµν +hµν
Sometimes it is more natural to consider gµν and ḡµν , rather than the pair (hµν , ḡµν) as the
independent variables on which the EAA depends, and to set
∣
Γk[gµν , ḡ ∣µν ]≡ Γk[hµν ; ḡµν ] h=g−ḡ
In this formulation the intrinsic bi-metric character of the EAA becomes manifest. We refer
to the second argument of Γk[gµν , ḡµν ] as its extra ḡµν -dependence since, contrary to the ḡ-
dependence within gµν ≡ ḡµν + hµν , this ‘extra’ dependence does not combine with hµν to
form a full dynamical metric.
The Functional Renormalization Group Equation governs the k-dependence of Γk, [110–
114], and in principle the functional integral over ĝµν can be evaluated indirectly by solving
the FRGE instead [121]. For this to be possible, and the Functional Renormalization Group
Equation to exist in the first place, it is unavoidable to employ a running action functional
which admits an arbitrary extra ḡµν -dependence and, in fact, depends also on Faddeev-Popov
ghost fields ξ µ and ξ̄µ , respectively: Γk[gµν , ḡ µµν ,ξ , ξ̄µ]. (See ref. [122] for further details.)
The approaches A. and B. have complementary advantages and disadvantages. In A. the
strategy of literally avoiding any background, and starting from a ‘vacuum’ state with no space-
time interpretation at all, it is comparatively easy to describe a possible phase of unbroken
diffeomorphism invariance. But the corresponding broken phase is a very hard problem since it
is due to the cooperative effect of a huge number of ‘atoms of spacetime’. Conversely, in B., the
broken phase is the easier one since, at least when the background is chosen self-consistently
so that gµν = ḡµν holds true, the quantum fluctuations can be relatively weak, with vanishing
expectation value hµν ≡ 〈ĥµν〉 = 0. The symmetric phase, on the other hand, requires huge
quantum effects as gµν ≡ ḡµν + 〈ĥµν〉 is supposed to vanish identically even though ḡµν stays
always non-degenerate.
308 Chapter 10. Background Independence
In the present context we realize Background Independence in the second way, i.e., loosely
speaking, by quantizing the fluctuations of the metric in all possible background spacetimes
at a time. We are going to explore the RG evolution of the EAA with a particular emphasis
on how the amount of ‘extra’ ḡ-dependence the functional Γk suffers from depends on the RG
scale k. The extra ḡ-dependence is an entirely unphysical artifact present only at intermediate
computational steps. To establish Background Independence it must be turned to zero at the end.
The extra ḡ-dependence is most conveniently discussed in terms of the background quantum
split-symmetry, its breaking, and its restoration. This symmetry transformation changes hµν
and ḡµν according to
δ splitε hµν = εµν(x) , δ
split
ε ḡµν =−εµν(x) (10.1)
and it has an obvious action on the functionals1 Γk[h; ḡ] ≡ Γk[g, ḡ]. Here εµν(x) is an arbi-
trary symmetric tensor field. Clearly, the full dynamical metric g = ḡ+ h is invariant under
δ splitε , while ḡ is not. As a consequence, Γk is invariant under split transformations, δ
split
ε Γk = 0,
precisely when the two-metrics functional Γk[g, ḡ] is actually independent of its second argu-
ment δδ Γk[g, ḡ] = 0, or equivalently, when Γk[h; ḡ] happens to be a functional of the sum of itsḡ
arguments only, viz. ḡ+h≡ g.
In order to understand were the unavoidable extra ḡ-dependence of the EAA comes from
recall that the EAA derives from a functional integral which, after the split ĝµν → ḡµν + ĥµν ,
has the form [122]
∫ ∫ ( )
Dĥ µµν Dξ Dξ̄µ exp −S[ḡ+ ĥ]−Sgf[ĥ; ḡ]−Sgh[ĥ,ξ , ξ̄ ; ḡ]−∆Sk[ĥ,ξ , ξ̄ ; ḡ] (10.2)
While the (arbitrary) bare action S[ḡ+ ĥ] is indeed δ splitε -invariant, the same is not true for the
gauge fixing term Sgf, the ghost action Sgh, and in particular the cutoff action ∆Sk; it implements
the IR cutoff in the familiar way by mode suppression terms which are quadratic in the quantum
fields [110–114]. This shows that the very concept of an EAA, namely the idea to ‘coarse grain’
(that is, smoothly regularize in the infrared) a gauge-fixed functional integral hinges in a crucial
way on the availability of ḡµν : neither the gauge fixing, nor the ‘cutting out’ of the IR modes
necessary to derive a functional RG equation could be implemented without promoting the
background metric to an independent entity, different from gµν − hµν in general. That this
is indeed necessary is easily traced back to the gravitational field’s special status among the
carriers of the fundamental interactions, namely its very relation to the geometry of spacetime.2
We also recall that the EAA employs a gauge fixing condition which belongs to the spe-
cial class of background gauges, as a result of which Sgf + Sgh is invariant under the so-called
background gauge transformations, δ Bv , not be confused with the ‘quantum gauge transforma-
tions’ to be fixed, of course. Furthermore, ∆Sk is constructed so as to enjoy the same invariance
property. For a diffeomorphism generated by any vector field v, the background gauge trans-
formation δ Bv acts as the Lie derivative Lv on both the dynamical fields (ĝ, η̂ ,ηˆ̄ ) or (ĥ,C,C̄),
and on ḡµν . Ultimately this leads to an EAA which is an invariant functional of its arguments,
δ Bv Γk[h,ξ , ξ̄ ; ḡ] = 0, and its RG flow preserves this property.
In the limit k → 0, where the IR regulator is removed, Rk and ∆Sk vanish by construc-
tion, and as a result Γ ≡ limk→0 Γk coincides with the ordinary effective action, for the specific
(background-type) gauge chosen.
1We suppress the ghosts when they are inessential for the discussion. On them, δ splitε ξ
µ = 0 and δ splitε ξ̄µ = 0.
2Note that there exists no analogous ‘Background Independence’ issue in Yang-Mills gauge theories on
Minkowski space. There, an ‘extra Āµ -dependence’ can always be avoided in a trivial way, namely by simply not
using a gauge fixing condition involving a background gauge field, whereupon no such field will appear anywhere.
Not so in gravity: even if we were to give up background gauge invariance and use a ḡµν -free gauge fixing condition,
a mode suppression term ∆Sk with the necessary properties could not even be written down without having a second
metric at our disposal [122].
10.1 Introduction 309
While Γ, like Γk at any k > 0, is perfectly δ Bv -invariant, it is still not δ
split
ε -invariant: While
one source of split-symmetry violation, the one due to ∆Sk, disappears at k = 0, the other, the
gauge fixing and ghost sector, still precludes complete δ splitε -invariance. However, in a sense,
this is a very weak violation since it concerns the gauge modes only, and should disappear, too,
upon going on-shell [68, 201–205].
Nevertheless, at intermediate scale k > 0 the RG flow generates in principle all possible,
generically δ splitε -violating functionals Γk[g, ḡ,ξ , ξ̄ ] of four independent arguments. They are
constrained only by their built-in δ Bv -invariance, and proper approximation schemes for solving
the functional flow equation such as truncations of theory space must take account of this fact.
To date, almost all available RG studies of the Asymptotic Safety scenario still involve the
same type of approximation to the exact EAA, the so-called single-metric truncation, which
was used very early on as the first testing ground for the gravitational Functional Renormaliza-
tion Group Equation [122],
[
1 ( ) ]−1
∂tΓk[g, ḡ,ξ , ξ̄ ] = STr Γ
(2)+Rk ∂tRk , (10.3)
2
Truncations of this general class project the RG flow implied by the (exact!) equation (10.3) on
the subspace spanned by functionals of the form
Γk[g, ḡ,ξ , ξ̄ ] = Γ
sm
k [g]+ (Sgf+Sgh)[g, ḡ,ξ , ξ̄ ] (10.4)
While Γsmk is a generic (diffeomorphism invariant) functional of the dynamical metric only,
Sgf and Sgh are the classical gauge fixing and ghost actions3, depending on the expectation
value fields now. An example of (10.4) is the Einstein-Hilbert tr∫u√ncation [1∫22√, 135, 191] in
which Γsmk [g] is specialized further to contain the two invariants gR and g, only, with
k-dependent pre-factors involving a running Newton constant Gk and cosmological constant λ̄k,
respectively.
Because of their quite substantial technical complexity, the work on more general trunca-
tions that would go beyond (10.4) started only recently. In [136] a first ‘bi-metric’ truncation
with separate gµν and ḡµν -dependence was analyzed in conformally reduced gravity and, in
[137], for matter induced gravity (in the large N-limit). The first bi-metric investigation of fully
fledged Quantum Einstein Gravity employed a truncation ansatz with two separate Einstein-
Hilbert terms for the dynamical (‘D’) and the background (‘B’) metric [172]. It consists of the
following ‘graviton’ (‘grav’) part, added to the classical gauge fixing and ghost terms whose RG
evolution is still neglected :
∫ ( ) ∫ √ ( )
Γgrav
1 √ 1
k [g, ḡ] =− ddx g R(g)−2λ̄D − ddk x ḡ R(ḡ)−2λ̄Bk (10.5)16πGDk 16πGBk
(Generalizing the truncation in a different direction, single-metric ansätze with a running ghost
sector were studied in [132–134].)
This chapter is devoted to a detailed investigation of the RG flow related to the bi-metric
Einstein-Hilbert truncation using the RG equations for the bulk invariants derived in the previ-
ous part and thus based on the (bulk) beta-functions listed in chapter 8. The analysis will be far
more comprehensive than the preliminary one in ref. [172]. We shall be specifically interested
in global properties of the flow, in particular the question as to whether Asymptotic Safety can
coexist with Backround Independence, that is, the restoration of split-symmetry at the phys-
ical level. This question is not easy to answer as it requires control over the fully extended
RG trajectories, their limits k → ∞ and k → 0 in particular. Coexistence of Asymptotic Safety
3With the running field renormalization of hµν as implied by Gk included usually.
310 Chapter 10. Background Independence
with Background Independence implies that there exists at least one RG trajectory k 7→ Γk[g, ḡ]
which is non-singular for all k ∈ [0,∞), and which approaches a non-Gaussian fixed point at its
upper end, i.e. for k→ ∞, while at the lower end, in the ‘physical’ limit k→ 0 where the EAA
equals the ordinary effective action, it is split-symmetry-restoring, i.e. there Γk looses its extra
ḡ-dependence.4 In particular on a truncated theory space it is by no means obvious that such
trajectories do indeed exist and both requirements can be met simultaneously.
One of our main results will be that within the bi-Einstein-Hilbert truncation Asymptotic
Safety and Background Independence can indeed coexist: Some, but not all RG trajectories
which emanate from a non-Gaußian fixed point in the UV also restore split-symmetry in the
IR. It will be instructive to uncover the very elegant mechanism of how the concrete RG
differential equations bring about this symmetry restoration; we shall see it involves a moving
attractor in the background part of theory space. We shall see that from a 4-parameter family
of asymptotically safe trajectories a 2-parameter subset is symmetry restoring.
The existence of this subset of symmetry restoring trajectories is good news for the Asymp-
totic Safety program for two independent reasons: First of all, it shows that at least in this
truncation Asymptotic Safety and Background Independence are not mutually exclusive. Sec-
ondly, since a physically meaningful theory can only be based on a RG trajectory from this
subset it follows that the predictive power of the Asymptotic Safety scenario is actually higher
than what one would expect by just counting relevant perturbations at the fixed point; the indis-
pensable subsidiary condition of Background Independence narrows down the possibilities of
constructing physically inequivalent theories at a given fixed point.
Recently the crucial importance of split-symmetry has also been demonstrated in a very im-
pressive way within a 3-dimensional scalar toy model [206]: being careless about its restoration,
one can even destroy the Wilson-Fisher fixed point!
Most of this chapter is devoted to an analysis of the beta-functions in d = 4 elaborately de-
rived in part II. In section 10.2 we first present a detailed analysis of the global properties of the
RG flow. In particular we demonstrate that Asymptotic Safety and Background Independence
are indeed compatible by explicitly constructing RG trajectories which restore split-symmetry
in the IR. We explain how the special properties of these trajectories come about and we un-
cover the role played by a moving attractor point in theory space.
Section 10.3 contains a very detailed comparison of the bi-metric Einstein-Hilbert trunca-
tion with its single-metric approximation. The goal of this section is to find out under what
conditions the latter is sufficiently reliable, and under what circumstances it is mandatory to
employ the former. We shall find the somewhat ‘miraculous’ result, unexplained by any gen-
eral principles, that the single-metric approximation seems to perform best near a non-Gaussian
fixed point. Since section 10.3 is rather technical, and mostly intended to establish the lessons
one can learn from the calculation of part II and which can help to design future truncations
optimally, this section can be skipped by the reader who is mostly interested in the new results.
Next, we consider a first application of the bi-metric RG equation in section 10.4: we
compute the scale dependent spectral dimension of the effectively fractal spacetimes QEG gives
rise to.
In section 10.5 we briefly comment on the bi-metric RG flow in d = 2+ ε and d = 3
spacetime dimension. Near 2 dimensions, where all Newton constants become dimensionless,
their beta-functions contain a universal leading term which we compute. The resulting universal
discrepancies between the single- and the bi-metric results are rather striking. In this setting it
becomes particularly obvious that single-metric approximations can be very misleading even at
the qualitative level. Subsequently, we conclude this chapter by giving a short summary and
discussion of the presented results.
4Except the one due to the gauge modes, to be precise.
10.2 Can split-symmetry coexists with Asymptotic Safety? 311
10.2 Can split-symmetry coexists with Asymptotic Safety?
We are now going to study the fully fledged RG flow on the four dimensional theory space of
the bulk sector by analyzing the following system of coupled differential equations with both
analytical and numerical methods:
[ ]
∂ gD = βD(gD,λD)≡ d−2+ηD(gD,λD) gDt k g k k k k k (10.6a)
∂ λD = βD(gD Dt k λ k ,λk ) (10.6b)[ ]
∂ gBt k = β
B
g (g
D,λDk k ,g
B
k )≡ d−2+ηB(gD Dk ,λk ,gB) gBk k (10.6c)
∂ λ B B D D B Bt k = βλ (gk ,λk ,gk ,λk ) (10.6d)
In this section we are mostly interested in the global properties of the RG flow. In particular we
investigate to what extent split-symmetry at k = 0 can be realized by a judicious choice of the
trajectory’s ‘initial’ conditions. In practice they are imposed at an intermediate scale k0, and the
differential equations are solved then both in the upward and downward direction. The all deci-
sive question we try to answer is: Do there exist RG trajectories for which split-symmetry,
i.e. Background Independence for k→ 0 coexists with Asymptotic Safety? What makes this
question hard is its global nature: Background Independence and Asymptotic Safety concern
exactly the opposite ends of the RG trajectories, the limits of very small and very large scales,
respectively.
Since the D-couplings are not affected by the B-sector but, conversely, enter the B-beta-
functions, we first solve the eqs. (10.6a) and (10.6b) for gDk and λ
D
k , and then substitute the
solutions into the beta-functions of gB and λ Bk k in eqs. (10.6c) and (10.6d) to obtain the ‘B’-
components of the trajectory. We describe the results of the two steps in turn.
10.2.1 Trajectories of the ‘D’ sub-system
The set of solutions for the B-independent (gDk ,λ
D
k )-system (10.6a), (10.6b) decomposes into a
subset with positive gDk for all k, one with g
D
k < 0 always, and a single trajectory with g
D
k = 0
∀k that separates the two regions. The sign of the Newton coupling never changes along any
trajectory. From chapter 9 we know already that the D-system allows for three fixed points: the
Gaussian one, GD-FP, which is located on the separating trajectory, and two non-Gaussian ones,
NGD D−-FP and NG+-FP, that lie below (g
D < 0) or above it (gD > 0), respectively. We will focus
in the sequel on the positive gD domain.
Fig. 10.1 shows the phase portrait on the gD-λD plane which we obtained numerically.5 We
see that it is impressively similar to the well-known phase portrait of the single-metric Einstein-
Hilbert truncation [191].
Following ref. [191] the trajectories in the upper half-plane in Fig. 10.1 can be classified
in the same way as their single-metric relatives, namely as of type (Ia)D, type (IIa)D, and type
(IIIa)D respectively, depending on whether the cosmological constant λDk approaches −∞, 0,
or +∞ in the far IR, i.e. for k → 0. The type (IIa)D solution separates the two regimes and
is, henceforth, also called separatrix. (In Fig. 10.1 it is represented by a dashed line, and we
have also highlighted a representative of the other types, a (Ia)D and a (IIIa)D trajectory.) The
separatrix ‘crosses over’ from the NGD D D+-FP in the UV to the G -FP in the IR. Notice that NG+-
FP is UV-attractive in both directions, and thus all trajectories are pulled into this fixed point
when k → ∞. Due to the imaginary part of the critical exponents they form spirals. Further
details on the fixed point structure can be found in chapter 9.
5Here and in all similar diagrams the arrows always point from the UV towards the IR.
312 Chapter 10. Background Independence
gD
1.2
1.0
0.8
0.6
0.4
0.2
D
0.0 λ
-0.1 0.0 0.1 0.2 0.3 0.4
Figure 10.1: The phase portrait of the bi-metric ‘D’-sector. The vertical (horizontal) axis corre-
sponds to gD (λD). Note the remarkable similarity with the phase portrait of the single-metric
Einstein-Hilbert truncation [191].
10.2.2 Solving the non-autonomous ‘B’ system
Each one of the D trajectories obtained above can now be substituted into the two RG equations
of the B- or level-(0) couplings. After fixing initial conditions they can be solved to give the
B/(0)
k-dependence of the two remaining coordinates of the 4-dimensional trajectories, namely gk
B/(0)
and λk .
Let us start by investigating their qualitative behavior for arbitrary initial conditions in the
B-sector. Once a solution (gDk ,λ
D
k ) is picked, the beta-functions in eqs. (10.6c), (10.6d) for the
B-couplings are polynomials with known, but scale dependent coefficients AB(k)≡ AB(gDk ,λDk )
and BB1(k)≡ BB(gD1 k ,λDk ):
∂ gB B B B B B B 2t k = βg (gk ,λk ;k) = 2gk +B1(k)(gk ) (10.7a)
∂ λ Bt k = β
B
λ (g
B
k ,λ
B
k ;k) =−2λ Bk +AB(k)gBk +BB1(k)gBk λ Bk (10.7b)
It is important to appreciate that when the functions AB(k) and BB1(k) are given, the eqs. (10.7)
form a closed coupled system for the two remaining (gBk , λ
B
k ) which, however, contrary to that
for (gDk , λ
D
k ), is not autonomous. The beta-functions on the RHS of eqs. (10.7a) and (10.7b)
possess an explicit dependence on k. Hence the vector field on the gB-λ B-plane they give rise to
and, as a consequence, the entire phase portrait on this plane are RG-time dependent. This will
complicate the analysis considerably.
The other fact to be appreciated is that the beta-functions of (10.7) are polynomial in the
unknowns gB and λ B. This is in sharp contradistinction to the dynamical sector which involves
complicated threshold functions pΦ Dn (−2λ ). For later use we also mention that, in terms of the
dimensionful quantities 1/GB and λ̄ B/GB, the system (10.7) has the following general solution:
10.2 Can split-symmetry coexists with Asymptotic Safety? 313
∫
1 1 k
= − dk′ k′BB(k′1 ) (10.8a)GB Bk Gk k0 0
λ̄ B λ̄ B
∫ k
k k= 0 + dk′ k′3AB(k′) (10.8b)
GB Bk Gk k0 0
Here, as always, k0 denotes an arbitrary (initial, or intermediate) scale somewhere along the
trajectory.
The ‘B’ Newton constant. Returning to dimensionless quantities the hierarchy among the
two equations (10.7) allows to first solve (10.7a) for gBk , and then determine the k-dependence
of λ Bk from (10.7b). The differential equation (10.7a) is of Bernoulli type. It determines g
B
k and
is independent of λ Bk . Besides the trivial solution g
B
k = 0 there exists a non-trivial one, namely
k2 gB
gB
k0
k = ∫ (10.9)k
k2−gB ′ B ′ ′0 k k k B (k )dk0 0 1
Let us demonstrate that the solution (10.9) is regular everywhere. The denominator in eq.
(10.9) does not vanish at any k. For this to happen the k′-integral in (10.9) would have to
be positive. It turns out however that BB(k) = BB(gD1 1 k ,λ
D
k ) in∫the integrand is positive in the
small interval λD ∈ [0.213,0.5] only; furthermore the integral k0 k′BB(k′k 1 )dk′ in (10.9) remains
negative even in the corresponding possibly ‘dangerous’ regimes, as for example the IR branch
for the type (IIIa)D∫ trajectories or the spirals in the vicinity of NG
D
+-FP. Fig. 10.2 shows the
decrease of k k′BB(k′)dk′k 1 as a function of k towards the UV, and the fact that it is negative in0
the far IR, for the three different types of trajectories in the D-sector. In fact, in the UV – when
0 0
-5000
-5000
-10000
-10000
∫ -15000 ∫
-15000 kBB B1 (k) kB1 (k)
-20000
IR k UV IR k UV
(a) Type (Ia)D (b) Separatrix
0
-5000
-10000
∫
-15000 kBB1 (k)
IR k UV
(c) Type (IIIa)D
∫
Figure 10.2: The function →7 kk ′ Bk k B1(k′)dk′ for representative examples of the three types0
of trajectories. The function is seen to be negative throughout, demonstrating that the solution
(10.9) is regular everywhere.
314 Chapter 10. Background Independence
the D-trajectories spiral into NGD+-FP – the denominator in (10.9) becomes very large so that
the k20 term can be neglected. Then (10.9) approaches
k2
B UVgk = ∫ −
k−→−→∞ 8.18 ≡ gB (10.10)
− k ′ ′ ′ ∗k k BB1(k )dk0
This limit equals precisely the NGB+-FP-value of g
B
k for any initial value g
B
k . This shows that in0
the UV, and for any trajectory,
lim BB ′1(k ) =C with |C|< ∞ and C < 0 (10.11)
k′→∞
should hold true, where C is some finite constant.
The ‘B’ cosmological constant. Upon inserting the solutions of the D-sector and gBk we
obtain from (10.7b) a single linear, inhomogeneous ODE with scale-dependent coefficients
which determines λ Bk :
[ ]
∂tλ
B = ABk (k)g
B B
k + B1(k)g
B
k −2 λ Bk (10.12)
The coefficient functions AB(k) and BB1(k) are fixed once initial conditions are imposed on g
D
k
and λDk . The solution to eq. (10.12) reads then
( )
B 4 [ gB ∫g k ]
λ B = k
k0 k
λ B 0 ′ ′3 B ′k gB k k
+ 4 dk k A (k ) (10.13)0
k k k0 0 0
where for gBk the expression (10.9) is to be inserted. Inside the square brackets on the RHS of
(10.13) we can distinguish two contributions, of first and of zeroth order in λ Bk , respectively. The0
moment the denominator in (10.9) starts increasing rapidly, gBk becomes almost independent of
the initial value gBk . This rende(rs the te)rm in (10.13) which is of zeroth order in λ
B
0 k
completely
0
independent of the initial data λ Bk , g
B
0 k
.
0
In Fig. 10.3 we display the k-dependence of both contributions separately. From the dia-
grams we conclude that the function k→7 λ Bk is indeed independent of the initial value gBk over0
a large range of scales. For those∫k-values the constant contribution, in the approximation of eq.3 B
(10.10) given by the fraction − k∫ A (k)2 B , starts dominating over the λ Bk -linear term already atk kB1 (k) 0
small scales k and finally approaches the NGB+⊕NGD+-FP value of λ B in the UV.
Summary. Recalling that above the same qualitative property was found for gBk also, we
see that under upward evolution, all solutions k 7→ (gB,λ Bk k ) of the B-sector ‘forget’ their values
at k = k0 and converge to a single trajectory ultimately hitting NG
B
+⊕NGD+-FP when moving
towards the UV.
This behavior is related to the observation that for k→ ∞ all solutions in the D-sector spiral
into NGD+-FP (assuming, as always, g
D
k > 0). This fact is a global, and nonlinear extension of0
the above linear analysis, yielding 2 attractive directions at NGD+-FP. Since all such solutions
share the same fate in the UV, the differential equations for the B-couplings, too, become
independent of (gD Dk , λk ) in the UV. In principle the B-couplings could still depend on their0 0
own initial values, (gB , λ Bk k ). However, we saw that the influence of g
B
k and λ
B
k is significant0 0 0 0
only in the IR. Hence, we discover that the UV attractivity of NGB+⊕NGD+-FP in all 4 directions
which previously was established on the basis of the critical exponents at the linearized level
only, possesses a nonlinear extension: The RG flow on the 4-dimensional theory space has
10.2 Can split-symmetry coexists with Asymptotic Safety? 315
0. 0.
( ) 0. 0.0
0.1 B ∝λB 0.1 ( )- ∝ λ - 0
k k0 0 -0.00005 ∝ λB -0.05k0 ∝λB
k0
-0.2 -0.2 -0.0001 -0.1
-0.00015 -0.15
-0.3 -0.3
-0.0002 -0.2
-0.4 -0.4
-0.00025 -0.25
IR k UV IR k UV
(a) Type (Ia)D (b) Separatrix
0. 0.
-0.025 ( )0 -0.05
∝ λB
k0
-0.05 -0.1
∝λB
-0.075 k0 -0.15
-0.1 -0.2
-0.125 -0.25
-0.15 -0.3
IR k UV
(c) Type (IIIa)D
Figure 10.3: The two contributions to the function λ Bk which are constant and linear in λ
B
k ,0
respectively, see eq. (10.13). Shown is the regime of scales when the denominator in (10.9)
becomes large. The vertical scales on the LHS (RHS) of the diagrams correspond to the linear
(constant) contribution.
the global property that all trajectories approach the doubly non-Gaussian fixed point under
upward evolution:
(gD D B Bk ,λk ,gk ,λk ) −
k−→−→∞ NGB+⊕NGD+-FP for all (gDk > 0, λD , gB > 0, λ B )0 k0 k0 k0
To be precise, the fixed point’s ‘basin of attraction’ consists of all points with positive dynamical
and background Newton constants.
10.2.3 The running UV attractor in the B-sector
We saw that in the D-sector the differential equations are autonomous: The beta-functions are
invariant under translations in the RG-time t, and we obtain a time independent phase portrait
in the D-sector, see Fig. 10.1. However, after choosing initial values (gDk , λ
D ) and inserting the
0 k0
resulting Dyn-solution into the ‘B’-equations, their translation invariance gets broken and the
B-system becomes non-autonomous, depending explicitly on k. As a result, the 2 dimensional
phase portrait on the gB-λ B plane is explicitly ‘time’ dependent.
We shall nevertheless be able to deduce the essential qualitative properties of the phase por-
trait by analytical methods. Its structure is essentially determined by the k-dependent analogue
of fixed points in the B-system.
The ‘moving fixed point’. We consider the two equations (10.7) for gBk and λ
B
k with given,
externally prescribed coefficient functions AB(k) and BB1(k) and search for k-dependent points
(gB,λ B• • ) on the g
B-λ B-plane at which both beta-functions vanish simultaneously:
( )
β Bg g
B
•(k),λ
B
• (k);k = 0( )
β B gBλ •(k),λ
B
• (k);k = 0 (10.14)
316 Chapter 10. Background Independence
As long as BB1(k) 6= 0, there exist two solutions to these equations.
A. The first one is trivial, gB• = 0 = λ
B
• , and happens to be independent of k. As a con-
sequence, we expect a 4-dimensional Gaussian fixed point (gD, λD∗ ∗ , g
B B
∗, λ∗ ) = 0 to be
present in all phase diagrams.
B. The second solution is non-trivial and explicitly k-dependent:
gB B•(k) =−2/B1(k) and λ B• (k) =− 12A
B(k)/BB1(k) (10.15)
Its k-dependence is shown in Fig. 10.4 for three typical D-trajectories, one of each
type, which determine AB(k) and BB1(k). Thus, for k → ∞ the ‘running fixed point’
(gB(k),λ B• • (k)) approaches a true one, namely NG
B
+-FP.
1.25 12.5
← → ←IR UV→IR UV
gB1. (k)
g•B
10. •
(k) 0.1 10.
0.75 7.5
0.05 5.
0.5 5.
0.25 2.5 0. 0.
0. 0. λ•B0.05 (k)- -5.
λ•B(k)
-0.25 -2.5
-0.1 -10.
k k
(a) Type (Ia)D (b) Separatrix
1.25 12.5
←IR UV→
1. 10.
0.75 7.5
g•B0.5 (k) 5.
0.25 2.5
0. 0.
λ•B(k)
-0.25 -2.5
k
(c) Type (IIIa)D
Figure 10.4: The k-dependence of the running UV attractor (gB•(k),λ
B
• (k)) for three repre-
sentative D trajectories. The left (right) scale corresponds to λ B• (g
B
•). The running attractor
(gB•(k),λ
B
• (k)) approaches the NG
B
+⊕NGD+-FP coordinates in the UV and finite, non-vanishing
values in the IR. In between the running UV attractor touches the boundary of theory space
as can be seen from the divergent coordinates. Notice that (gB(k),λ B• • (k)) is the position of the
‘sink’ the inverse RG flow in the B-sector is pointing to, but not a solution to the RG equations.
The UV behavior of (gB•(k),λ
B
• (k)) is seen to be the same for all underlying D trajectories
(gD,λDk k ), namely
( )
gB
k→∞
(k),λ B(k) −−−→ NGB-FP for all (gD• • + k > 0,λDk ,gBk > 0,λ Bk ) (10.16)0 0 0 0
Let us consider the embedding of the running fixed point into the 4-dimensional theory
space. It moves along a curve parametrized by
( )
u (k)≡ gD,λD• k k ,gB•(k),λ B• (k) (10.17)
Note that the curve k 7→ u•(k) is not an RG trajectory. Since all D trajectories approach NGD+-
FP for k→ ∞, it is clear that u•(k) approaches NGB+⊕NGD+-FP in the UV. However, its global
properties depend significantly on the type of the D-trajectory:
10.2 Can split-symmetry coexists with Asymptotic Safety? 317
A. The type (IIa)D-trajectory, the separatrix, for instance, describes a cross-over:
D ←k→−−0 k→∞G -FP (gDk ,λDk )−−−→ NGD+-FP (10.18)
Hence the resulting curve (10.17) connects two fixed points, namely the ‘doubly non-
Gaussian’ one in the UV, and the mixed NGB+⊕GD-FP in the IR:
NGB+⊕
k→0 k→∞
GD-FP←−− u•(k)−−−→ NGB+⊕NGD+-FP (10.19)
B. For type (Ia)D and (IIIa)D trajectories, where λD for k→ 0 goes to −∞ and +∞6, respec-
tively, the point (10.15) approaches, in the IR,
( ) ( )λD→±
gB, λ B• • −−−−−→
∞ 3π 2
,− (10.20)
5 5
At a certain intermediate scale (0 < k < ∞), the running attractor touches the boundary of
theory space, i.e. it is pulled to infinity7 , then returns to the interior of theory space, and finally
moves towards NGB+-FP, under upward evolution. This behavior is clearly seen in the diagrams
of Fig. 10.4.
Stability of the ‘moving fixed point’. Let us linearize the B-system (10.7) about (gB•,λ
B
• )
and deduce a k-dependent analogue of a stability matrix, B(gB•,λ
B
• ;k). This matrix turns out
(1/2)
to have two k-dependent eigenvectors V• (k), associated to the ‘would-be critical exponents’,
(1) (2)
i.e. its negative eigenvalues θ• = 2 and θ• = 4, respectively:
( )
−2 0
B(gB,λ B
(1) (2)
• • ;k) = , V
B B B B
1 − •
(k) = 4êg +A (k) êλ , V• (k) = êλ (10.21)
2A
B(k) 4
(1) (2)
Here, êB Bg and êλ are unit vectors in the g
B- and λ B-direction, respective(ly. As)both θ• and θ•
are found to be positive we may conclude that at all scales the point gB B•,λ• is UV-attractive
in both B-directions. No matter which initial conditions we choose for the B-couplings, under
upward evolution the B-trajectories k 7→ (gB,λ Bk k ) are always pulled towards this point for k→ ∞.
Therefore we shall refer to it as the running UV attractor and denote it AttrB or AttrB(k) in the
following.
Global structure of the B-flow. For k → 0 the attractor property AttrB implies that all
B-trajectories converge to the values (10.20) in the (Ia)D and (IIIa)D cases, and to NGB⊕GD+ -FP
in the case of the separatrix (IIa)D.
Under upward evolution, when AttrB(k)moves due to an increasing k, the B-trajectories try
to follow its motion until they all end up in the doubly non-Gaussian fixed point NGB+⊕NGD+-FP
for k→ ∞. This behavior is shown in the phase portraits of Figs. 10.5 - 10.7 for representative
type (Ia)D, (IIa)D, and (IIIa)D trajectories, respectively.
Each one of the ‘snapshots’ displayed in Figs. 10.5 - 10.7 is structured as follows. The
‘current RG-time’ can be inferred from the position of the five-pointed star on the underlying
D-trajectory; it is sketched inside the small box on the right of th(e phase)portrait. The arrows in
this larger diagram represent the 2-component vector field ~βB ≡ β Bg ,β Bλ on the gB-λ B-plane at
6For a moment we ignore the singularity in the beta-functions at λD = 1/2.
7Note that there is nothing wrong with diverging values of gB• and λ
B
• at some k since, as we stressed already, the
curve followed by the ‘running fixed point’ is not an RG trajectory. A divergent gB• and /or λ
B
• simply means that at
this special value of k there exists no such fixed point.
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) ÷ SolB (k)(0,0) (0,0)
6 ì SolB• (k) 6 ì SolB• (k)
gB gB
4 4
ø
ø
2 2
ì ì
0 ÷ 0 ÷
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) B(0,0) ø ÷
Sol (k)
(0,0)
6 ì Sol•B(k) 6 ì SolB• (k) ø
gB gB
4 4
2 2
ì
ì
0 ÷ 0 ÷
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8 ìì ÷
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) SolB (k)(0,0) ÷ (0,0)
6 ì Sol•B(k) 6 ì SolB• (k)
ø ø
gB gB
4 4
÷
2 2
0 0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
Figure 10.5: The B-phase portraits at increasing scales k. The underlying type (Ia)D trajectory
in the D-sector is shown in the inset on the right, and the current RG time is marked with a
star therein. The arrows point towards the IR and picture the instantaneous vector field in the
B-sector. The (red) solid and the (gray) dashed curve highlight two important solutions in the
B-sector, namely SolB• (k) and Sol
B
(0,0)(k), respectively. Their current position is indicated by the
(green) diamond and the (violet) six-pointed star, respectively.
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k)(0,0) ÷
SolB (k)
(0,0)
6 ì SolB• (k) 6 ì SolB• (k)
gB gB
4 4
ø
2 2
ì ø ì÷
÷
0 0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) SolB (k)(0,0) ÷ø (0,0)
6 ì SolB• (k) 6 ì SolB• (k) ø
gB gB
4 4
2 2 ì÷
ì÷
0 0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8
ì÷
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) SolB (k)(0,0) ÷ (0,0)
6 ì SolB• (k) 6 ì SolB• (k)
ø ø
B ì÷g gB
4 4
2 2
0 0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
Figure 10.6: The B-phase portraits at increasing scales k. The underlying separatrix in the D-
sector is shown in the inset on the right, and the current RG time is marked with a star therein.
The arrows point towards the IR and picture the instantaneous vector field in the B-sector. The
(red) solid and the (gray) dashed curve highlight two important solutions in the B-sector, namely
SolB• (k) and Sol
B
(0,0)(k), respectively. Their current position is indicated by the (green) diamond
and the (violet) six-pointed star, respectively.
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) ÷ SolB (k)(0,0) (0,0)
ø
6 ì SolB• (k) 6 ì SolB• (k)
gB gB
4 4
2 2
ì ø ì
0 ÷ 0 ÷
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8
ø RG-‘time’ k ø RG-‘time’ k
÷ SolB (k) ÷ SolB (k)(0,0) ø (0,0)
6 ì SolB• (k) 6 ì Sol•B(k) ø
gB gB
4 4
2 2 ì
ì
0 ÷ ÷0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
8 8 ì÷
ø RG-‘time’ k ø RG-‘time’ k
ì
÷ SolB (k) SolB (k)(0,0) ÷ (0,0)
6 ì Sol•B(k) 6 ì SolB• (k)
ø ø
gB gB
4 4
÷
2 2
0 0
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
λ B λ B
Figure 10.7: The B-phase portraits at increasing scales k. The underlying type (IIIa)D trajectory
in the D-sector is shown in the inset on the right, and the current RG time is marked with a
star therein. The arrows point towards the IR and picture the instantaneous vector field in the
B-sector. The (red) solid and the (gray) dashed curve highlight two important solutions in the
B-sector, namely SolB• (k) and Sol
B
(0,0)(k), respectively. Their current position is indicated by the
(green) diamond and the (violet) six-pointed star, respectively.
10.2 Can split-symmetry coexists with Asymptotic Safety? 321
this particular instant of time. The instantaneous integral curves of this vector field are shown,
too; because of its time dependence, those integral curves are no RG trajectories, however.
Furthermore, information about a single, especially interesting RG trajectory is provided
by the time dependent location of the six-po(inted s)tar which can be seen in all snap shots. It
indicates the current position of the solution gBk , λ
B
k of the RG equations which is fixed by the
‘final condition’ lim → (gBk 0 k ,λ
B
k ) = (0,0). This solution is denoted Sol
B
(0,0)(k) in the diagrams.
The dashed curve, for clarity shown in the plots at any time, is the set of points visited by
SolB(0,0)(k) for 0≤ k < ∞. It is a true RG trajectory, i.e. a solution to the eqs. (10.7).
Similarly, we included in all phase portraits a (red) curve that shows another g(enuine R) G
trajecto(ry, denoted Sol
B
) • (k). It is the solution picked by the ‘final condition’ limk→0 g
B B
k , λk =
lim → gB(k), λ Bk 0 • • (k) . In other words, under upward evolution it grows out of the UV attractor,
with which it coincides at k = 0.
As time elapses from k = 0 to k = ∞, we expect SolB B• (k) and Attr (k) to follow different
paths, since the former curve is an RG trajectory, while the latter is not. This characteristic
feature can be clearly observed in the diagrams. The trajectory SolB• (k) ultimately gets pulled
into the doubly non-Gaussian fixed point for k→ ∞:
AttrB(0)⊕ (· · · )D ←k→−−0 k→∞SolB• (k)−−−→ NGB+⊕NGD+-FP
Here (· · · )D stands for the various possible IR regimes of the underlying dynamical trajectory.
The RG-trajectory SolB• (k) is an especially important one since all trajectories converge to-
wards SolB• (k) when k is increased.
In the snapshots of Fig. 10.6, obtained from the D-separatrix, this convergence is clearly
seen to occur for the trajectory which, under upward evolution, begins in the doubly Gaussian
fixed point GB⊕GD-FP at k= 0. This solution coincides with the attractor’s position already at
a rather low RG-scale where (gB•,λ
B
• ) has not yet moved much and seems to be k-independent.
Only after both trajectories have merged, (gB• ,λ
B
• ) actually starts running rapidly, and the re-
maining evolution towards k→ ∞ can be entirely described by the red curve that sits on top of
the dashed one.
In the plots the attraction towards (gB B•,λ• ) is well visible since the running UV attractor is
first heading for infinity in the vertical direction, then returns, moves to λD ≈ 0, and lowers its
gD value until it ultimately reaches NGB+-FP. This motion of the running attractor reflects itself
in the bow of the trajectories.
In Fig. 10.5, pertaining to a (Ia)D trajectory, this feature is less pronounced: The SolB• -
solution remains close to the Gaussian fixed point value for a long period of RG-time, and the
confluence of both curves takes place in the far UV only.
Finally, Fig. 10.7 for the (IIIa)D case looks very similar to the type (Ia)D result and most of
the significant running in the B-sector takes place in the UV only. In the IR we went down only
to RG-scales which are such that λDk is still well separated from the singular boundary of theory
space at λD = 1/2.
Summary of the attractor mechanism. In Fig. 10.8 we give a schematic description of
the attractor mechanism which we uncovered in the ‘snapshots’. It is sufficient to focus on the
(upward!) evolution of AttrB(k) and its most interesting ‘follower’ the RG trajectory SolB• (k).
The dashed (black) and solid (red) curves describe the footprints of AttrB(k) and SolB• (k), re-
spectively, during their full evolution from k = 0 to k → ∞. In the upper right part of the plot
the (orange) dotted curve adumbrates the boundary of (gB, λ B)-space where the B-couplings di-
verge. The clocks mark equal-time positions on both curves; their black filling indicates the RG
time elapsed since they left their common initial point, AttrB(k = 0). The attraction of SolB ′• (k )
to the current position of AttrB(k′) is indicated by the dashed (blue) arrows.
322 Chapter 10. Background Independence
gB ∂ (gB,λB)
NGFP
B
AttrB(k=0) λ
Figure 10.8: The dashed (black) and solid (red) curve show the k-dependent positions of
AttrB(k) and SolB(k) on the gB-λ B-plane. Recall that SolB• • (k) is an RG trajectory while Attr
B(k)
is not. The (orange) dotted curve indicates the boundary ‘∂ (gB, λ B)’ where the B-couplings
diverge. The clocks mark equal-time positions on the curves; the black filling indicates the
elapsed RG time for upward evolution. The dashed (blue) arrows indicate the direction in
which SolB• (k
′) is pulled at ‘time’ k′, namely the position of the attractor at this instant of time,
AttrB(k′). While initially, at k = 0, the trajectory SolB• (0) coincides with Attr
B(0) the curves
depart due to their different velocities. They meet again at NGB+⊕NGD+-FP for k→ ∞. However,
the motion of AttrB(k) at intermediate scales is encoded in the indentation of the SolB• (k) curve
before it approaches NGB⊕NGD+ +-FP.
Notice that the two tracks meet only in the IR, atAttrB(0) (red star on the bottom), and at
NGB⊕NGD+ +-FP in the UV. This is due to the following fact. While SolB• (k′) is an RG trajec-
tory whose velocity is determined by the beta-functions, AttrB(k) is a k-dependent solution to
a ‘non-evolution’ equation. The latter moves to the boundary of (gB, λ B)-space for some inter-
mediate scales, but rapidly turns back to finite values of gB and λ B, and then slowly approaches
NGB+⊕NGD+-FP from above. On the other hand, the RG trajectories, in particular SolB• (k), have
a smaller velocity and thus only see the ‘taillamp’ of AttrB(k) with which they try to catch up.
The journey of AttrB(k) to the boundary and back is reflected in the indentation of the SolB• (k)
curve before it approaches NGB+⊕NGD+-FP.
Summary: We have seen that the RG-evolution in the B-sector is crucially determined
by the scale dependent UV attractor AttrB(k). Whereas, in the UV, the RG-trajectories have
no other choice but ultimately run into NGB+⊕NGD+-FP for k → ∞, they differ strongly in their
IR behavior, in particular in the way they approach the physical point k = 0. F(ollowing th)e
trajectory backward, i.e. for increasing , the dependence on their ‘initial’ point D/B D/B( ) k ( g)k , λ0 k0
reduces the more the closer gB,λ Bk k gets to the running UV attractor g
B
•(k),λ
B
• (k) .
10.2.4 Split-symmetry restoration in the physical limit k → 0
We will now search for RG trajectories which comply with the requirement of split-symmetry
restoration in the IR, i.e. when k is lowered towards the physical point k = 0.
Recall that split-symmetry is a property of idealized solutions of the FRGE where Γk[h; ḡ]
reduces to a functional of a single field, gµν = ḡµν +hµν . If fully intact in the truncation ansatz
(p) (p)
(11.52), itsGk ’s and λ̄k ’s are the same then at all levels, p= 0,1,2, · · · . In B-D language, this
10.2 Can split-symmetry coexists with Asymptotic Safety? 323
is tantamount to saying that gravΓk [g, ḡ] looses its ‘extra ḡ-dependence’ so that, for p= 1,2,3, · · · :
1 ≡ 1 ! 1 ≡ 1 1 ⇔ 1 ≡ k
2
!
= + = 0 (10.22a)
(p)
G G
D (0) D B B
k G Gk Gk Gk g
B
k k k
λ̄ (p) D
(0)
λ̄ λ̄ D B B B≡ != k ≡ λ̄ λ̄+ k ⇔ λ̄k ≡ 4 λk !k = 0 (10.22b)
(p) GD (0) GD GB GB gBGk k Gk k k k k
Clearly we cannot expect those conditions to hold everywhere along a trajectory, at best, and
only approximately, in a restricted regime of scales. After all, split-symmetry is broken ex-
plicitly both by the gauge fixing and the cutoff term. It is desirable, however, to base the
construction of QEG on an asymptotically safe trajectory which reinstalls split-symmetry as
exactly as possible8 for k→ 0: In this limit the EAA approaches the standard effective action
whose n-point functions, taken on-shell, are related to observable S-matrix elements.
By eqs. (10.22), approximate split-symmetry demands GBk to be very ‘large’, and λ̄
B to be
very ‘small’, in an appropriate sense. Note that these are conditions on the B-couplings only,
the D ones are left unconstrained.
If the relations (10.22) indeed hold true for all k in some interval (k1,k2), the k-differentiated
relations are satisfied, too. They require the beta-functions of the dimensionful B-couplings to
vanish; from (10.8):
( )
∂ 1/GB =−k2BB(λD,gD !t k 1 k k ) = 0 (10.23a)
( )
∂ B B 4 B D D
!
t λ̄k /Gk = k A (λk ,gk ) = 0 (10.23b)
The conditions (10.23) guarantee the stability of (10.22) under the RG-evolution. Notice that,
by them, the running of the B-couplings is not explicitly restricted, rather they put constraints
on the D quantities λD and gDk k . This is just opposite as above.
! !
When we try to solve (10.23) by finding simultaneous zeros of BB B1 = 0 and A = 0 we find
that (on the gD > 0 half-plane) there exists only one such zero, namely the point (gD Dzero, λzero)≈
(0.708, 0.207). This shows clearly that we have to abandon the idea of finding a full trajectory
that preserves split-symmetry, but rather look for RG-trajectories that restore split-symmetry in
the physical limit k→ 0 at least, which is perfectly sufficient.
Nevertheless, it is quite remarkable that the point (gDzero, λ
D
zero) is strikingly close to the NG
D
+-
FP fixed point which we located at (gD∗ , λ
D
∗ ) ≈ (0.703, 0.207). There is no obvious general
reason for this ‘miracle’ to happen.
Split-symmetric ‘final conditions’ in the B-sector
From now on we shall be modest and try to establish split-symmetry at k = 0 only. In order to
explore the implications of (10.22) for this case let(us assu)me we have solved the differential
e(quation)s of theD-sector, found all trajectories k→7 g
D D
k , λk , and labeled them by their position
gDk , λ
D
k at some intermediate scale, 0 < k0 < ∞. Then, inserting the D-trajectories into the0 0
flow eqs. (10.8b) for gB Bk and λk , our task is to identify those initial, or more appropriately, final
conditions for the B couplings that lead to intact split-symmetry in the physical limit k→ 0.
8Which is not to say, fully. The gauge fixing dependent contents of Γk which never makes its way into observables
may remain split-symmetry violating. Note that in the present truncation this ‘gauge fixing dependent contents’ is
the Γgf+Γgh part of the EAA which never becomes split-symmetric, of course. Our discussion concerns only the
Γgrav-part of the EAA ansatz.
324 Chapter 10. Background Independence
Taking advantage of the explicit solution to the two B-equations given in (10.9) and (10.13)
the requirement of split-symmetry, at some k which is still arbitrary, assumes the form
2 ∫1 k
= k2
1 k
= 0 − ′ !k BB ′ ′
GB gB gB 1
(k )dk = 0 (10.24a)
k k k k0 0
B B λ B ∫λ̄ 4λ k= k k = k4 k0 !0 + dk′ k′3AB(k′) = 0 (10.24b)GBk gBk gBk k0 0
We want split-symmetry to be intact at k = 0, so we now let k → 0 in the conditions (10.24),
w( hile kee)ping k0 strictly nonzero throughout. Then these conditions uniquely fix ‘initial’ values
GB Bk , λ̄k for the dimensionful
9 background couplings at k = k0:0 0
(∫ k0 )
B B 2 −1Gk = gk /k0 =− k′BB ′1(k )dk′ (10.25a)0 0
∫ 0k0
λ̄ B = λ Bk k k
2
0 = G
B
k dk
′ k3 ′AB(k′) (10.25b)
0 0 0
0
Here BB(k)≡ BB(gD, λD1 1 k k ) and AB(k)≡ AB(gD Dk , λk ) depend manifestly on the D trajectory under
consideration.
This result is good news for the Asymptotic Safety program in a twofold way: First, there
does indeed exist a trajectory in the B-sector which complies with the requirement of split-
symmetry at k = 0, but second, there is only one such trajectory; as a consequence, when
solving the RG equations for the B-couplings there are no constants of integration that could be
chosen freely, and this increases the predictivity of the theory.
Another remarkable feature of the initial conditions (10.25) is their close relationship to the
running attractor AttrB(k), to which we turn next.
The k → 0 asymptotics of the D trajectories
As a necessary preparation for the exploration of the split-symmetry restoration of the full 4-
dimensional system and to demonstrate the role played by the running attractor we first summa-
rize the IR behavior of (gD,λDk k ) along trajectories of the three types, (Ia)
D, (IIa)D, and (IIIa)D,
respectively.
k→0
Type (Ia)D: This type of trajectories has the defining property that λDk −−→ −∞. If we
consider the D-system for very large negative (positive) values of λDk we obtain the following
asymptotic solutions (for gDk > 0):0
( 2πg
D 2
D kgk =
0)k k→
− −−→
0
0 (10.26)
gD 2 2k k0 k +2πk
2
0 ( ) 0
gD k4 4 4 D
D k(0 0− k −6πk0λλk = ( ) k0 ) −
k−→→0 ±∞ (10.27)
3k2 gD 2k k − k2 20 0 −2πk0
( )
Here the sign for the limit of λD0 agrees with that of λ
D
k sign 6πλ
D − gD . For type (Ia)D
0 k0 k0
trajectories the initial point (gD ,λDk k ) is chosen such that the IR-limit of λ
D
k is negative. (The0 0
9Note that because of the explicit factors of k2 and k4, respectively, which relate dimensionless and dimension-
ful quantities in eqs. (10.24) the implications of split-symmetry in the limit k → 0 are discussed most easily in
dimensionful terms.
10.2 Can split-symmetry coexists with Asymptotic Safety? 325
positive sign in (10.27) applies to the type (IIIa)D we will discuss below.) The corresponding
dimensionful D-couplings have the following asymptotic behavior:
( )
1 gD k2k 0− k2 +2πk20 0 −k−→→0 1 k
2
= + 0 (10.28a)
GDk 2πg
D
k G
D
k 2π0 0
λ̄D [ ( )] λ̄D 4k = k4 λD /gD 1 4 k→0 k0 k0
GD 0 k0 k
− 6π 1− (k/k0) −−→ − (10.28b)0
k G
D
k 6π0
The IR-limit of GDk in eq. (10.28a) will later on be used to define the Planck mass for the class
of type (Ia)D trajectories.
Type (IIa)D: Along the separatrix the dimensionless cosmological and Newton’s constant
k→0 k→0
approach zero in the IR: λDk −−→ 0 and gDk −−→ 0. Linearizing around the values gD = 0= λD
we find
→ 2 2D D k 0 k k→0 kgk = gk (k/k0)
2 −−→ 0 1/GD 0k = −−→ 0 (10.29a)0 gD Dk g0 k0
3 3
λD = gD
k→0 k→0
k k (k/k0)
2 −−→±0 λ̄Dk /GD 48π 0 k = k −−→ 0 (10.29b)8π
The initial values λDk and g
D
k , imposed near the Gaussian fixed point, are constrained by the0 0
‘separatrix condition’ λDk = (3/8π)g
D
k .0 0
Type (IIIa)D: The type (IIIa)D trajectories suffer from the well known problem that they
run into a divergence of the beta-functions and terminate at some low but nonzero scale kterm
when λDk reaches the value 1/2: Neither within the single- nor the bi-metric Einstein-Hilbert
truncation their full extension to k = 0 can be computed. Nevertheless, the situation is fairly
clear for the trajectories of interest, namely those that before hitting the singularity enjoy a
long classical regime [207–209] in which GD Dk ≡ Gcl.reg. and λ̄D ≡ λ̄Dcl.reg. are approximately con-
stant. Their dimensionless counterparts are then given by, for k ≫ kterm in the classical regime
(‘cl.reg.’),
gDk = G
D 2
cl.reg.k (10.30a)
λDk = Λ
D
cl.reg./k
2 (10.30b)
For the purposes of the present paper we hypothesize that in reality the classical regime even
extends to scales k→ 0. The running (10.30) corresponds to that found in (10.27) then.
The attractor mechanism of split-symmetry restoration ( )
Next we are going to insert the various types of asymptotic (for k→ 0) solutions k→7 gDk , λDk
into BB B D D B B D D1(k) ≡ B1(gk , λk ) and( A (k) ≡) A (gk , λk ). Then we use the resulting functions in eqs.
(10.25) for the initial values GB B ′k , λ̄0 k whereby we can perform the two k -integrals analytically.0
For the asymptotic formulae to be sufficiently good approximations in the integrands we must
choose k0 very close to zero (more precisely, much smaller than the Planck scale, ≪ ( D)−1/2k0 G0 ,
see below). To convince yourself that the use of the asymptotic solutions is indeed permissible
then, and to see what it entails, notice also the following facts:
A. Since gDk approaches zero in the IR for all three classes of D trajectories, contributions to
BB1 and A
B containing the anomalous dimension ηD ∝ gD produce only terms of subleading
order in the asy(mptotic) expansion, which we may neglect. This implies in particularp
q (−2λD)≈ pΦ D Dn k n −2λk for gk → 0.
326 Chapter 10. Background Independence
B. For the large values of λDk (which o)ccur in IR of type (Ia)
D and (IIIa)D trajectories we may
exploit that lim p DλD→±∞ Φn −2λk = 0 for n ≥ 1. Onl(y the g)host terms contribute to A
B
k
and BB1. For the separatrix we instead use lim
p
λD→0 Φn −2λD
p
k = Φn (0) in lowest order.k
C. The ghost contributions to both BB and AB1 are unaffected by any approximation in the
D-sector, simply because they do not depend on the D couplings at all and are thus k-
independent and independent of the D initial conditions.
Going through the explicit formulae for BB1 and A
B it is now easy to check that, as a conse-
quence of these three simplifying properties, we are entitled to perform the integrals (10.25) in
the far IR by substituting constant functions BB B B B1(k)≈ B1(0) and A (k)≈ A (0). In this manner
we find that the initial values of the B couplings that lead to a fully intact split-symmetry at
k = 0, are given by
( )
GB = −2/BB(0) /k2k 1 0 = gB•(0)/k20 (10.31a)0 ( )
λ̄ B = − 1ABk (0)/BB(0) k2 = λ B(0)k2 (10.31b)0 2 1 0 • 0
This is an important result, and various comments are in order here:
A. By comparing the initial values (10.31) to the coordinates (10.15) of AttrB(k) in the limit
λD →±∞ or λD → 0, respectively, we find that the split-symmetry restoring initial point
(gBk ,λ
B
k ) coincides exactly with the location of the running UV attractor (g
B B
•,λ• ) for0 0
k = 0.
B. In the first place, the result demonstrates that there exists indeed a fully extended RG tra-
jectory, well behaved at all scales between zero and infinity. It defines an asymptotically
safe theory, hitting a NGFP for k → ∞, and at the same time restores split-symmetry in
the physical limit k → 0 when all fluctuations are integrated out. At least numerically,
we can compute this trajectory for all k ∈ [0,∞). In this way we have verified that the
trajectory lies indeed on the UV critical hypersurface of the doubly non-Gaussian fixed
point, NGB D+⊕NG+-FP.
C. There exists one, and only one set of initial values in the B-sector, (gB Bk ,λk ), that leads0 0
to split-symmetry at k = 0. (The uniqueness follows from the fact that the running UV
attractor is IR repulsive in all directions.) This has the positive side effect that the num-
ber of free parameters that characterize the asymptotically safe quantum theories one can
construct does not increase when we generalize the 2-parameter single-metric Einstein-
Hilbert truncation to the 4-parameter bi-metric one. In fact, the newly introduced param-
eters immediately get ‘eaten up’ by the necessity to turn the split-symmetry violations to
zero at k = 0.
D. For type (Ia)D and (IIIa)D trajectories the presence of the ghost contributions (ρgh = 1)
is found to be essential for the existence of the symmetry restoring point. Making ρgh
explicit, the corresponding IR solutions assume the form
1 −k−→→0 1 − 5ρgh k20 (10.32a)GBk GBk 3π0
λ̄ B −k−→→0 λ̄k0 2ρgh+ k4
GB GB 3π 0
(10.32b)
k k0
From here the split-symmetry restoring initial values GB = 3π/(5k2 ρ ) and λ̄ Bk 0 gh k =−2/50 0
can be deduced. Omitting the ghost terms (ρgh = 0) split-symmetry restoration in the IR
would require GBk → ∞ at a nonzero k0! Note also that the corresponding dimensionless0
initial data, for ρgh = 1, are (gB Bk , λk ) = (3π/5, −2/5) which, by (10.21), is exactly the0 0
IR position of the attractor, AttrB(k→ 0).
10.2 Can split-symmetry coexists with Asymptotic Safety? 327
E. For the separatrix there are additional graviton contributions shifting the initial data to-
wards smaller values:
1 −k−→→0 1 − (7+20ρgh) k20 (10.33a)GB GBk k 12π0
λ̄ B λ̄ B−k−→→0 k0 − (5−8ρgh) k4 (10.33b)
GBk G
B
k 12π
0
0
( )
‘(Switching on)’ the ghosts (ρgh = 1) the split-symmetry restoration happens at g
B B
k , λ =0 k0
4π/9, −1/9 . These are precisely the B-coordinates of the fixed point NGB+⊕GD-FP,
which in turn equals the k→ 0 limit of AttrB(k) for the type (IIa)D trajectory.
Summary: For every RG trajectory of the D-sector there exists precisely one associ-
ated trajectory of the B couplings which restores split-symmetry for k → 0. In the IR, the
B trajectory approaches the (UV attractive, i.e., IR repulsive) running attractor AttrB(k). For
k → ∞, the combined 4-dimensional trajectory is asymptotically safe and runs into the fixed
point NGB+⊕NGD+-FP.
10.2.5 All classes of split-symmetry restoring trajectories
In this subsection we provide a survey of all classes of RG trajectories with restored split-
symmetry in the IR, and we present explicit examples. In the D-sector, we do have the freedom
of choosing initial data gDk and λ
D
k at some k= k0, and so we will select a typical representative0 0
of each type (Ia)D, (IIa)D, and (IIIa)D, respectively. Once such a solution is picked we have no
further freedom: the values of gBk and λ
B
k are then uniquely determined by the requirement of0 0
split-symmetry at k = 0, and so there is a unique ‘lift’ of the 2-dimensional D trajectory to the
4-dimensional theory space. The resulting trajectories will be referred to as of type (Ia)D-AttrB,
(IIa)D-AttrB, and (IIIa)D-AttrB, respectively. Clearly the last type is the most interesting one
since it is closest to real Nature, presumably.
In the following we shall always express the RG-scale k and the dimensionful couplings in
units of the Planck mass defined by the dynamical Newton constant: mPl = [GD ]−1/2k . For theIR
infrared normalization scale kIR we choose kIR = 0 for the type (Ia)D and (IIa)D trajectories. In
the (IIIa)D case were we cannot follow the evolution down to k = 0 we choose for kIR a value
in the (semi-)classical regime where GDk and λ̄
D
k are approximately constant (lower horizontal
branch of the trajectory). In principle the notion of a ‘Planck mass’ depends on the level of
the Newton constant used to define it. In our case there is no ambiguity since we enforce split-
(p)
symmetry in the IR. Hence all Gk give rise to the same Planck mass:
( √ ) √ √(p) ≡ (0)m DPl = lim 1/ G 1/ G ≡ 1/ G
k→0 k 0 0
This definition depends on the chosen trajectory, however.
Next we present the k-dependence of all running couplings, both in dimensionless and di-
mensionful form, for one representative numerical solution in each class of trajectories.10 For
comparison we include the single-metric result for the Einstein-Hilbert truncation [122] using
the same initial data as for the D couplings. The diagrams in Figs. 10.9-10.11, 10.12-10.14, and
10.15-10.17, respectively, are devoted to the (Ia)D-, (IIa)D-, and (IIIa)D-AttrBtrajectories.
10For the ‘initial’ conditions at the scale k0 = 1 we set (g
D D
k ,λk ) = (0.3,−0.01), (gD ,λD 3πk k ) = ( 8 10−4, 10−4),0 0 0 0
and (gD Dk ,λk ) = (0.05, 0.01) for the representative of type (Ia)
D, (IIa)D, and (IIIa)D, respectively.
0 0
328 Chapter 10. Background Independence
R In all plots we employ the following color / style-coding to distinguish the couplings g
I ,
GI , λ I , and λ̄ I , for I ∈ {D, B, (0), sm}:
dashed (red): I = sm (single-metric)
solid (dark-blue): I = D≡ (p) for p≥ 1
solid (light-blue): I = (0)
dot-dashed (blue): I = B
2.5 0.20
0.15
2.0
0.10
1.5
0.05
1.0
0.00
0.5 -0.05
-0.10
0.0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
k/mPl k/mPl
Figure 10.9: Type (Ia)D-AttrBtrajectory: dimensionless couplings.
2.5 2.5
2.0 2.0
1.5 1.5
1.0 1.0
0.5
0.5
0.0
0.0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
k/mPl k/mPl
Figure 10.10: Type (Ia)D-AttrBtrajectory: dimensionful couplings.
14 25
12
20
10
15
8
6
10
4
5
2
0 0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mPl k/mPl
Figure 10.11: Type (Ia)D-AttrBtrajectory: the coefficients as they appear in the EAA. Note the
perfect split-symmetry restoration in the IR: 1/GBk and λ̄
B/GBk vanish for k→ 0, implying that
grav
Γk looses its extra ḡµν dependence.
Dimensionless couplings. Let us first consider the running dimensionless couplings shown
in Figs. 10.9, 10.12, and 10.15. The following features are shared by all three types of trajecto-
1/GI Ik [
I
mPl] Gk [mPl] gk
λ̄ Ik/G
I
k [mPl] λ̄
I I
k [mPl] λk
10.2 Can split-symmetry coexists with Asymptotic Safety? 329
3.0
0.20
2.5
0.15
2.0
0.10
1.5
0.05
1.0 0.00
0.5 -0.05
-0.10
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5
k/mPl k/mPl
Figure 10.12: Type (IIa)D-AttrBtrajectory: dimensionless couplings.
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5 0.5
0.0
0.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
k/mPl k/mPl
Figure 10.13: Type (IIa)D-AttrBtrajectory: dimensionful couplings.
14
12 20
10
15
8
6 10
4
5
2
0 0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mPl k/mPl
Figure 10.14: Type (IIa)D-AttrBtrajectory: the coefficients as they appear in the EAA. Note the
split-symmetry restoration for k→ 0.
3.0
0.20
2.5
0.15
2.0
0.10
1.5 0.05
0.00
1.0
-0.05
0.5
-0.10
0.0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
k/mPl k/mPl
Figure 10.15: Type (IIIa)D-AttrBtrajectory: dimensionless couplings.
ries:
The solutions for the single-metric, the level-(0) and the D couplings do not agree in any ap-
gI 1/GI Ik k [
I
mPl] G [mPl] gk k
λ I λ̄ I/GI [ ] λ̄ I Ik k k mPl k [mPl] λk
330 Chapter 10. Background Independence
proximate sense, but differ quite significantly for most k. In the IR, we imposed the requirement
of split-symmetry and this is clearly seen even in the results for the dimensionless quantities:
For k → 0, in the classical regimes, the p = 0 and p ≥ 1 curves overlap basically. Likewise,
in the UV, we observe that, consistent with the analysis in Section 10.3, there is a remarkable
similarity of the single- and bi-metric curves in the vicinity of their non-Gaussian fixed points.
All plots confirm this numerical ‘miracle’ which, as we emphasized already, is not due to any
general principle. (But it is highly welcome of course.) At intermediate scales the single-metric
and the bi-metric solutions are found to be rather different, even qualitatively. This is precisely
the symptom of the broken split-symmetry.
In conclusion we can say in comparison with the bi-metric truncation, the single-metric
treatment seems to be a good approximation in the far IR and UV, but at the quantitative level
it does not account for what happens in between. It must be said that the single-metric results
convey the correct qualitative picture, nevertheless.
Dimensionful couplings. Next we take a look at the dimensionful couplings, expressed
in units of the Planck mass. Figs. 10.10, 10.13, and 10.16 show the results for the Newton and
cosmological constants for the three classes. As for the asymptotic k-dependence of the Newton
constants GIk, I ∈ {D, B, (0), sm} we observe that all of them vanish for k→ ∞.
Thus we recover gravitational anti-screening in the bi-metric setting, but only at a high
(Planckian) scale. In fact, it is quite impressive to see that, for k below the Planck scale, the
dynamical Newton constant GDk actually increases with k then, assumes a maximum near k ≈
mPl, and finally decreases for k & mPl. On the other hand, the single-metric Newton constant
Gsmk decreases with k at all scales k ≥ 0. Clearly this behavior of GDk is a consequence of the
sign-flip of BD1(λ
D), and therefore ηD, at λD = λDcrit.
Coefficients. For k → 0 the background Newton constant GBk diverges, and λ̄ Bk vanishes,
exactly as it should be in order to make 1/GBk vanish in this limit, which is necessary for split-
symmetry. This is best seen in Figs. 10.11, 10.14, and 10.17. There the dependence of the
2.5
2.5
2.0
2.0
1.5
1.5
1.0 1.0
0.5 0.5
0.0
0.0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
k/mPl k/mPl
Figure 10.16: Type (IIIa)D-AttrBtrajectory: dimensionful couplings.
pre-factors of the B-type invariants in the truncation ansatz on k is shown, namely 1/GBk and
λ̄ B/GBk , respectively. Remember that the (blue) dot-dashed line is related to the B-sector, it is
very impressive to see how close to zero it stays in the IR for all three types of trajectories.
These plots confirm that we were indeed successful in combining Background Independence
with Asymptotic Safety.
Notice that for moderately large values of k the B-pre-factors increase. Their deviation from
zero is relatively small when compared with theD- or level-(0) sector, and this implies that split-
symmetry is intact at least approximately. For intermediate scales we again find considerable
violation of split-symmetry, which manifests itself by B-coefficients which are now of the same
GIk [mPl]
λ̄ Ik [mPl]
10.3 Single-metric vs. bi-metric truncation: a confrontation 331
order as the D- and level-(0) ones. Therefore the single-metric (red, dashed line) only converges
to the bi-metric curves for k→ 0 and k→ ∞.
14
12 20
10
15
8
6 10
4
5
2
0 0
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mPl k/mPl
Figure 10.17: Type (IIIa)D-AttrBtrajectory: the coefficients as they appear in the EAA. Note
again the vanishing 1/GBk and λ̄
B/GBk , indicative of split-symmetry restoration in the limit k→ 0.
From the differences between the p= 0 and p≥ 1 curves, too, we see again that for all three
trajectory types split-symmetry is apparently well restored in the IR and the UV, but in between
it suffers from a considerable breaking; the D couplings show a more pronounced k-dependence
than the level-(0) couplings, whereas the single-metric functions are monotone.
10.3 Single-metric vs. bi-metric truncation: a confrontation
In this section we perform an in-depth analysis of the differences between the bi-metric Einstein-
Hilbert truncation and its single-metric approximation. We shall describe how precisely their
results are interrelated on general grounds, and what can be learned from the numerical compar-
ison about the validity of the single-metric truncation. As we shall see, its degree of reliability
varies considerably over the theory space. For future work it will be important to know of course
where, and to what extent it can be trusted. In particular we shall also understand why in the
past it has always been notoriously difficult to obtain accurate and stable results for the critical
exponents.
In this section, in subsection 10.3.7, we shall also critically examine how our new method
based on the ‘deformed α = 1 gauge’ compares to the bi-metric calculation in ref. [172] which
employed the transverse-traceless approach.
This section is of a somewhat technical nature, and can be skipped by the reader who is
mostly interested in the results.
10.3.1 Collapsed level hierarchies
Let us write the level-expanded EAA symbolically as Γ (p)k[h; ḡ] = ∑
∞ F [ḡ] (h )pp=0 k µν where the
(p)
Fk ’s depend on the background metric only. When we insert this expansion into the FRGE
and project on a fixed level p we see that ∂ (p)tFk which appears on its left-hand-side (LHS)
gets equated to an expression exclusively involving { (q)Fk |q = 2, · · · , p+ 2}. Hence the scale
derivative of all (dimensionful) level-(p) couplings is given by a beta-function depending on the
level-(q) couplings, with q = 2, · · · , p+2, only.11 This is the generic situation when no special
restrictions on the form of the EAA are assumed: the FRGE amounts to an infinite hierarchy
of equations ∂ (p)tFk = · · · for p = 0,1,2,3, · · · which does not terminate at any finite level and
couples all levels therefore. Only if it was possible to realize split-symmetry exactly this tower
of equations collapses to a single equation that governs all levels.
11The dimensionless couplings also contain trivial canonical terms in their beta-functions, of course. They play
no role in this discussion and are ignored here.
1/GIk [mPl]
λ̄ I Ik/Gk [mPl]
332 Chapter 10. Background Independence
The bi-Einstein-Hilbert truncation used in the present paper involves the assumption that
split-symmetry is broken only weakly, and that differences among the ‘higher’ levels p =
1,2,3, · · · are sufficiently small so that they may be ignored. The lowest level, p = 0, how-
ever is dealt with separately and is allowed to show a RG behavior different from p ≥ 1. As
always, the p= 0 couplings have beta-functions which depend on the p= 2 couplings. For the
present truncation the latter happen to be equal to those at all non-zero levels p≥ 1.
Stated more abstractly, what reduced the infinite hierarchy of RG equations to just 2 equa-
tions was an additional hypothesis about the RG flow, namely that the split-symmetry breaking
is such that it lifts only the degeneracy between levels with p= 0 and p > 0, while those with
p> 0 remain degenerate among themselves.
The logical status of the familiar single-metric truncations can be characterized analogously.
Here the additional hypothesis invoked is even stronger: one pretends that the solutions to the
FRGE exhibit exact split-symmetry so that all levels p = 0,1,2,3, · · · undergo an equivalent
RG evolution. It is sufficient then to retain the RG equations for the lowest level at p= 0 to fix
the k-dependence of all running couplings.
10.3.2 Relating single- and bi-metric beta-functions
Let us return to the concrete example of the bi-Einstein-Hilbert truncation with its 4 independent
couplings {g(0),λ (0),g(1),λ (1)} and let us see in which way precisely its beta-functions are related
to those of the standard single-metric Einstein-Hilbert truncation with only 2 running couplings.
Recall that after the conformal projection g = e2Ωµν ḡµν the EAA of the former equals that
of the latter for Ω = 0. In the bi-metric case, the flow equation is expanded in powers of Ω,
whereby the zeroth and first orders in Ω correspond to the level-(0) and level-(1) couplings,
respectively. Structurally the single-metric beta-functions β sm and β smg λ thus coincide with the
beta-functions of the level-(0) couplings, however only after we have identified all couplings of
different orders.
Even though this sounds trivial it changes the form of the beta-functions quite significantly
so that the new differential equations are of a rather different type, with qualitatively new prop-
erties. For example, the anomalous dimension ηD that (contrary to ηB!) can appear on the
RHS of the bi-metric flow equation is no longer related to an independent coupling g(2) then,
but in fact to the ηB-related g(0). This transition changes the simple, Bernoulli-type differential
equation (10.7) into a much more complicated non-polynomial (but autonomous) one.
In detail, we have to apply the following identifications:
∣
β sm sm sm (0)g/λ (g ,λ ;d)≡ β /λ ({g
(q) = gsm,λ (q) = λ sm};d)∣ p ∈ {0,1,2, · · · }
g η (p)≡η sm
(10.34a)
g(0) = g(1) = · · ·= gsm , λ (0) = λ (1) = · · ·= λ sm . (10.34b)
The running of all couplings at higher levels is pretended to be described by the two beta-
functions from level-(0).
10.3.3 Conditions for the reliability of a single-metric calculation
To check whether the single-metric truncation is a good approximation to the bi-metric one, we
must study the beta-functions of the higher levels, the conditions (10.34), and their implications:
sm != (0)
!
gk gk =
(1) (0)
g = · · · , λ smk k = λk = λ
(1)
k = · · · , (10.35a)
β sm sm sm
! (p)
g/λ (g ,λ ;d) = β /λ ({g
(q) = gsm,λ (q) = λ sm};d) p,q ∈ {0,1,2, · · · } (10.35b)
g
10.3 Single-metric vs. bi-metric truncation: a confrontation 333
Notice that contrary to split-symmetry requirement where it was more natural to consider di-
mensionful couplings, the requirements (10.35) are constraints on the dimensionless couplings.
But of course as long as we are not taking the k→ 0 or k→ ∞ limit we can simply strip off the
explicit k-factors from the split-symmetry condition and end up with the conditions (10.35):
!
Gsm = k−(d−2)gsm = k−(d−2) (0) = −(d−2) (1)k k gk k gk = · · ·=
(p)
Gk (10.36a)
λ̄ sm = 2λ sm
!
= 2λ (0) = 2λ (1)k k k k k k k = · · ·= λ̄
(p)
k (10.36b)
Though in principle there is the possibility that the explicit k-dependence of the dimensionful
couplings gives rise to split-symmetry for k = 0 or k → ∞ only, we never relied on this possi-
bility, and we shall never do it in what follows. This puts the respective requirements for intact
split-symmetry and a valid single-metric approximation on an equal footing.
10.3.4 The anomalous dimensions: structural differences
In the remainder of this section we restrict the discussion to the Newton couplings. This cov-
ers already all subtleties that arise in concretely working out the conditions (10.35) and their
generalization to the full 4-dimensional theory space.
The beta-functions of the Newton constants in the level-description were found in eqs.
(8.73) and (8.82) of chapter 8, respectively:
[ ( ) ]
β (0)(gD,λD,g(0); (0) (0)g d) = d−2+ B1 (λD;d)+ηDB D2 (λ ;d) g(0) g(0) (10.37a)[ ( ) ]
βDg (g
D,λD;d) = d−2+ BD D1(λ ;d)+ηDBD2(λD;d) gD gD (10.37b)
Due to the back-reaction of the D-couplings, i.e. those with p ≥ 1, especially via ηD, eqs.
(10.37a) and (10.37b) amount to structurally quite different expressions for the anomalous di-
mensions:
[ ]
η (0)(gD,λD,g(0);d) = (0)B (λD; )+ηD (0)1 d B2 (λ
D;d) g(0) (10.38a)
BD(λD;d)gD
ηD(gD,λD;d) = 1− (10.38b)1 BD D D2(λ ;d)g
The ‘confusion’ of different levels by the single-metric truncation yields a formula for its anoma-
lous dimension η sm that combines the non-polynomial gD dependence (10.38b) at the levels
p ≥ 1, with the dependence on the cosmological constant from level-(0), the latter given by
(0)
B1/2(λ ;d). Explicitly, we can extract the single-metric beta-functions for Newton’s coupling
by applying (10.34) to eq. (10.37):
∣
β sm(gD,λD,g(0);d) = β (0)(gsm,λ sm,gsm;d)∣g [g ηD=η sm( ) ]
= d−2+ (0)B sm sm (0) sm sm sm1 (λ ;d)+η B2 (λ ;d) g g (10.39)
The identification of ηD with η sm leads to an implicit equation for η sm from which we obtain
(0)
B (λ sm;d)gsm
η sm(gsm,λ sm;d) = 1 (10.40)
1− (0)B2 (λ sm;d)gsm
Eq. (10.40) highlights the limitations of the single-metric formula in approximating the full
bi-metric RG flow: Even though η sm inherits the dynamical g-dependence and thus reproduces
all findings based on its non-polynomial form, it completely looses any information on the
dynamical λ -dependence. It is thus rather non-trivial that our bi-metric results, even at the
numerical level, in many cases stayed very close to the single-metric ones.
334 Chapter 10. Background Independence
10.3.5 (Un-)Reliable portions of the single-metric theory space
Since intact split-symmetry is closely related to the reliability of the single-metric truncation,
we expect the IR regime of the trajectories — where we explicitly restored split-symmetry —
to be well approximated by the trajectories of gsm and λ sm. The plots presented in section 10.2.5
actually confirm this expectation, see Figs. 10.9, 10.12, and 10.15.
Nevertheless, we already pointed out that there exists no fully extended solution {Γk, k ∈
[0,∞)} along which split-symmetry would be intact everywhere. As can be seen in the diagrams
of section 10.2.5, only in the extreme IR and UV, that is only when k/mPlanck ≪ 1 or k/mPlanck ≫
1 the dimensionless bi-metric couplings are satisfactorily approximated by the single-metric
ones, in between the curves disagree even qualitatively.
Instead of focusing on a single trajectory only, we next investigate the validity of the single-
metric approximation in an extended region of the bi-metric theory space. For this purpose
we determine the set R of all pairs (g,λ ) with g > 0 such that at the points of theory space
associated to them via (g(0),λ (0),gD,λD) = (g,λ ,g,λ ) ∈ T the ‘single-metric condition’ (10.35)
is satisfied. Those points form a subset of the 4-dimensional theory space, denoted
TR = {(g,λ ,g,λ ) | (g,λ ) ∈R} ⊂ T
We may think of this submanifold as set of initial points for RG trajectories at which the single-
metric approximation is exact. The crucial question is to what extent this submanifold is invari-
ant under the 4-dimensional RG flow.
If we start from a point with (g(0),λ (0)) = (g,λ ) = (gD,λD), and insist that this equality is
preserved under an infinitesimal RG transformation, we must require that the corresponding
!
anomalous dimensions agree: η (0)(g,λ ,g;d) = ηD(g,λ ;d). Using (10.38) this entails a con-
straint for the pair (g,λ ) which involves the B1/2-functions:
[ ] [ ]
!
0= (0)g B1 (λ ;d)−BD1(λ ;d) +g2 BD1(λ ;d)
(0)(λ ; )− (0)B2 d B1 (λ ;d)BD2 (λ ;d) (10.41)
This equation is a necessary condition for (g,λ ) ∈R.
!
Besides the requirement η (0) =ηD, leading to (10.41), the ‘single-metric validity conditions’
!
(10.35) also include the constraint arising from η sm(g,λ ;d) = ηD(g,λ ;d). Expressed in terms
of the B1/2-functions it is found to be exactly identical with eq. (10.41).
12 As a result, equation
(10.41) is a sufficient condition for the validity of the single-metric truncation within R.
Let us now solve eq. (10.41) explicitly to determine the pairs (g,λ ) ∈R.
Solution class (g,λ R). A first class of solutions consists of points (g,λR) with g arbitrary
and λR satisfying both
(0)
B1 (λ ;d) =
(0)
BD1(λ ;d) and B2 (λ ;d) = B
D
2(λ ;d).
Are there λ -values for which these conditions are satisfied, or satisfied approximately at
least? The explicit comparison of (0)B D1/2 and B1/2 with respect to their λ -dependence in d = 4 is
depicted in Fig. 10.18 and Fig. 10.19, respectively. The dashed vertical line marks the fixed
point value of the dynamical cosmological constant, λD∗ . The dark line gives the dependence of
BD1/2 on λ , whereas the light one shows the functions
(0)
B1/2. In the inset pictures we plotted the
difference between the D- and level-(0) functions.
We observe that while (0)B1/2(λ ) and B
D
1/2(λ ) have a qualitatively similar λ -dependence, the
exact equality (0)B1/2(λ ) = B
D
1/2(λ ) holds only at a single value of λ . However, what comes as
a real surprise is that this distinguished λ -value at which the single-metric truncation performs
12The coincidence of the η (0) = ηD and η sm = ηD conditions is owed to the fact that Bsm1/2 ≡
(0)
B1 2, which reflects/
the general relationship between the single-metric and the level-(0) conformally projected bi-metric results.
10.3 Single-metric vs. bi-metric truncation: a confrontation 335
0
-1
-2
1.0
-3
0.5
0.0
-0.5
-4
-1.0
-1.5
0.00 0.05 0.10 0.15 0.20 0.25
-5
0.00 0.05 0.10 0.15 0.20 0.25
λ
Figure 10.18: This figure depicts the dependence of BD1(λ ;4) and
(0)
B1 (λ ;4) on their argument
λ by the dark, respectively light, blue line. The difference of the functions is shown as the
(red) curve inserted in the lower left corner. In both plots the fixed point value λD∗ is marked
with the dashed vertical line. The smaller the difference of BD1(λ ;4) and
(0)
B1 (λ ;4) the better
is the single-metric approximation to the bi-metric truncation. While in the vicinity of the
Gaußian fixed point there is a large discrepancy, the point of best approximation is seen to be
(miraculously) close to the NGFP value λD∗ .
0.4 0.00
-0.02
-0.04
0.3
-0.06
-0.08
0.2
0.00 0.05 0.10 0.15 0.20 0.25
0.1
0.0
0.00 0.05 0.10 0.15 0.20 0.25
λ
Figure 10.19: The second condition for the fulfillment of eq. (10.41) is the agreement of
BD2(λ ;4) with
(0)
B2 (λ ;4). Their dependence on λ is given by the dark, respectively light, blue
line. The difference of the functions is shown in the upper left corner. The fixed point value λD∗
is marked with the dashed vertical line. Notice that the agreement of BD2(λ ;4) with
(0)
B2 (λ ;4) is
best in the vicinity of the NGFP. There, the single-metric truncation is a good approximation
to the bi-metric one. Away from the NGFP its quality deteriorates considerably.
best is impressively close to the λ -coordinate of NGD+-FP, λ
D
∗ . This ‘miracle’ is not explained
by any general principle. Its implication is clear though: The single-metric approximation to the
bi-Einstein-Hilbert truncation is most reliable precisely in that region of theory space where the
non-Gaußian fixed point is located. This discovery may be seen as an a posteriori justification
D/(0) D/(0)
B2 (λ ;4) B1 (λ ;4)
(0)
BD2 (λ ;4)−B2 (λ ;4) BD1 (λ ;4)−
(0)
B1 (λ ;4)
336 Chapter 10. Background Independence
of the single-metric investigations of the Asymptotic Safety conjecture.
1.0
0.8
0.6
gD
0.4
0.2
0.0
0.150 0.175 0.200 0.225 0.250 0.275
λD
Figure 10.20: The two solutions to the split-symmetry condition (10.41), given by pairs (g,λR)
and (gR(λ ),λ ), respectively, are shown by the two light, and the dark (red) curve. They are su-
perimposed on the phase portrait of the ‘D’ sector. In the vicinity of these curves split-symmetry
is approximately intact. The ‘miraculous’ result is that both of them, the almost linear (dark red)
curve, and the (light red) exactly vertical line, pass very close to the NGFP (gray disc). This jus-
tifies the use of the single-metric truncation in the vicinity of the NGFP. On the other hand, it is
also apparent that every RG trajectories stays only briefly within a neighborhood of (g,λR) or
(gR(λ ),λ ). Away from these regions the single-metric truncation might become problematic.
Solution class (gR(λ ),λ ). Up to now we solved the condition (10.41) by setting to zero
the coefficients of g and g2 separately. There is a second type of solutions consisting of pairs
(gR(λ ),λ ) where
(0)
≡ B1 (λ ;d)−B
D
1(λ ;d)gR(λ ) (10.42)(0)
B1 (λ ;d)B
D
2(λ ;
(0)
d)−BD1(λ ;d)B2 (λ ;d)
In Fig. 10.20 the function gR(λ ), along with the above (g,λR) solution, is superimposed on the
phase portrait of the ‘D’ sector which, as we know, is qualitatively similar to the ‘sm’ one.
The good news we learn from this plot is that the NGFP is situated very close to the curve
(gR(λ ),λ ). This is a second ‘miracle’, again unexplained by any general argument, and inde-
pendent of the first one. So the single-metric approximation seems indeed most reliable when
it comes to locating the NGFP and exploring its properties.
The bad news is that there does not exist a single RG trajectory that would stay on, or
close to the (gR(λ ),λ )-line for all scales; the trajectories intersect it at most once or twice.
The consequence is that every ‘sm’ trajectory unavoidably contains segments where it differs
substantially from its bi-metric, i.e. ‘D’ analogue.
10.3 Single-metric vs. bi-metric truncation: a confrontation 337
Summary: The general picture which emerges is a as follows. In the extreme UV, all
RG trajectories start out from initial points which are infinitesimally close to the non-Gaußian
fixed point which is at the heart of the Asymptotic Safety construction. In this region, the bi-
metric description is well approximated by the single-metric truncation. Once the trajectories
escape from the non-Gaußian fixed point regime and enter an intermediate region where λ .
0.15, say, split-symmetry is significantly violated, and the single-metric approximation becomes
unreliable. This can be seen in the Figs. 10.18, 10.19, and the explicit RG trajectories given in
the diagrams of section 10.2.5.
10.3.6 The (un-)reliability of the critical exponent calculations
What remains to be investigated is the precise relation between the single- and bi-metric results
concerning the non-Gaußian fixed point, in particular its location and critical exponents. Figs.
10.18 and 10.19 already shed some light on this question since ‘miraculously’ the best agree-
ment of single-metric and bi-metric results was found to be close to λD∗ . In this subsection we
concentrate on the implications of this observation and, most importantly, try to understand why
the predictions for the critical exponents differ so much in the two truncations.
A. Consider the single-metric and the bi-metric non-Gaußian fixed point regimes. Exactly
at the NGFP all anomalous dimensions are η I∗ = −(d−2). We first turn our attention to
the bi-metric Newton couplings at the fixed point:
(d−2)
gD∗ = (10.43a)BD2(λ∗
D;d)(d−2)−BD1(λ∗D;d)
( ) (d−2)g 0∗ = (10.43b)(0)
B D2 (λ∗ ;d)(d−2)−
(0)
B D1 (λ∗ ;d)
It is obvious that the better (0)B (λD;d) ≈ BD (λD1/2 ∗ 1/2 ∗ ;d) is satisfied the closer are
(0)
g∗ and
D = (p)g∗ g∗ , p ≥ 1, and the better is the split-symmetry. At the NGD+-FP the deviation is
small but non-zero; we expect (0)g∗ to be at best approximately equal to gD∗ , which in fact
was found in section 9.2. The main reason for the differing fixed point values is the
difference in the (0)/DB1 -functions. Both, B
D
1(λ ;d) and
(0)
B1 (λ ;d) decrease for increasing λ ,
and so does their difference. For λ < λD∗ the difference B
D
1(λ ;d)−
(0)
B1 (λ ;d) is positive,
and thus (0)g∗ < gD∗ .
(0)
B. The relation of g∗ and gD∗ to their single-metric cousin g
sm
∗ is more involved:
sm (d−2)g∗ = (10.44)(0)(λ sm; )( −2)− (0)B2 ∗ d d B sm1 (λ∗ ;d)
The difference of sm and (0)g∗ g∗ is a consequence of the differing fixed point values of the
cosmological constants, in particular we find λ sm∗ < λ
D
∗ . This discrepancy has a balancing
effect such that gD ≈ gsm while (0)g∗ < gsm∗ ∗ ∗ owed to the fact that (0)B1 (λ sm∗ ; ) >
(0)
d B1 (λ
D
∗ ;d)
for λ sm∗ < λ
D
∗ .
C. Moving away from the NGFPs the first question that arises is whether the respective
linearized flows in their vicinity are similar. The solution for the Newton constants in the
bi-metric truncation reads
( )θ ′ ( )
(1) k0
gD(k) = gD∗ +2c
(1) ′′
1V1 cos ϑc+ϑ +θ ln(k0/k) (10.45a)k
( )θ ′ ( ) ( )θ3
(0)( ) = (0) +2 (1)
k0 k0
g k g∗ c1V3 cos ϑc+ϑ
(1) +θ ′′ ln( / ) + (3)k0 k c3V
k 3 k
(10.45b)
338 Chapter 10. Background Independence
Here, , ϑ are real constants of integration, (j)c j c Vr is the rth component of the jth (real)
eigenvector, and ϑ (j) is its phase. The set of critical exponents consists of the complex
pair θ ′ ′′ ′1/2 = θ ± ıθ , where θ > 0, and the purely real, positive critical exponent θ3. in
addition to the spiral motion into the non-Gaußian fixed point present for the dynami-
cal coupling, there is an additional k2-term (θ3 = −2) that contributes to the running of
g(0)(k).
In the single-metric truncation the linearization yields instead
( )θ ′
k sm ( )sm( ) = sm+2 sm (1;sm) 0g k g c V cos ϑ sm +ϑ (1;sm)∗ 1 1 c +θ ′′k sm ln(k0/k) (10.46)
We see that qualitatively gsm(k) has the same kind of running as gD(k). Again, there is
a pair of complex conjugate critical exponents with a positive real part θ ′sm and a non-
vanishing imaginary part θ ′′sm that produces spirals. From a quantitative perspective the
results differ considerably, however, and it is instructive to understand why.
D. Despite the small difference of the respective fixed point coordinates, the critical expo-
nents disagree substantially in the single- and bi-metric truncation, as we discuss next.
Considering the single-metric and the dynamical bi-metric sector the two critical expo-
nents are given by
( ) √1 1
θ1/2 =− B11+B22 ∓ 4B12B21+(B11−B22)2 (10.47)2 2
where Brs is the r-s entry of the respective stability matrix:
( ) ( ) ( )
∂gβg ∂λ βg ⇒ −2.32 −27.8 −2.34 −10.4B= BD = , Bsm =
∂gβλ ∂λ βλ 0.72 −4.9 0.96 −0.61
(10.48)
Comparing BD1 to B
B
1 we observe that their first columns agree quite well, but the second
columns are rather different. The main difference between the matrices BD and Bsm is
the way they depend on the cosmological constant. Whereas a change in the Newton
couplings has, even quantitatively, a similar impact on BD and Bsm1 , a change in the cos-
mological constant affects the bi-metric system much more strongly than the ‘sm’ one.
Since in both cases the product 4B12B21 is negative and much larger than (B 211−B22) ,
the square root in (10.47) is purely imaginary, yielding
( ) √
θ ′ =−1 1B11+B ′′22 , θ = −4B12B21− (B 211−B22) (10.49)
2 2
The differences of the single-metric and dynamical bi-metric critical exponents are there-
fore approximately
θ ′ ′
1( )−θ ≈− BDsm 22−Bsm2 22 = 2.2 , (10.50a)√
θ ′′
√
−θ ′′sm ≈ −BD D12B21− −Bsm12Bsm21 = 1.3 . (10.50b)
So, numerically the error due to the single-metric approximation is indeed considerable,
more than two units (one unit) in the real (imaginary) part.
Taking the explicit structure of the beta-functions into account it thus becomes clear that it
is the slope of the functions B1/2(λ ;d) and A1/2(λ ;d) at λ = λ∗ that is mainly responsible
for the differences. It is apparent from Figs. 10.18 and 10.19 that these slopes are indeed
quite different for the ‘D’ and the ‘sm’ case, even at the intersection points where the
functions themselves agree. These λ -derivatives are the main reason for the quantitative
differences
10.3 Single-metric vs. bi-metric truncation: a confrontation 339
Summary: In conclusion we can say that the vicinity of the non-Gaußian fixed point is
sufficiently well described within the single-metric approximation if we are satisfied with ‘semi-
quantitative’ results. It correctly captures all qualitative properties of the flow. Our experience
with the present truncation suggests however that it will hardly be possible to perform precision
calculations of critical exponents in a single-metric truncation, not even with a very general
ansatz for Γk.
10.3.7 Comparison with the TT-based approach of ref. [172]
At this point it is worthwhile to check how our present bi-metric results compare to those ob-
tained in ref. [172] where a similar truncation ansatz including two Einstein-Hilbert actions for
gµν and ḡµν was used, and the same running couplings were investigated. Both calculations
rely on the conformal projection technique, actually first employed in [172], however with two
main differences:
A. The gauge choice: While in ref. [172] the ‘anharmonic gauge’, ϖ = 1/d, with gauge
parameter α → 0 was chosen, the present calculation uses the harmonic gauge fixing
condition, ϖ = 1/2, with gauge parameter α = 1− (d−6)Ω+O(Ω2). The dependence
of the ‘sm’ results on the gauge fixing parameter had already been investigated in ref.
[123, 135, 147], and the changes between α = 0 and α = 1 were found to be of the order
of a few percent only, so we expect the choice of α to be of minor importance.
B. Uncontracted derivatives: In the present calculation the uncontracted derivative terms
cancel due to the gauge choice. Therefore the heat kernel expansion becomes straight-
forward. Instead, in ref. [172], one had to deal with contracted as well as uncontracted
derivative terms, and it was necessary to apply a TT-decomposition to project the traces
in the FRGE onto the truncated theory space. This led to a set of new field variables
(irreducible component fields) and made the heat kernel expansion by far more involved
and lengthy.
Recall that a truncation of theory space is not specified by the ansatz for the EAA alone, but
in addition by a prescription for the projection on the field monomials. Variation of either ingre-
dient may alter the results. The difference of our present calculation to [172] can be understood
as stemming from the Γk-ansatz only, namely in the gauge fixing part of the EAA. Furthermore,
the cutoff action ∆Sk is slightly different in the two cases, since in [172] it is formulated in terms
of the irreducible (TT) components of hµν , while in our new approach simply in terms of the
undecomposed hµν . The comparison can help distinguishing between artifacts of the specific
truncation and robust, truncation-independent, results.
In what follows we focus on the properties of the doubly non-Gaußian fixed point in d = 4,
and compare the results of the TT-based bi-metric calculation in [172] with our present one, as
well as with the single-metric approximation.
Existence and location of non-Gaußian fixed points
In ref. [172] the same number of fixed points, in the same six categories as in our present
approach has been found, namely three different D fixed points, and for each of them two B
fixed points. The most important one for Asymptotic Safety is NGD D+-FP, having g∗ > 0. In
Tab. 10.1 we summarize its properties. Notice that the new and the old bi-metric results in
the D sector and their single-metric counterparts share the same qualitative as well as semi-
quantitative features, the B-sector differs in that the two approaches lead to λ B∗ and g
B
∗ values,
with the opposite signs even. However, the B-couplings describe only differences between level-
(0) and the D or p≥ 1 couplings and thus their precise values and signs are not meaningful as
such; they are an indication for the degree of split-symmetry, however. (The negative gB∗ found
in [172] only tells us that in this case (0)g∗ > gD∗ .)
340 Chapter 10. Background Independence
Bi-metric [172] Bi-metric (present) Single-metric [122]
NGB−⊕NGD+-FP NGB+⊕NGD+-FP NG-FP
(gD∗ = 1.05, λ
D
∗ = 0.22) (g
D
∗ = 0.70, λ
D
∗ = 0.21) (g
sm
∗ = 0.71, λ
sm
∗ = 0.19)
(gB∗ =−41.6, λ B = 0.58) (gB = 8.2, λ B∗ ∗ ∗ =−0.01)
(0)
NG+ ⊕
(0)
NGD+-FP NG+ ⊕NGD+-FP NG-FP
(gD D D∗ = 1.05, λ∗ = 0.22) (g∗ = 0.70, λ
D
∗ = 0.21) (g
sm
∗ = 0.71, λ
sm
∗ = 0.19)
( (0)g∗ = 1.08, λ
(0)
∗ = 0.21) (
(0)
g∗ = 0.65, λ
(0)
∗ = 0.19)
θ± = 4.5±4.2ı θ± = 3.6±4.3ı θ± = 1.5±3.0ı
sUV = 2D +2(0) = 4 sUV = 2D+2(0) = 4 sUV = 2
Table 10.1: The properties of the doubly non-Gaußian fixed points are listed for the bi-metric
calculation in [172], for the present one, and for the single-metric approximation [122]. The
dynamical fixed point properties obtained by the two bi-metric approaches are seen to be qual-
itatively equivalent, and also well approximated by the single-metric truncation. In the back-
ground sector, the results for gB B∗ and λ differ by sign, but only their combination with the ‘D’
parameters yielding the level-(0) couplings is numerically meaningful, and those are indeed
qualitatively similar. In addition the critical exponents for the dynamical sector are given. They
are complex, with a positive real part.
Impact of α and ϖ on the NGFPs
The impact of changing the functional form of the gauge fixing condition, concretely the param-
eter ϖ , and the parameter α in its pre-factor can be observed in the fixed point coordinates. The
single-metric, and the present bi-metric truncation employ the same harmonic gauge fixing and
α = 1 choices; in Table 10.1 we see that their fixed point values almost coincide. The TT-based
bi-metric calculation [172] which uses the ‘anharmonic’ choice for ϖ together with α = 0 leads
to a somewhat different fixed point value gD∗ > 1, with about the same λ
D
∗ . This is a small, but
visible effect due to the different gauge choices.
For the design of future, more advanced truncations it is also instructive to monitor how the
gauge choice influences the ghost sector and the beta-functions. In Fig. 10.21 we therefore plot
the ρgh-dependence of the fixed point coordinates obtained with the old approach [172]. For
comparison our findings from Fig. 9.2 are indicated as light gray lines. The diagrams show
basically overlapping curves for most coordinates. If ρgh . 2 the positive gD∗ values, though
qualitatively displaying the same increasing behavior for decreasing ρgh, differ quantitatively,
but by much less than one order of magnitude. The standard choice ρgh = 1 is within this
regime and thus reflects the small, but visible difference in the fixed point values of Tab. 10.1.
At large ρgh, the influence of ϖ in the ghost sector becomes completely negligible, at least in
the non-Gaußian fixed point regime.
We can get an indication for the quality of the split-symmetry in the vicinity of the NGFP
by the size of its gB∗ value, or more appropriately, by its inverse. The smaller 1/g
B
∗ , the better
is the coincidence of (0)g∗ and gD∗ . In Fig. 10.22 the dependence of 1/g
B
∗ on ρgh is shown for
both bi-metric calculations, that is [172] and the present one. As already pointed out, they yield
different signs for gB∗, but this is irrelevant.
The good news we learn from Fig. 10.22 is that 1/gB∗ possesses a zero, and that this zero
occurs in both calculations very close to ρgh = 1, that is, to the actually implemented value of
the ghost normalization! There, the magnitude of gB∗ diverges, and this in turn forces
(0)
g∗ and
10.3 Single-metric vs. bi-metric truncation: a confrontation 341
gD∗ λD∗
0.2
ρ
2 4 6 8 10 gh
0.1
-1 ρ
2 4 6 8 10 gh
GD-FP -0.1
-2 NGD−-FP
NGD+-FP -0.2
-3
-0.3
(a) The dependence of gD∗ on ρgh. (b) The dependence of λ
D
∗ on ρgh.
Figure 10.21: The dependence of gD∗ and λ
D
∗ on ρgh for the three fixed points, G
D-FP, NGD−-
FP, and NGD+-FP according to the approach of [172] (dark lines). The gray lines indicate the
corresponding results of the present calculation. The qualitative agreement of these results for
all fixed points is obvious. Except for very small values of ρgh, even quantitatively the results
are seen to be almost equal, indicating that the impact of ϖ on the ghost sector is relatively
small, and is probably negligible compared to the other sources of uncertainty.
1/gB∗ 1/g
B
∗
1.5
NGB⊕GD− -FP 5
NGB−⊕NGD−-FP1.0
NGB⊕NGD− +-FP 4 NGB⊕GD+ -FP
NGB+⊕NGD−-FP
0.5
3 NGB⊕NGD+ +-FP
0.0 ρ
1 2 3 4 gh 2
-0.5 1
0 ρgh
-1.0 1 2 3 4
(a) The dependence of 1/gB∗ on ρgh in [172]. (b) The dependence of 1/g
B
∗ on ρgh in the present cal-
culation.
Figure 10.22: The dependence of 1/gB∗ on ρgh is shown for the three ‘B’ non-Gaußian fixed
points. The smaller 1/gB∗ , the better split-symmetry is realized at the respective non-Gaußian
fixed point. We see that the curves for the physically most relevant one, based upon NGD+-FP,
displays a zero which in both calculations is located very close to ρgh = 1.
gD = (p)∗ g∗ , p ≥ 1 to be equal. The privileged status of a choice near ρP = 1 seems to be at least
one of the reasons for the ‘miraculously’ good agreement of the single- and bi-metric truncation
in the vicinity of the NGFP.
Critical exponents and UV-critical hypersurface
Turning next to the linearized flow near the (doubly) non-Gaußian fixed point, Tab. 10.1 shows
that the critical exponents for the two bi-metric calculations are more similar among themselves
than in comparison to the single-metric results. In fact, for the spiral motion they predict almost
the same frequency.
Still there is a difference between the bi-metric truncations, the origin of which can again
be traced back to the λD-dependence of the beta-functions. The stability matrix implied by the
differential equations in [172] reads:
( )
−2.37 −46.0
B= (TT-approach) (10.51)
0.49 −6.6
342 Chapter 10. Background Independence
While the first column, corresponding to the derivatives with respect to gD, shows only relatively
small deviations from BD1 in eq. (10.48), the source of the quantitative change in the critical ex-
ponents again originates in the λD-derivatives in the second column, which in turn are governed
by the λD-dependence of BD1(λ
D) in the vicinity of the non-Gaußian fixed point.
The agreement of the dimensionality of the critical hypersurface as predicted by both bi-
metric calculations is an additional point in support of the following general picture: The results
of the present, new bi-metric approach and the earlier TT-based one in [172] are in close analogy,
at least in the vicinity of the non-Gaußian fixed point. Deviations in the UV are of a minor
numerical kind, but most importantly all qualitative results for physically essential quantities
agree among the two approaches.
10.4 An application: the running spectral dimension
It has been observed very early on that the effective spacetimes described by the EAA, in partic-
ular those along asymptotically safe RG trajectories display self-similar properties reminiscent
of fractals, with a k-dependent effective dimensionality deff ≡ 4+ηN which interpolates be-
tween deff = 4 macroscopically and deff = 2 microscopically [123, 135]. This observation gave
rise to the development of a general scale dependent analog of Riemannian geometry for those
spacetimes [210, 211], and the discovery of a dynamically generated minimum length, a notion
that turned out surprisingly subtle [210].
Computing the spectral dimension Ds of those spacetimes [99] revealed the same crossover
from 4 dimensions in the IR to 2 in the UV which was observed on the basis of deff . It is an
exact, truncation independent prediction of asymptotically safe gravity, with or without matter.
More recently [100] the scale dependence of Ds was reconsidered under the more restrictive
assumptions of (i) pure gravity, (ii) the validity of the (single-metric) Einstein-Hilbert trunca-
tion, and (iii) the choice of an RG trajectory which admits a long classical regime [209, 212].
Under these conditions, the EAA predicts in addition an extended semiclassical regime at in-
termediate scales in which the spectral dimension assumes the rational value Ds = 4/3. It has
been argued [100] and substantiated by a detailed comparison with Monte Carlo data that the
dimensional reduction observed in numerical CDT simulations [213–215] actually originates
in this semiclassical regime rather than the asymptotic scaling region of a fixed point. (See [100,
168, 216] for further details, and [125, 217] for extensions.)
The scale dependent spectral dimension Ds(k) ≡ Ds(gk,λk) derived in [100], under the
above conditions, reads (for d = 4)
8
Ds(g,λ ) = − (10.52)4+λ 1βλ (g,λ )
where λ was the dimensionless cosmological constant of the single-metric truncation, λ sm ≡
λ̄ sm/k2. From the EAA-based derivation of eq. (10.52) it is obvious [99] that λk enters this
formula via the (contracted) Einstein-equation R(〈g〉 ) = 4λ̄ sm = 4k2λ smk k k which describes how
the effective metric responds to the scale dependence of the cosmological constant.
When we now go over from the single- to the bi-metric Einstein-Hilbert truncatio(n we
inte)r∣pret 〈g〉k as a self-consistent background metric. It is given by the tadpole equation
13 δΓk/
δh ∣ = 0 which, for the present truncation, has again the same structure as the classical
h=0
vacuum Einstein equation, but now containing the running level-(1) cosmological constant:
= −λ̄ (1)Ḡµν k ḡµν . Going through its derivation [100] it is therefore easy to see that the above
formula for the spectral dimension remains correct for the bi-metric truncation of the present
13See Section 4 of ref. [137] for a detailed discussion.
10.4 An application: the running spectral dimension 343
paper provided we set λ ≡ λ (1) in eq. (10.52), and interpret βλ as the beta-function of λ (1),
depending on the level-(1) couplings:14
( ) ( 8D (g 1 ,λ 1)s ) = (10.53)
4+(λ (1))−1β (1) (1) (1)λ (g ,λ )
Note that Ds is a scalar function on the g(1)-λ (1) theory space: it depends only on the value of
the couplings, but not the scale at which the RG trajectory passes there. It is well-defined if the
denominator on the RHS of (10.53) is always positive. This is indeed the case for the separatrix
and all type (IIIa)D trajectories, but not for those of type (Ia)D. They pass through a point where
the assumptions behind (10.52) and (10.53) do not apply [122] and Ds diverges.
It is instructive to insert solutions of the RG equations, 7→ ( (1), λ (1)k gk k ) = (gDk , λDk ), into eq.
(10.53) and determine the resulting scale-dependence of Ds. For each class of trajectories we
take one representative and depict the result in the following. While Ds is actually independent
of the B-couplings, it is most natural to think of these trajectories as 4-dimensional ones whose
B-sector is chosen in the split symmetry-restoring way.
Type (Ia)D trajectories.
For this class of trajectories the cosmological constant turns negative in the IR. This entails a
6
4
2
0
-2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mDPlanck
Figure 10.23: The spectral dimension along a typical (Ia)D trajectory. The interpretation as a
spectral dimension is lost near the singularity where the cosmological constant turns negative.
Besides a short semiclassical plateau at Ds = 4/3, disturbed by the divergence, the IR and the
UV limits at 4 and 2, respectively, are well visible.
divergence of Ds at a certain scale where the denominator of eq. (10.52) vanishes. In the vicin-
ity of this scale, the function Ds admits no physical interpretation as a ‘k-dependent spectral
dimension’, since this would require a very slow (‘adiabatic’) dependence on k. Apart from
that, we find in the corresponding Fig. 10.23 qualitatively precisely the same k-dependence of
Ds, which had been obtained within the single-metric approximation [100, 125]: In the IR the
running spectral dimension approaches Ds = 4, there exists a semiclassical plateau at Ds = 4/3,
here relatively short because it is disturbed by the divergence, and in the UV the running Ds
converges to its fixed point value, 2.
14In a truncation that distinguishes between g(p) for p≥ 1 the spectral dimension inherits an implicit dependence
on higher level couplings since those appear in β (1)λ .
Ds
344 Chapter 10. Background Independence
Type (IIa)D trajectory.
The separatrix in the D-sector gives rise to the scale-dependent Ds shown in Fig. 10.24. The
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mDPlanck
Figure 10.24: Spectral dimension along the (IIa)D trajectory, the separatrix in the dynamical
sector. The jump at k = 0 is a computational artifact; in reality the semiclassical regime with
Ds = 4/3 extends down to k = 0.
semiclassical plateau at Ds = 4/3 is quite pronounced in this case, and it seems to terminate
in a sudden jump to Ds = 4 for k → 0. Actually this jump is a computational artifact since in
a numerical calculation one is never able to find the separatrix exactly; rather, almost always
the computer will generate a type (Ia) or (IIIa) trajectory. Hence it is ultimately pushed away
from the GFP along the λ -direction, at some very low scale k/mPlanck ≪ 1, and this is exactly
what caused the apparent jump to Ds = 4 in Fig. 10.24. In reality, for the perfect separatrix
solution (and in absence of matter!) the semiclassical regime with Ds = 4/3 extends down to
k = 0; there exists no genuinely classical regime with Ds = 4 [100]. Towards the UV, we find
a smooth cross over of the semiclassical plateau to Ds = 2, as expected in the NGFP regime.
Type (IIIa)D trajectories.
For this type of trajectories, Fig. 10.25 shows the running spectral dimension along a typical
example. All three plateaus are well visible here, with Ds = 2 in the UV, Ds = 4 in the IR,
and an intermediate plateau, well below the Planck scale, with the semiclassical value Ds = 4/
3. This, again, is in accord with the results obtained in [100] by means of a single-metric
truncation.
Summarizing this subsection we can say that as far as the running spectral dimension is
concerned, the single-metric Einstein-Hilbert truncation is a fully reliable approximation to its
bi-metric generalization. By its very definition, a k-dependent spectral dimension makes sense
only if Ds(k) changes with k at most ‘adiabatically’ [100]. Basically Ds can be interpreted
meaningfully only when it develops a plateau. Yet, for all three types of trajectories, the single-
and bi-metric truncations agree on the respective plateau structures, and on the values which Ds
assumes there.
10.5 A brief look at d = 2+ ε and d = 3
Gravity in, or near two dimensions has always been an important theoretical laboratory for
quantum gravity. In particular, the Asymptotic Safety scenario was first proposed in the 2+ ε
Ds
10.5 A brief look at d = 2+ ε and d = 3 345
4.0
4.0
3.5
3.5
3.0
2.5
3.0
2.0
0.05 0.10 0.15 0.20
D
2.5 k/mPlanck
2.0
1.5
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
k/mDPlanck
Figure 10.25: The spectral dimension along a type (IIIa)D trajectory. All three plateaus are well
developed: we find Ds = 2 in the UV, Ds = 4 in the IR, and in between a semiclassical regime
with Ds = 4/3. The insert shows the classical regime at k≪ mDPlanck.
dimensional Einstein-Hilbert theory. In this section we re-analyze this theory in the bi-metric
setting. Because of its universality properties, absent in higher dimensions, our findings about
the relation between the single- and bi-metric treatment are particularly clearcut, and in fact
quite striking.
Also three dimensions are of special interest since in d = 3 the metric and ghost fluctuations
compensate exactly on shell. For the EAA which is a typical off shell object this implies by no
means that there is no RG flow in d = 3 (as is often believed wrongly). Rather, while certain
characteristic terms indeed disappear from the beta-functions, there is still a non-trivial RG
running which needs to be taken seriously, for instance, when one uses the EAA to construct the
continuum limit of a regularized functional integral for gravity. In d = 3 we have the advantage
that this characteristic ‘off-shell running’ can be studied in isolation.
In the following two subsections we discuss d = 2+ ε and d = 3 in turn; a complete list of
the pertinent beta-functions can be found in chapter 8.
10.5.1 Near dimension two
The case of d = 2+ ε dimensions is special in that all Newton constants become dimensionless
for ε → 0. In the lowest nontrivial order in ε , and for vanishing (dynamical) cosmological
constant, all anomalous dimensions have the structure
η I =−bI gI +O(ε) , I ∈ {D, B, (p)} (10.54)
with certain constants bI . For ε ց 0, the leading term in η I is of order ε0. Within our ap-
proximation of retaining in all formulae only the lowest nontrivial order the condition for a
non-Gaussian fixed point (ε +η I = 0) has a solution which is linear in ε , namely gI = ε/bI∗ .
Note also that in the approximation (10.54) the general relationship (10.22b) connecting the
level- to the D-B-language boils down to the simple statement
b(0) = bB+bD and b(p) = bD for p≥ 1. (10.55)
In the single-metric Einstein-Hilbert truncation [122], too, the anomalous dimension is well
known to have the structure (10.54). In its dia- vs. para-magnetic decomposed form, the perti-
Ds
Ds
346 Chapter 10. Background Independence
nent coefficient bI ≡ bsm was found to be [165]
[ ]
bsm
2 − 2 38= ( 3) + (4) + (6)para+(12) = × [+19] = (10.56)
3 dia gh−dia gh−para 3 3
All four contributions to bsm are separately universal, i.e. independent of the cutoff shape func-
tion (0)Rk .
Guided by our experience from 4 dimensions we might expect that the result of the single-
metric calculation approximately agrees at some level of accuracy with that in the D sector of
the bi-metric computation. In particular bsm of (10.56) should not be very different from bD, the
crucial coefficient in the anomalous dimension ηD of the dynamical Newton constant GDk . What
we actually find reads as follows:
2[ ( ) ( ) ]
bD = −2 3+2ρgh+15ρP Φ21 (0)+12 7−2ρgh ρP Φ32 (0)3
2[ ]
= −6−4ρgh+12ρP−12ρghρP (10.57)
3
The diamagnetic contributions are obtained from this expression by setting ρP = 0, and the
graviton part by letting ρgh = 0; furthermore, the paramagnetic contributions are those propor-
tional to ρP, and the ghost part is proportional to ρgh. Hence, when written in the style of the
single-metric result, eq. (10.57) reads
2[ ] 2 20
bD = (−6)dia+(−4)gh−dia+(12)para+(−12)gh−para = × [−10] =− (10.58)3 3 3
Obviously bsm and bD are quite different, not even their signs are in agreement so that screen-
ing and anti-screening behavior get interchanged. While the single-metric calculation predicts
bsm > 0, hence a NGFP at gsm∗ > 0 (for ε > 0), the bi-metric analogue has b
D < 0 with a corre-
sponding fixed point at a negative Newton constant, something one normally considers unphysi-
cal. This clash is particularly striking since, like in bsm, all 4 separate contributions appearing in
the dia/para, metric/ghost decomposition of bD are separately scheme independent, but none of
them agrees with its single-metric analogue. Indeed, in the second line of (10.57) we exploited
that all threshold functions of the type Φn+1n (0), for vanishing argument, assume the universal,
i.e. (0)Rk -independent values Φ
n+1
n (0) = 1/Γ(n+ 1), n ≥ 0. In this sense, both the single- and
the bi-metric results can be considered particularly robust and ‘clean’.
Turning to the anomalous dimension of the B-sector we find likewise
2[ ( ) ( ]
bB = (4ρgh −3)Φ10 (0)+2 3+2ρ 2 3gh+6(3+ρgh)ρP Φ1 (0)−12 7−2ρgh)ρPΦ3 2 (0)
2[ ]
= 3+8ρgh−6ρP+24ρghρP . (10.59)
3
This expression, again, contains only universal values of the threshold functions. Casting
(10.59) in a more instructive way, we obtain
2[ ] 2 58
bB = (3)dia+(8)gh−dia+(−6)para+(24)gh−para ≡ × [+29] = (10.60)
3 3 3
Notice that in total bB is positive while bD was negative.
What makes these findings particularly alarming, or at least puzzling at first sight is that they
show a considerable degree of internal consistency. To see this, let us add up the b-coefficients
of the background and the dynamical sector, thereby maintaining the dia/para, metric/ghost
decomposition:
2[ ]
bB +bD = (3−6)dia+(8−4)gh−dia+(−6+12)para+(24−12) sm3 gh−para = b
(10.61)
10.5 A brief look at d = 2+ ε and d = 3 347
Remarkably enough, not only does the sum bB + bD exactly agree with the old single-metric
result, even all 4 terms of the decomposition separately do so. The ‘miracle’ behind (10.61)
finds its explanation when we evaluate the level-(0) anomalous dimension, the corresponding
coefficient b(0) being
2[ ( ) ]
b(0) = (4ρgh −3)Φ10 (0)+6 1+2ρgh ρ Φ2P 1 (0)3
2[ ]
= −3+4ρgh+6ρP+12ρghρP (10.62)
3
This result coincides, not only as a sum but even term by term, exactly with the single-metric
result (10.56):
[ ]
b(
2
0) = (−3) + (4) + (6) + (12) = bsmdia gh−dia para gh−para (10.63)3
From our discussion in Section 10.3 of the relation between single- and bi-metric beta-functions
the equality b(0) = bsm was indeed to be expected; there we demonstrated that quite generally the
single-metric RG equations are closely related to the level-(0) ones in the bi-metric computation.
It is reassuring to see this rule at work here, despite potential subtleties related to the limit ε → 0.
On the other hand, from (10.55) we know that at the level of the b-coefficients the translation
rule from theD-B to the level description is simply b(0) = bB+bD. Combining this with b(0) = bsm
from (10.63) we obtain bB+bD = bsm, and this is precisely what we found in eq. (10.61)!
At this point several important remarks are in order:
A. The main message conveyed by the above universal numbers is that the leading order in
ε is suffering from a significant violation of split-symmetry, as is testified by the values
b(0) = bsm = + 383 at level zero, and b
(1,2,3,··· ) = bD = − 203 at the higher levels. Equivalent
to that, there is a corresponding disagreement between the single- and bi-metric results,
even at the level of signs.
B. In order to judge to what extent these somewhat disconcerting findings might carry over
to higher dimensions, d = 4 in particular, the (perhaps) worried reader should recall the
following facts.
B.1. In general dimensions, ηD(gD,λD) is given by eq. (8.33), and it involves two func-
tions of the dynamical cosmological constant, BD1 and B
D
2 . Expanding this anoma-
lous dimension in gD, and retaining the leading, i.e. linear term only, it reads
ηD(gD,λD;d) = BD1(λ
D;d)gD +O(gD2) (10.64)
Upon an additional expansion of BD1 for small λ
D the approximation for ηD boils
down to
ηD(gD,λD;d) =−(d−2)ωD gD +O(gD2)+O(λDd ) (10.65)
It is characterized by a single coefficient only, ωD Dd ≡−B1(0;d)/(d−2).
B.2. If we set d ≡ 2+ ε and then expand ηD of eq. (8.33) in powers of ε w(e obtain)
ηD =−bD gD+O(ε) for the lowest non-trivial order in ε , with bD = lim → ε ωDε 0 2+ε
a well defined, and non-zero number. Notice that this simple equation for ηD has the
same structure as the doubly expanded (in gD and λD) general result (10.65) valid
for all d. Here, however, it results from nothing more than the ε → 0 limit alone
[122]; no separate assumption about the smallness of the couplings and, based on
that, expansions with respect to gD, λD are invoked. Stated differently, if we retain
only at the lowest order in ε , it is unavoidable that ηD becomes linear in gD and,
more importantly, that the coefficient function BD(λD1 ) automatically always gets
348 Chapter 10. Background Independence
evaluated at the point λD = 0 only. So, the essential conclusion for the present
discussion is that in the limit ε → 0, to leading order in ε , the RG flow completely
‘forgets’ about how a non-zero value of λD affects the RG running of gD.
B.3. In accord with this last remark, even the NGFP at gD∗ = ε/b
D is fully determined by
bD, i.e. by BD1(λ
D = 0). For ε ց 0, the very same coefficient which for general d
controls the semiclassical regime near the GFP only also decides about whether or
not there exists the desired NGFP on the gD > 0 half-space. We saw that this led
to a clash between the single- and bi-metric calculation, since bsm > 0, but bD < 0.
However, in subsection 9.2.4 we also saw already that in d = 4 the situation is
different in a crucial way. There BD(λD1 ) changes its sign between λ
D = 0 and
λD = λD∗ ; as a result, the semiclassical regime on the g
D > 0 half-space exhibits
gravitational screening (ηD > 0), as in d = 2+ ε , while for larger values of λD the
sign of ηD turns negative, ultimately approaches ηD∗ =−2, and a NGFP forms at a
positive value of Newton’s constant, exactly where we would like it to be.
C. Summarizing the above remarks, we can say that the most problematic property of the
(2+ε)-dimensional theory, the dynamical Newton constant having a negative fixed point
value15, is an artifact of the ε-expansion. In d = 4, instead, screening at small λD can
coexist with anti-screening and a NGFP at larger values of the cosmological constant
thanks to the λD-dependence of ηD, a property the leading order of the ε-expansion is
completely insensitive to.
10.5.2 Three dimensions
Next, let us turn our attention to spacetimes of dimension d = 3. The full set of results is given
in section 8.B of chapter 8. First we focus on the anomalous dimensions in the semiclassical
regime which, for all d, read η I = −(d− 2)ω I gI +O(gI 2,λDd ). In d = 2+ ε the coefficients
ω Id were universal, in d = 3 they explicitly depend on the cutoff
(0)
Rk instead. To make our
main point as clear as possible, it is instructive to write down the dia/para and metric/ghost
decomposed form of the coefficients appearing in η I =−ω I gI3 +O(gI 2,λD) for I = {D, B, (0)}.
In the D and B sector we find, respectively:
1 [( ) ( ) ]
ωD3 =−√ 3+2ρgh+2ρP(9−4ρ ) Φ2gh 3/2 (0)−6ρP 9−4ρgh Φ35/2 (0) (10.66)π
√1
[ ( )
ωB3 = 2(ρ
1 2
gh −1)Φ1/2 (0)+ 3+2ρgh+4ρP(6−ρgh) Φπ 3/2 (0)
( ) ]
−6ρP 9−4ρgh Φ35/2 (0)
Notice that ωD B3 and ω3 are perfectly generic, in the sense that they receive contributions of both
paramagnetic and diamagnetic origin, from both ghost and metric fluctuations.
The expected ‘magic’ happens only in the case of the level-(0) coefficient: From the general
identity
η (0) ηD ηB η (p) ηD
= + and = for all p≥ 1 . (10.67)
(0) gD gB (p) Dgk k k g
g
k k
we infer ω (0) = ωD+ωB3 3 3 in the present approximation, yielding
[ ]
ω (0) = √2 (ρ −1)Φ1 (0)+ρ (3+2ρ )Φ23 gh 1/2 P gh 3/2 (0) (10.68)π
15We assume ε > 0 here, as always.
10.6 Summary and conclusion 349
Upon ‘switching on’ the ghosts by setting ρgh = 1, the entire coefficient ω
(0)
3 is seen to be pro-
portional to ρP. Hence
(0)
the level-(0) anomalous dimension η (0) ∝ ω3 is of purely paramagnetic
origin.
The non-zero diamagnetic contributions from ωD3 and ω
B
3 have canceled precisely. At the
level of the bi-metric EAA, for generic arguments (‘off shell’ in particular), this cancellation is
the only immediate reflection of the fact that classical Einstein-Hilbert gravity has no propagat-
ing modes in d = 3. There is no comparable compensation of metric and ghost modes at higher
orders.
The observed perfect cancellation of all diamagnetic terms in ω (0)3 is fully consistent with
earlier results on the single-metric case [165] where the same cancellation was found to occur
in the analogous coefficient ω sm3 . Indeed, it can be verified again that ω
(0)
3 = ω
sm
3 , as it should be
for general reasons.
Finally, let us leave the semiclassical regime and consider the possibility of non-Gaussian
fixed points in 3 dimensions. Using the full fledged beta-functions of section 8.B we do indeed
find such fixed points. The results can be summarized as follows:
d=3 NGD-FP NGD+ −-FP G
D-FP
(gD∗ =0.13,λ
D D
∗ =0.2) (g∗ =−0.38,λD∗ =−0.25) (gD∗ =0,λD∗ =0)
NGB+-FP NG
B
+⊕NGD+-FP NGB+⊕NGD−-FP NGB⊕GD+ -FP
(gB=1.3,λB
(10.69)
∗ ∗ =−0.97) (gB∗=0.17,λB∗ =−0.14) (gB∗=0.18,λB∗ =−0.15)
GB-FP GB⊕NGD B+-FP G ⊕NGD−-FP GB⊕GD-FP
(gB=0,λB=0) (gB=0,λB∗ ∗ ∗ ∗ =0) (g
B
∗=0,λ
B
∗ =0)
We find, as in d = 4, a total of six fixed points, five of which are non-Gaussian. Remarkably
enough, the qualitative picture is exactly the same as in the four dimensional case, displaying a
non-Gaussian fixed point both in the upper and lower half-plane of gD, as well as a Gaussian one
at gD∗ = 0. Instead, the background coordinate value g
B (λ B∗ ∗ ) is found to be positive (negative).
Furthermore, a relatively large hierarchy gD ≈ 10gB∗ ∗ for the doubly non-Gaussian NGB⊕NGD+ +-
FP, indicating approximate split-symmetry, is found also in d = 3.
10.6 Summary and conclusion
In this section we start with a summary of this chapter by means of a brief and concise list dis-
playing the main results. Therefore, it is convenient to also recapitulate the findings of chapter
9 that includes the fixed points of the RG flow and its properties, thus addressing the question
of Asymptotic Safety within the present truncation. We then close with a number of general
conclusions and the essential lessons for future investigations, which we learned here.
10.6.1 Summary of the main results
A. On the technical side, we employed and tested a new method for dealing with operator traces
involving uncontracted covariant derivatives which is much simpler than those used before. We
found that, within the expected truncation uncertainty and cutoff dependence, the resulting RG
flow matches the results obtained with the conventional method based on a York decomposition,
even though, mathematically, the pertinent beta-functions are quite different. We hope that
similar methods will be helpful also in future{b(i-metric calcul)at}ions.
B. On the 4-dimensional theory space T≡ gD, λD, gB( ) , λ
B the RG flow was found to de-
compose hierarchically according{t(o g
D,λD → gB → λ B)} . This allows us to compute the flow
on the dynamical subspace TD ≡ gD, λD without reference to the background couplings.
350 Chapter 10. Background Independence
C. The main characteristics of the 2-dimensional flow on TD are as follows:
C.1. There exist two non-Gaußian and one Gaußian fixed point on TD, namely NG
D
−-FP,
NGD+-FP, and G
D-FP. They are located at a negative, positive, and vanishing coordinate
gD∗ , respectively.
C.2. Reliability analyses reveal that NGD+-FP is very robust, while NG
D
−-FP is likely to be
a truncation artifact.
C.3. The critical exponents of NGD+-FP are given by a complex conjugate pair, with a
non-zero imaginary part leading to spiral-shaped trajectories. The fixed point is UV
attractive in both directions.
C.4. There are no RG trajectories crossing the gD = 0 line, in particular there exists no
cross-over trajectory connecting NGD−-FP to NG
D
+-FP. We may therefore restrict TD to
the half-plane with gD > 0 which is invariant under the flow.
C.5. The phase portrait on TD is very similar to the one obtained with the single-metric
truncation. In particular the properties of NGD+-FP are numerically similar to those of
the single-metric fixed point, and the RG trajectories admit exactly the same (‘type Ia,
IIa, IIIa’) classification that has been introduced for the single-metric case.
C.6. In contrast to all single-metric based predictions, gravitational anti-screening is lost
near the Gaußian fixed point GD-FP.
D. Each RG trajectory on TD, together with initial conditions for g
B and λ B, gives rise to a
4-dimensional trajectory. We analyzed the corresponding flow on the complete theory space T.
D.1. On T, there exists a total of 6 fixed points, each one of the three D-fixed points above
can be combined with both a Gaußian and a non-Gaußian fixed point of the background
couplings.
D.2. The doubly non-Gaußian fixed point on T, NGB D+⊕NG+-FP, is the natural candidate
for the construction of an asymptotically safe infinite cutoff limit. It possesses 4 rele-
vant directions, so the UV critical hypersurface associated to it, SUV, has maximum
dimension within the present truncation.
E. In order to define a quantum field theory, we need an RG trajectory which, at ‘k = ∞’, starts
out infinitesimally close to the fixed point, always runs on SUV, and thereby gradually becomes
split-symmetric, at the very least in the physical limit when k approaches zero. This is the
crucial requirement of Background Independence. Within the bi-Einstein-Hilbert truncation it
requires gravΓk to loose its ‘extra ḡ-dependence’, that is, 1/G
B B B
k and λ̄ /Gk must vanish in the IR
limit of low scales k approaching zero.
E.1. For the first time, it was possible to demonstrate explicitly there do indeed exist
trajectories which meet both of the two key requirements, Asymptotic Safety and Back-
ground Independence, simultaneously. Those trajectories are labeled by only two free
parameters. Thus the theory’s predictivity is actually higher than expected on the basis
of a 4-dimensional SUV.
E.2. The RG trajectories which restore split-symmetry in the IR (at k= kIR) were found to
be precisely those which merge with the ‘running UV attractor’ at low scales, AttrB(k).
F. The relevance of the running UV attractor AttrB(k) to the problem of split-symmetry restora-
tion was uncovered by the following sequence of results(and obs)ervations.
F.1. For every fixed trajectory UD on TD, i.e. k→7 gD D Dk , λk ≡U (k) the background cou-
plings gBk and λ
B
k are governed by a non-autonomous, i.e. explicitly R{G(-time d)e}pendent
system of differential equations on the 2-dimensional subspace TB ≡ gB, λ B . It can
be visualized as a time dependent vector field ~βB(k) on TB, which also depends on the
D-trajectory chosen.
F.2. For every trajectory UD( , and at ev)ery fixed RG time k, the vector field ~βB(k) has an
(instantaneous) zero at gB•(k), λ
B
• (k) ∈ TB. The (likewise instantaneous) stability anal-
10.6 Summary and conclusion 351
y(sis reveals that it is U)V attractive in both B-directions, hence the curve k→7 Attr
B(k)≡
gD, λDk k , g
B
•(k), λ
B
• (k) acts as a ‘running UV attractor’. It is not an RG trajectory by
itself.
F.3. The UV limit of the running attractor is precisely the doubly non-Gaußian fixed
point: limk→∞ Attr
B(k) = NGB+⊕NGD+-FP.
F.4. Picking a D-trajectoryUD we are equipped with a k-dependent vector field ~βB on TB.
Integral curves of ~β (k′B ) at any fixed moment of ‘time’, k′, are not projections onto TB
of an RG trajectory on the full theory space in general; they correspond to ‘snapshots’
of the phase portrait on TB (at this v)ery moment.
F.5. For every ‘initial’ point gBin, λ
B
in specified on TB at the time kin there exi(sts a uni)que
solution of the RG equations through this point, k 7→UB(k), withUB(kin) = gBin, λ Bin . It
is obtained by integrating the (now explicitly k dependent) equation ∂ UBt (k)=~βB(U
B(k);k)
both upward and downward. Making all input data explicit, we denote this solution
as UB{UD;k ;gB( in in, λ
B
in})(k). It gives rise to an RG trajectory on the full theory space:
U(k) = UD(k),UB(k) .
F.6. We find that in the limit k ≫ kin (of a lon)g upward evolution, U
B{UD;k B Bin;gin, λin}
looses its memory of the initial point gB Bin, λin , and a universal limit curve is obtained:
lim Bk≫kinU {UD;kin;gBin, λ Bin}(k) =UB• (k)
F.7. Every D trajectory UD implies a specific limit curveUB( • (k), and to)gether they define
an RG trajectory on the full theory space: k→7 U•(k) = UD(k),UB• (k) ). This trajectory
is precisely the one which, in the I(R, ends o)n the( running attracto)r Attr
B(k).16 It is
determined by the ‘final condition’ gBk , λ
B
k = g
B B
IR IR •(kIR), λ• (kIR) with kIR → 0. In
short, U•(kIR) = Attr
B(kIR), while in the opposite limit U•(k) approaches the doubly
non-Gaußian fixed point: U•(k→ ∞) = NGB+⊕NGD+-FP.
F.8. We found that the class of all trajectories U•(k), obtained by varying the underlying
UD(k), are of special importance: The trajectories in this class, and only those, restore
split-symmetry in the IR, being at the same time asymptotically safe. This explains the
relevance of the running UV attractor to the problem of split-symmetry restoration.
G. Performing a detailed comparison of the single- and the bi-metric Einstein-Hilbert trunca-
tion we found that, quite unexpectedly, the former is a rather precise approximation to the latter
in the vicinity of the non-Gaußian fixed point. In the far IR the split-symmetry restoring tra-
jectories, by construction, give rise to another regime in which the two truncations agree well.
However, in between there are strong qualitative differences, for instance with respect to the
sign of the dynamical anomalous dimension ηD. Furthermore, the quantitative differences of
the critical exponents in both settings are quite significant, despite the ‘miraculous’ precision
of the single-metric truncation near the NGFP. They clearly show the limitations of the single-
metric approximation. Results from an independent calculation using the TT-decomposition
techniques [172] support these findings.
H. As a concrete application of the bi-metric flow in d = 4 we computed the running spectral
dimension Ds(k) of the emergent fractal spacetimes according to the definition proposed in
[100]. We found that the bi- and single-metric results agree on all universal predictions this
definition can give rise to, namely the formation of plateaus on which Ds(k) assumes the values
Ds = 4, Ds = 4/3, and Ds = 2, respectively.
I. We saw that in d = 3 the bi-metric flow is very similar to the one in d = 4, even though the
classical Einstein-Hilbert theory in d = 3 has no propagating modes. This makes it particularly
16Note that numerically constructing the trajectory reaching BAttr (kIR) involves fine-tuning, in the sense that the
desired final point is IR repulsive in both B-directions. From this perspective the UV attractor is better called an ‘IR
repeller’.
352 Chapter 10. Background Independence
clear that, in any dimension, due to the off-shell nature of the EAA not only the ‘radiative’ field
modes but also the ‘coulombic’ ones play an important role in driving the RG evolution. This
is in accord with the physical picture of ‘paramagnetic dominance’ put forward in [165, 195].
J. The most striking and even qualitatively essential discrepancies between the single- and
the bi-metric predictions we encountered in the leading order of the ε-expansion about two
dimension (d = 2+ε , ε > 0). Disentangling metric /ghost and dia- /paramagnetic contributions,
both the anomalous dimension ηD and its single-metric approximation η sm are characterized by
4 separately universal coefficients. All 4 of them were found to disagree between ηD and η sm.
Thus, contrary to the single-metric prediction, the bi-metric RG flow possesses no NGFP at
positive Newton constant. We were able to show that this discrepancy and the related strong
breaking of split-symmetry are an artifact of the leading order ε-expansion, which does not
generalize to higher dimensions.
10.6.2 Conclusions and outlook
The main message of the present investigations for future work on the Asymptotic Safety pro-
gram is quite clear: From a certain degree of precession onward it seems to make little sense
to keep including further invariants in the truncation ansatz that are built from the dynamical
metric alone. Rather, to the same extent we allow the effective average action to depend on
gµν in a more complicated way, also its dependence on the background metric ḡµν must be
generalized. While the picture conveyed by the single-metric truncations is qualitatively correct
often, they are certainly insufficient for quantitatively precise calculations, the determination of
critical exponents for example.
Taking the bi-metric structure of the EAA seriously we find ourselves in a situation which is
very widespread in quantum field theory: One starts from a classical field theory with a certain
symmetry, tries to quantize it, thereby discovers that some sort of regularization is needed,
then picks a regulator which spoils the symmetry, perhaps because there exists no invariant
regularization, and finally after the quantization one tries to re-establish this symmetry at the
observable level, or close to it. A well known example of this situation is the quantization of
the electromagnetic field in a scheme which violates gauge invariance. Computing radiative
corrections one then might encounter a non-zero photon mass at the intermediate steps of the
calculation. At the end, however, it must always be possible to choose the bare parameters in
such a way that the renormalized, physical photon mass turns out exactly zero.
In quantum gravity, the background quantum split-symmetry should be seen as analogous
to gauge invariance in this example. Split-symmetry is a formal way to express the arbitrariness
of the background on which we quantize the metric fluctuations, and unbroken split-symmetry
is tantamount to Background Independence. The effective average action is not invariant per se,
but the space of RG trajectories contains special solutions which re-establish split-symmetry at
the physical level.
In this paper we showed explicitly that the restoration of split-symmetry is indeed possible,
and how it can be achieved in practice. In future bi-metric analyses of Quantum Einstein Gravity
this will always be a central and important step. The analogy with the photon mass makes it very
clear that the implementation of split-symmetry deserves considerable attention since otherwise
we never can be sure to deal with the right ‘universality class’. After all, comparing QEG
to the familiar matter field theories on flat space, its most momentous distinguishing feature
is Background Independence; it implies in particular the necessity of an ab initio derivation
of the arena all non-gravitational physics takes place in, namely spacetime. Clearly this has
much more profound consequences for the general structure of the theory than its notorious
perturbative non-renormalizability, for example, which shows up at a secondary technical level
only.
11.1 Introduction
11.2 The effective average action as a ‘C-
function’
Counting field modes
Running actions and self-consistent back-
grounds
FIDE, FRGE, and WISS
Pointwise monotonicity
Monotonicity vs. stationarity
The proposal
The mode counting property revisited
11.3 Asymptotically safe quantum gravity
The single- and bi-metric Einstein-Hilbert trun-
cations
Gravitational instantons
Numerical results
Crossover trajectories and their mode count
11.4 Discussion and outlook
11.A The special status of Faddeev-Popov
ghosts
11.B The trajectory types Ia and IIa
11. A GENERALIZED C-THEOREM
C This chapter follows closely ref. [193].
We develop a generally applicable method for constructing functions, C , which have properties
similar to Zamolodchikov’s C-function, and are geometrically natural objects related to the
theory space explored by non-perturbative functional Renormalization Group (RG) equations.
Employing the Euclidean framework of the Effective Average Action (EAA), we propose a
C -function which can be defined for arbitrary systems of gravitational, Yang-Mills, ghost, and
bosonic matter fields, and in any number of spacetime dimensions. It becomes stationary both
at critical points and in classical regimes, and decreases monotonically along RG trajectories
provided the breaking of the split-symmetry which relates background and quantum fields is
sufficiently weak. Within the Asymptotic Safety approach we test the proposal for Quantum
Einstein Gravity in d> 2 dimensions, performing detailed numerical investigations in d= 4. We
find that the bi-metric Einstein-Hilbert truncation of theory space, here discussed for the results
of II and for those of ref. [172], is general enough to yield perfect monotonicity along the RG
trajectories, while its more familiar single-metric analog fails to achieve this behavior which
we expect on general grounds. Investigating generalized crossover trajectories connecting a
fixed point in the ultraviolet to a classical regime with positive cosmological constant in the
infrared, the C -function is shown to depend on the choice of the gravitational instanton which
constitutes the background spacetime. For de Sitter space in 4 dimensions, the Bekenstein-
Hawking entropy is found to play a role analogous to the central charge in conformal field
theory. We also comment on the idea of a ‘λ̄ -N connection’ and the ‘N-bound’ discussed
earlier.
11.1 Introduction
One of the most remarkable results in 2-dimensional conformal field theory is Zamolodchikov’s
c-theorem [218, 219]. It states that every 2D Euclidean Quantum Field theory with reflection
positivity, rotational invariance, and a conserved energy momentum tensor possesses a function
354 Chapter 11. A generalized C-theorem
C of its coupling constants, which is non-increasing along the renormalization group trajecto-
ries and is stationary at fixed points where it equals the central charge of the corresponding
conformal field theory.
After the advent of this theorem many authors tried to find a generalization that would be
valid also in dimensions greater than two [220–229]. This includes, for instance, suggestions
by Cardy [220] to integrate t∫he trac√e anomaly of the energy-momentum tensor 〈Tµν〉 over a
4-sphere of unit radius, C ∝ S4 d
4 µx g〈Tµ 〉, the work of Osborn [221, 222], and ideas based
on the similarity of C to the thermodynamical free energy [223], leading to a conjectural ‘F-
theorem’ which states that, under certain conditions, the finite part of the free energy of 3-
dimensional field theories on S3 decreases along RG trajectories and is stationary at criticality
[224]. Cappelli, Friedan and Latorre [225] proposed to define a C-function on the basis of the
spectral representation of the 2-point function of the energy-momentum tensor.
While these investigations led to many important insights into the expected structure of the
hypothetical higher-dimensional C-function, the search was successful only recently [230, 231]
with the proof of the ‘a-theorem’ [220, 229]. According to the a-theorem, the coefficient of
the Euler form term in the induced gravity action of a 4D theory in a curved, but classical,
background spacetime is non-increasing along RG-trajectories.
Clearly theorems of this type are extremely valuable as they provide non-perturbative infor-
mation about quantum field theories or statistical systems in the strong coupling domain. They
constrain the structure of possible RG flows on theory space, and they rule out exotic behavior
such as limit cycles, for instance (at least for a suitable class of beta-functions [232]).
In this paper we describe and test a broadly applicable search strategy by means of which
generalized ‘C-like’ functions could be identified under rather general conditions, in particular
in cases where the known c- and the a-theorems do not apply. Our main motivation is in fact
theories which include quantized gravity, in particular those based upon the Asymptotic Safety
construction [122, 145, 168–171, 233].
In a first step, we try to generalize only one specific feature of Zamolodchikov’s C-function
for a generic field theory in any number of dimensions, namely its ‘counting property’: the
sought-after function should roughly be equal to (or at least in a known way be related to) the
number of degrees of freedom that are integrated out along the RG trajectory when the scale is
lowered from the ultraviolet (UV) towards the infrared (IR). Technically, we shall do this by
introducing a higher-derivative mode-suppression factor in the underlying functional integral
which acts as an IR cutoff. We can then take advantage of the well established framework of
the Effective Average Action (EAA) to control the scale dependence [110–114].
In a generic theory comprising a certain set of dynamical fields, Φ, and corresponding back-
ground fields, Φ̄, the EAA is a ‘running’ functional Γk[Φ,Φ̄] similar to the standard effective
action, but with a built-in IR cutoff at a variable mass scale k. Its k-dependence is governed by
an exact functional RG equation the FRGE we discussed in detail in chapter 4.
The specific property of the EAA which will play a crucial role in our approach is the
following: For a broad class of theories, those that are ‘positive’ in a sense we shall explain, the
very structure of the FRGE implies that the EAA is a monotonically increasing function of the
IR cutoff:
∂kΓk[Φ,Φ̄]≥ 0 ∀(Φ,Φ̄) (11.1)
We shall refer to this property as the pointwise monotonicity of the EAA since it applies at all
points (Φ,Φ̄) of field space independently. Thus the EAA provides us with even infinitely many
monotone functions of the RG scale k, one for each field configuration (Φ,Φ̄). So one might
wonder if those functions, or a combination thereof, deserve being considered a generalization
of Zamolodchikov’s C-function.
11.1 Introduction 355
Unfortunately the answer is negative, for the following reason. The pointwise monotonicity
property refers to Γk when it is evaluated at dimensionful field arguments, and is parametrized
by dimensionful running coupling constants. However, the c-theorem and its generalizations
apply to the RG flow on theory space, T, a manifold which is coordinatized by the dimension-
less couplings. The latter differ from the dimensionful ones by explicit powers of k fixed by
the canonical scaling dimensions. As a consequence, when rewritten in terms of dimensionless
fields and couplings, the property (11.1) does not precisely translate into a monotonicity state-
ment about the de-dimensionalized theory space analog of Γk, henceforth denoted Ak. Rather,
when the derivative ∂k hits the explicit powers of k, additional canonical scaling terms arise
which prevent us from concluding simply ‘by inspection’ that Ak is monotone along RG trajec-
tories.
Nevertheless, our main strategy will be to take maximum advantage of the pointwise mono-
tonicity of Γk as the primary input, and then try to get a handle on the monotonicity violations
that occur in going from Γk to Ak. We shall see that by evaluating the action functionals on
special field configurations this difficulty can be reduced to a far more tractable level.
A related issue is that Γk, while having attractive monotonicity properties, does not in addi-
tion also become stationary at fixed points, as a sensible generalization of the C-function in 2D
should. But, again for the above reason, this is not necessarily a drawback since at an RG fixed
point it is anyhow not the dimensionful couplings but rather the dimensionless coordinates of
theory space that are supposed to assume stationary values, i.e. actually it is Ak that approaches
a ‘fixed functional’, A∗.
When we look at Γk and Ak pointwise, or equivalently, at the infinitely many k-dependent
couplings they parametrize them independently, then there is a clash between stationarity at
fixed points and monotonicity along the RG flow: Ak is stationary at fixed points, but not mono-
tone, while for Γk the situation is the other way around. However, by adopting the pointwise
perspective we are expecting by far too much, namely that all dimensionless couplings individ-
ually behave like a C-function. Presumably we can hope to find at best a single, or perhaps
a few, real valued quantities with all desired properties. We shall denote such a hypothetical
function by Ck in the following. Assuming it exists, the quantity Ck is one function depending
on infinitely many running couplings along the RG trajectory, so the transition from Γk to Ck
amounts to a tremendous ‘data reduction’.
Thus, within the EAA framework, the central question is whether there exists a kind of
‘essentially universal’ map from k-dependent functionals Γk to a function Ck that is monotone
along the flow and stationary at fixed points. Here the term ‘universal’ is to indicate that we
would require only a few general properties to be satisfied, comparable to reflection positivity,
rotational invariance, etc. in the case of Zamolodchikov’s theorem. The reason why we believe
that there should exist such a map is that the respective monotonicity properties of Γk and the
C-function in 2D have essentially the same simple origin. They both ‘count’ in a certain way
the degrees of freedom (more precisely: fluctuation modes) that are integrated out at a given
RG scale intermediate between the UV and the IR.
We begin by considering a class of Ck-candidates which are obtained by evaluating Γk[Φ,Φ̄]
at a particularly chosen pair of arguments (Φ,Φ̄) that will have an explicit dependence on k in
general. In this chapter we propose to use self-consistent background field configurations for
this purpose. We evaluate the EAA at a scale dependent point in field space, namely Φ =
Φ̄ ≡ Φ̄s.c.. By definition, a background field Φ̄ ≡ Φ̄s.c.k k is said to be self-consistent (‘s.c.’) if
the equation of motion for the dynamical field Φ that is implied by Γk admits the solution
Φ = Φ̄. With other words, if the system is put in a background which is self-consistent, the
fluctuations of the dynamical field, ϕ ≡ Φ− Φ̄, have zero expectation value and, in this sense,
do not modify this special background. As we shall demonstrate in detail in section 11.2, the
356 Chapter 11. A generalized C-theorem
proposal Ck = Γ [Φ̄s.c. s.c.k k ,Φ̄k ] is indeed a quite promising candidate for a generalized C-function.
It is stationary at fixed points and it is ‘close to’ being monotonically decreasing along the flow.
The phrase ‘close to’ requires an explanation. Especially in quantum gravity, Background
Independence is a central requirement [73, 234]. While in the causal dynamical triangulation
approach [94–98] or in loop quantum gravity [72, 93], for instance, this requirement is met by
strictly not using a background spacetime structure at all, the EAA framework uses the back-
ground field technique [27]. At the intermediate steps of the quantization one does introduce a
background spacetime, equipped with a non-degenerate background metric in particular, but at
the same time one makes sure that no observable prediction will depend on it.1 This can be done
by means of the Ward identities pertaining to the split-symmetry [136, 235, 236] which governs
the interrelation between ϕ and Φ̄. This symmetry, if intact, ensures that the physical contents
of a theory is independent of the chosen background structures. Usually, at the ‘off-shell’ level
of Γk, in particular when k > 0, the symmetry is broken by the gauge fixing and cutoff terms
in the bare action. Insisting on unbroken split-symmetry in the physical sector restricts the
admissible RG trajectories the EAA may follow, see chapter 10 for more details.
The wording ‘close to’ used above has the precise meaning that in the idealized situation
of negligible split-symmetry violation the monotonicity of Ck is manifest for the entire class of
theories with pointwise monotonicity of the EAA. This can indeed be seen without embarking
on any complicated analysis, whose outcome could then possibly depend on the type of theory
under consideration. Instead, in reality where the breaking of split-symmetry often is an issue,
such an analysis is indeed necessary in order to check whether or not the split-symmetry vio-
lation is strong enough to destroy the monotone behavior of Ck. We believe that under weak
conditions whose precise form needs to be found, the monotonicity is not destroyed so that the
proposed Ck indeed complies both with the stationarity and the monotonicity requirement.
In this chapter, rather than attempting a general proof we investigate a concrete system,
asymptotically safe gravity in d dimensions, d = 4 in particular, determine its RG flow, and
explore the properties of the resulting function Ck. Clearly, for practical reasons we can study
the flow only on a truncation of the a priori infinite dimensional theory space. Nevertheless, we
shall be able to demonstrate that, provided the truncation is sufficiently refined, the correspond-
ing approximation to the exact Ck is indeed a non-decreasing function of k. It will be quite
impressive to see how non-trivially the various components of the truncation ansatz for Γk must,
and actually do conspire in order to produce this result.
Concretely we shall explore Quantum Einstein Gravity (QEG) both within the familiar
(‘single-metric’) Einstein-Hilbert-truncation [122] in which the background metric appears only
in the gauge fixing part of the action, as well as within a more refined ‘bi-metric truncation’ [136,
137] where the EAA can depend on it via a second Einstein-Hilbert term, over and above the
one for the dynamical metric. The beta-functions for this ‘bi-metric Einstein-Hilbert truncation’
have been derived in [172] using TT-decomposition techniques and in part II utilizing a Ω-
deformed gauge sector. Those results form the basis of the present analysis.
We emphasize that the goal of the present investigation is not, or at least not primarily, to
reanalyze or reformulate the known c- and a-theorems within the EAA approach. (Work along
these lines has been reported in [237, 238].) Rather, we want to investigate the properties of
a candidate for a generalized C-function which is distinguished and natural in its own right,
namely from the perspective of the EAA. As such it can be tentatively defined under condi-
tions that are far more general than those leading to the a- and c-theorems (with respect to
dimensionality, field contents, symmetries, etc.). We expect that after restricting this generality
appropriately Ck can be given properties similar to a C-function. The long-term objective of
1The construction in section 11.2 will force us to deal with a background metric and include it into the set Φ̄
even when analyzing pure matter theories on a classical (for instance, flat) spacetime.
11.2 The effective average action as a ‘C-function’ 357
this research program is to find out which restrictions precisely lead to interesting properties of
Ck. Here we provide a first example where this strategy can be seen to actually work, namely
(truncated) QEG in 4 dimensions.
The structure of this chapter is as follows. In section 11.2 we develop the general theory
for using the EAA and Ck as a counting device2, after recalling some necessary background
material. Then, in section 11.3 we apply the resulting framework to the particularly important
case of asymptotically safe Quantum Einstein Gravity. Our setting will apply to an arbitrary
dimensionality of spacetime; the numerical calculations needed to verify the claimed properties
of Ck are performed for the most interesting case of 4 dimensions though. We conclude this
chapter in section 11.4 by discussing related topics, as the N-bound or the Bekenstein-Hawking
entropy, and present an outlook for future work in this direction. This chapter contains a small
appendix, including the special status of Faddeev-Popov ghost and some remark on the results
for different classes of RG trajectories.
11.2 The effective average action as a ‘C-function’
In this section we develop a generally applicable framework for constructing functions Ck which
have properties similar to aC-function, and at the same time are ‘geometrically natural’ objects
from the perspective of the theory space explored by the EAA. To prepare the ground, and to
fix various notations, it is unavoidable to embark on some special aspects of the EAA technique
first.
11.2.1 Counting field modes
Eigenfunction representation of the path integral. We consider a general quantum field
theory on a d dimensiona∫l Euclidean spacetime, either rigid or fluctuating, that is governed by
a functional integral Z = [dΦ̂]e−S[Φ̂,Φ̄]. The bare action S depends on a set of commuting and
anticommuting dynamical fields, Φ̂, and on a corresponding set of background fields, Φ̄. (Here
and in the following we use a compact notation, leaving implicit all field indices and possible
sign factors depending on the Grassmann parity of the field components.) We assume that the
functional integral is regularized in the UV in some way; we shall be more precise about this
point in a moment.
In the case of a Yang-Mills theory, Φ̂ contains both the gauge field and the Faddeev-Popov
ghosts, and S is understood to include gauge fixing and ghost terms. Furthermore, the corre-
sponding background fields are part of Φ̄. As a rule, the fluctuation field ϕ̂ ≡ Φ̂− Φ̄ is always
required to gauge-transform homogeneously, i.e. like a matter field. Henceforth we regard ϕ̂
rather than Φ̂ a∫s the true (dynamical)variable and interpret Z as an integral over the fluctuation
variables: Z = [dΦ̂] exp −S[ϕ̂;Φ̄] .
For conceptual reasons that will become apparent below, the set of background fields, Φ̄,
always contains a classical spacetime metric ḡµν . In typical particle physics applications on a
rigid (flat, say) spacetime for instance one would not be interested in how Z depends on the
background metric and one might set ḡµν = δµν throughout. But in quantum gravity, when
Background Independence is an issue, one wants to know Z ≡ Z[ḡµν ] for any background. In
fact, employing the background field technique to implement Background Independence [27]
one represents the dynamical metric as ĝµν = ḡµν + ĥµν and requires split-symmetry at the
level of observable quantities, see chapter 10. When the spacetime is dynamical, ĝµν and ĥµν
are special components of Φ̂ and ϕ̂ , respectively.
2A brief discussion and an application of the method to the example of black hole physics appeared already in
[166] and will be revised in chapter 13. (See also ref. [239].)
358 Chapter 11. A generalized C-theorem
Next we pick a basis in field space, {ϕω}, and expand the fields that are integrated over.
Then, symbolically, ϕ̂(x) = ∑ω aω ϕω(x) where ∑ω∫ stands for a summation and/or integration
over all labels carried by the basis elements, and [dΦ̂] is now interpreted as the integration
over all ∫possible valu(es that ca)n be assumed by the expansion coefficients a ≡ {aω}. Thus,
= ∞Z ∏ω −∞ daω exp −S[a;Φ̄] .
To be more specific, let us assume that the basis {ϕω} is constituted by the eigenfunctions
of a certain differential operator, L, which may depend on the background fields Φ̄, and which
has properties similar to the negative Laplace-Beltrami operator, −D̄2, appropriately general-
ized for the types of (tensor, spinor, · · · ) fields present in ϕ̂ . We suppose that L is built from
covariant derivatives involving ḡµν and the background Yang-Mills fields, if any, so that it is
covariant under spacetime diffeomorphism and gauge-transformations. We assume an eigen-
value equation Lϕω = Ω2ωϕω with positive spectral values Ω
2
ω > 0. The precise choice of L is
arbitrary to a large extent.
The only property of Lwe shall need is that it should associate small (large) distances on
the rigid spacetime equipped with the metric ḡµν to large (small) values of Ω2ω . A first (but for
us not the essential) consequence is that we can now easily install a UV cutoff by restricting the
ill-defined infinite product ∏ω to only those ω’s which satisfy Ωω < Ωmax. This implements a
UV cutoff at the mass scale Ωmax.
More importantly for our purposes, we also introduce a smooth IR cutoff at a variable scale
k ≤ Ωmax into the integral, replacing it with
∫
′ ∞
Zk = ∏ daω e−S[a;Φ̄]e−∆Sk (11.2)
ω −∞
where the prime indicates the presence of the UV cutoff, and
1
∆S 2k ≡ ∑Rk(Ωω)a2ω (11.3)2 ω
implements the IR cutoff. The extra piece in the bare action, ∆Sk, is designed in such a way that
those ϕω -modes which have eigenvalues Ω2ω ≪ k2 get suppressed by a small factor e−∆Sk ≪ 1
in eq. (11.2), while e−∆Sk = 1 for the others. The function Rk is essentially arbitrary, except for
its interpolating behavior between Rk(Ω2ω)∼ k2 if Ω 2ω ≪ k and Rk(Ωω) = 0 if Ωω ≫ k.
The operator L defines the precise notion of ‘coarse graining’ field configurations. We re-
gard the ϕω ’s with Ωω > k as the ‘short wavelength’ modes, to be integrated out first, and those
with small eigenvalues Ωω < k as the ‘long wavelength’ ones whose impact on the fluctuation’s
dynamics is not yet taken into account. This amounts to a diffeomorphism and gauge covari-
ant generalization of the standard Wilsonian renormalization group, based on standard Fourier
analysis on Rd , to situations with arbitrary background fields Φ̄ = (ḡµν , Āµ , · · · ).
While helpful for the interpretation, for most practical purposes it is often unnecessary to
perform the expansion of ϕ̂(x) in terms of the L-eigenfunctions explicitly. Rather, one thinks
of (11.2) as a ‘basis independent’ functional integral
∫
Z = [d′ϕ̂ ]e−S[ϕ̂ ;Φ̄]e−∆Sk[ϕ̂ ;Φ̄]k (11.4)
for which the L-eigenbasis plays no special role, while the operator L as such does so, of
course. In particular the cutoff action ∆Sk is now rewritten with Ω2ω replaced by L in the
argument of Rk:
∫
1 √
∆Sk[ϕ̂ ;Φ̄] = d
dx ḡ ϕ̂(x)Rk(L) ϕ̂(x) (11.5)
2
Note that at least when k > 0 the modified partition function Zk depends on the respective
choices for L and Φ̄ separately.
11.2 The effective average action as a ‘C-function’ 359
The generality of the framework. In this chapter we propose to use the one-parameter
family of partition function k 7→ Zk, for k ∈ [0,∞), as a diagnostic tool to investigate the RG
flow between different quantum field theories. This is a quite general and flexible framework.
There is considerable freedom in choosing the cutoff operator, and even when L≡ L(Φ̄) is
fixed we may still choose Φ̄ in a large variety of different ways so as to ‘project out’ different
information from the partition function. However, as we shall discuss in the next subsection,
there exists a natural, almost ‘canonical’ choice of background configurations Φ̄.
Monotonicity of Zk Our discussion in the following sections is based upon the key obser-
vation that Zk enjoys a simple property which is strikingly reminiscent of the C-theorem in 2
dimensions.
Let us assume for simplicity that all component fields constituting ϕ̂ are commuting, and
that Φ̄ has been chosen k-independent. Then (11.4) is a (regularized, and convergent for ap-
propriate S) purely bosonic integral with a positive integrand which, thanks to the suppression
factor e−∆Sk , decreases with increasing k. Therefore, Zk and the ‘entropy’ lnZk, are monotoni-
cally decreasing functions of the scale:
∂k lnZk < 0 (11.6)
The interpretation of (11.6) is clear: Proceeding from the UV to the IR by lowering the infrared
cutoff scale, an increasing number of field modes get un-suppressed, thus contribute to the
functional integral, and as a consequence the value of the partition function increases. Thus,
in a not too literal sense of the word, lnZk ‘counts’ the number of field modes that have been
integrated out already.
Before we can make this intuitive argument more precise and explicit we must introduce a
number of technical tools in the following subsections.
11.2.2 Running actions and self-consistent backgrounds
The generating functional. Introducing a source term for the fluctuation fields turns the
partition functions into the generating functional
∫ ( ∫ )
Z [J;Φ̄]≡ eWk[J;Φ̄] = D′
√
k ϕ̂ exp −S[ϕ̂;Φ̄]−∆Sk[ϕ̂ ;Φ̄]+ ddx ḡJ(x)ϕ̂(x) (11.7)
Hence the Φ̄- and k-dependent expectation value 〈ϕ̂〉 ≡ ϕ reads
ϕ(x)≡ 〈 1 δWk[J;Φ̄]ϕ̂(x)〉 = √ (11.8)
ḡ(x) δJ(x)
If we can solve this relation for J as a functional of Φ̄, the definition of the Effective Average
Action (EAA), essentially the Legendre transform ofWk, may be written as
∫ √
Γk[ϕ ;Φ̄] = d
dx ḡϕ(x)J(x)−Wk[J;Φ̄]−∆Sk[ϕ ;Φ̄] (11.9)
with the solution to (11.8) inserted, J ≡ Jk[ϕ ;Φ̄]. (In the general case, Γk is the Legendre-
Fenchel transform ofWk, with ∆Sk subtracted.)
The EAA gives rise to a source-field relationship which includes an explicit cutoff term
linear in the fluctuation field:
√1 δΓk[ϕ ;Φ̄] +Rk[Φ̄]ϕ(x) = J(x) (11.10)
ḡ δϕ(x)
360 Chapter 11. A generalized C-theorem
Here and in the following we write Rk ≡Rk(L), and the notation Rk[Φ̄] is used occasionally
to emphasize that the cutoff operator may depend on the background fields. The solution to
(11.10), and more generally all fluctuation correlators 〈ϕ̂(x1) · · · ϕ̂(xn)〉 obtained by multiple
differentiation of Γk, are functionally dependent on the background, e.g. ϕ(x)≡ ϕk[J;Φ̄](x).
The bi-field variant. For the expectation value of the full, i.e. un-decomposed field Φ̂ =
Φ̄+ ϕ̂ we employ the notation
Φ = Φ̄+ϕ with Φ ≡ 〈Φ̂〉 and ϕ ≡ 〈ϕ̂〉 . (11.11)
Using the complete field Φ instead of ϕ as the second independent variable, accompanying Φ̄,
entails the ‘bi-field’ variant of the EAA,
∣
Γk[Φ,Φ̄]≡ Γk[ϕ ;Φ̄]∣ (11.12)ϕ=Φ−Φ̄
which, in particular, is always ‘bi-metric’: Γk[gµν , · · · , ḡµν , · · · ].
The level representation. Later on it will often be helpful to organize the terms contribut-
ing to Γk[ϕ ;Φ̄] according to their level which, by definition, is their degree of homogeneity in
the ϕ’s. The underlying assumption is that the EAA admits a level expansion of the form
∞
p
Γk[ϕ ;Φ̄] = ∑ Γ̌k [ϕ ;Φ̄] (11.13)
p=0
where p pΓ̌k [cϕ ;Φ̄] = c
p Γ̌k [ϕ ;Φ̄] for c > 0. If Γk[ϕ ;Φ̄] admits a Taylor expansion in ϕ about
ϕ = 0, this expansion exists, of course, with the level-(p) contribution pΓ̌k being its p-derivative
term, but this is not guaranteed in general.
The self-consistent backgrounds. We are interested in how the dynamics of the fluctua-
tions ϕ̂ depends on the environment they are placed in, the background metric ḡµν , for instance,
and the other classical fields collected in Φ̄. It would be instructive to know if there exist special
backgrounds in which the fluctuations are particularly ‘tame’ such that, for vanishing external
source, they amount to only small oscillations about a stable equilibrium, with a vanishing
mean: ϕ ≡ 〈ϕ̂〉= 0. Such distinguished backgrounds Φ̄ ≡ Φ̄s.c. are referred to as self-consistent
(s.c.) since, if we pick one of those, the expectation value of the field 〈Φ̂〉= Φ = Φ̄ does not get
changed by any violent ϕ̂-excitations that, generically, can shift the point of equilibrium.
From eq. (11.10) we obtain the following condition Φ̄s.c. must satisfy (since J = 0 by as-
sumption):
δ ∣
Γk[ϕ ;Φ̄]∣ = 0 (11.14)
δϕ(x) ϕ=0,Φ̄=Φ̄
s.c.
k
This is the tadpole equation from which we can compute the self-consistent background con-
figurations, if any. In general Φ̄s.c. ≡ Φ̄s.c.k will have an explicit dependence on k. A technically
convenient feature of (11.14) is that it no longer contains the somewhat disturbing Rkϕ-term
that was present in the general field equation (11.10). Self-consistent backgrounds are equiva-
lently characterized by eq. (11.8),
δ ∣
W ∣k[J;Φ̄] s.c. = 0 (11.15)
δJ(x) J=0,Φ̄=Φ̄k
which again expresses the vanishing of the fluctuation’s one-point function.
11.2 The effective average action as a ‘C-function’ 361
Note that provided the level expansion (11.13) exists we may replace (11.14) with
δ ∣
Γ̌1k[ϕ ;Φ̄]∣ϕ=0,Φ̄=Φ̄s.c. = 0 (11.16)δϕ(x) k
which involves only the level-(1) functional Γ̌1k . Later on in the applications this trivial observa-
tion has the important consequence that self-consistent background field configurations Φ̄s.c.k (x)
can contain only running coupling constants of level p= 1, that is, the couplings parameterizing
the functional Γ̌1k which is linear in ϕ .
3
The level-dependence of EAA for self-consistent backgrounds. In our later discussions
the value of the EAA at ϕ = 0 will be of special interest. While it is still a rather complicated
functional for a generic background where Γk[0;Φ̄] = −Wk[Jk[0;Φ̄];Φ̄], the source which is
necessary to achieve ϕ = 0 for self-consistent backgrounds is precisely J = 0, implying
Γk[0;Φ̄
s.c.
k ]≡ Γ̌0[0;Φ̄s.c.k k ] =−W [0;Φ̄s.c.k k ] (11.17)
Here we also indicated that in a level expansion only the p = 0 term of Γk survives putting
ϕ = 0.
So we can summarize saying that the value of Γ [0;Φ̄s.c.k k ] can contain only the running
couplings of the levels p= 0 and p = 1, respectively, the former entering via Γ̌0k , the latter via
Φ̄s.c.k .
11.2.3 FIDE, FRGE, and WISS
The EAA satisfies a number of important exact functional equations which include a Func-
tional Integro-Differential Equation (FIDE), the Functional Renormalization Group Equation
(FRGE), the Ward Identities for split-symmetry (WISS), and the Becchi, Rouet, Stora and
Tyutin (BRST)-Ward identity.
Functional Integro-Differential Equation
The FIDE is obtained by substituting (11.9) in (11.7), using (11.10), and reads
∫ ( ∫ )
e−Γk[ϕ ;Φ̄] = [d′ϕ̂] exp − δΓkS[ϕ̂ ;Φ̄]−∆Sk[ϕ̂ ;Φ̄]+ ddx ϕ̂(x) [ϕ ;Φ̄] (11.18)
δϕ(x)
Here, as always, summation over field components and their tensor, spinor, internal symmetry,
etc. indices is understood. For the purposes of the present investigation, the most important
property of (11.18) is that the last term on its RHS, the one linear in ϕ̂ , vanishes if the back-
ground happens to be self-consistent, and at the same time the argument ϕ = 0 is inserted on
both sides of the FIDE:
( ) ∫ ( )
exp −Γ [0;Φ̄s.c.k k ] = [d′ϕ̂ ] exp −S[ϕ̂;Φ̄s.c.k ]−∆Sk[ϕ̂ ;Φ̄s.c.k ] (11.19)
We shall come back to this identity soon.
Functional Renormalization Group Equation
Another important exact relation satisfied by the EAA is the Functional Renormalization Group
Equation (FRGE)),
[
1 ( ) ](2) −1
k∂kΓk[ϕ ;Φ̄] = STr Γk [ϕ ;Φ̄]+Rk[Φ̄] k∂kRk[Φ̄] (11.20)2
3Notice that the k-derivative of Φ̄s.c.k (x) is in general governed also by higher level couplings due to their appear-
ance in the beta-functions of the level p= 1 couplings.
362 Chapter 11. A generalized C-theorem
(2)
with the Hessian matrix of the fluctuation derivatives Γk ≡ δ 2Γk/δϕ2. The supertrace ‘STr’ in
(11.20) provides the additional minus sign which is necessary for the ϕ-components with odd
Grassmann parity, Faddeev-Popov ghosts and fermions.
Ward Identities for split-symmetry
The EAA, written as Γk[Φ,Φ̄], satisfies the following exact functional equation which governs
the ‘extra’ background dependence it has over and above the one which combines with the
fluctuations to form the full field Φ:
[ ]
δ 1 ( )(2) −1 δ (2)
Γk[Φ,Φ̄] = STr Γk [Φ,Φ̄]+Rk[Φ̄] Stot [Φ,Φ̄] (11.21)δ Φ̄(x) 2 δ Φ̄(x)
Here (2)Stot is the Hessian of Stot = S+∆Sk with respect to Φ, where S includes gauge fixing
and ghost terms. The equation (11.21) is the Ward identity induced by the split-symmetry
transformations δϕ = ε , δ Φ̄ =−ε , hence the abbreviation WISS. It was first obtained in [235]
in the context of Yang-Mills theory. In quantum gravity, extensive use has been made of (11.21)
in ref. [136] were it served as a tool to assess the degree of split-symmetry breaking and, related
to that, the reliability of certain truncations of QEG.
For a discussion of the modified BRST-Ward identity enjoyed by the EAA we refer to
[235] and [122].
11.2.4 Pointwise monotonicity
Our search for a generalized C-type counting function which depends monotonically on k along
the RG trajectories will be based upon the following structural property of the FRGE (11.20).
From the very definition of the EAA by a Legendre transform it follows that for all Φ̄ the sum
(2)
Γk+∆Sk is a convex functional of ϕ , and that Γk +Rk is a strictly positive definite operator
therefore which can be inverted at all scales k ∈ (0,∞). Now let us suppose that the theory under
consideration contains Grassmann-even fields only. Then the supertrace in (11.20) amounts to
the ordinary, and convergent trace of a positive operator so that the FRGE implies
k∂kΓk[ϕ ;Φ̄]≥ 0 at all fixed ϕ , Φ̄. (11.22)
Thus, at least in a class of distinguished theories the EAA, evaluated at any fixed pair of argu-
ments ϕ and Φ̄, is a monotonically increasing function of k. With other words, lowering k from
the UV towards the IR the value of Γk[ϕ ;Φ̄] decreases monotonically.
We refer to this property as pointwise monotonicity in order to emphasize that it applies
at all points of field space, (ϕ ,Φ̄), separately. In particular this means that the argument of
Γk[ϕ ,Φ̄] is assumed to have no k-dependence of its own here.
In presence of fields with odd Grassmann parity, fermions and Faddeev-Popov ghosts, the
RHS of the FRGE is no longer obviously non-negative. However, if the only Grassmann-odd
fields are ghosts the pointwise monotonicity (11.22) can still be made a general property of the
EAA, the reason being as follows. At least when one implements the gauge fixing condition
strictly, it cuts-out a certain subspace of the space of fields Φ̂ to be integrated over, namely the
gauge orbit space. Hereby the integral over the ghosts represents the measure on this subspace,
the Faddeev-Popov determinant. The subspace and its geometrical structures are invariant under
the RG flow, however. Hence the EAA pertaining to the manifestly Grassmann-even integral
over the subspace is of the kind considered above, and the argument implying (11.22) should
therefore be valid again. For a more explicit version of this reasoning we refer to appendix
11.A.
11.2 The effective average action as a ‘C-function’ 363
11.2.5 Monotonicity vs. stationarity
The EAA evaluated at fixed arguments shares the monotonicity property with aC-function. One
of the problems is however that Γk[ϕ ;Φ̄] is not stationary at fixed points of the RG flow. In order
to see why, and how to improve the situation, some care is needed concerning the interplay of
dimensionful and dimensionless variables, to which we turn next.
The FRGE in component form. Let us assume that the space constituted by the func-
tionals of ϕ and Φ̄ admits a basis {Iα} so that we can expand the EAA as
Γk[ϕ ;Φ̄] = ∑ ūα(k) Iα [ϕ ;Φ̄] (11.23)
α
with dimensionful running coupling constants ū≡ (ūα). They obey a FRGE in component form,
k∂kūα (k) = b̄α(ū(k);k), whereby the functions b̄α are defined by the expansion of STr[· · · ] with
respect to the basis:
[( )− ]1 1(2)
STr ∑ ūα(k) Iα [ϕ ;Φ̄]+Rk k∂kRk =: ∑ b̄α(ū(k);k) Iα [ϕ ;Φ̄] (11.24)2 α α
Note that the statement of monotonicity in (11.22), when it holds true, translates into the point-
wise positivity of the sum ∑α b̄α Iα .
The dimensionless fields. Denoting the canonical mass dimension4 of the running cou-
plings by [ūα ]≡ dα , their dimensionless counterparts are defined by u −dαα ≡ k ūα . In terms of
the dimensionless couplings the expansion of Γk reads
Γk[ϕ ;Φ̄] = ∑uα(k)kdα Iα [ϕ ;Φ̄] (11.25)
α
[ ]
Now observe that since Γk is dimensionless the basis elements have dimensions Iα [ϕ ;Φ̄] =
−dα . Purely by dimensional analysis, this implies that5
I [ϕ ]α [c ϕ ;c
[Φ̄]Φ̄] = c−dα Iα [ϕ ;Φ̄] for any constant c> 0. (11.26)
This relation expresses the fact that the nontrivial dimension of Iα is entirely due to that of
its field arguments; there are simply no other dimensionful quantities available after the k-
dependence has been separated off. Using (11.26) for c= k−1 yields
kdα I [ϕ ;Φ̄] = I [k−[ϕ ]ϕ ;k−[Φ̄]α α Φ̄]≡ I ˜̄α [ϕ̃ ;Φ] (11.27)
Here we introduced the sets of dimensionless fields,
ϕ̃(x)≡ k−[ϕ ]ϕ(x), Φ˜̄ (x)≡ k−[Φ̄]Φ̄(x) (11.28)
which include, for instance, the dimensionless metric and its fluctuations:
h̃µν(x) ≡ k2hµν(x), g˜̄µν(x)≡ k2ḡµν(x) (11.29)
4Our conventions are as follows. We use dimensionless coordinates, [xµ ] = 0. Then [ds2] = −2 implies that
all components of the various metrics have [ĝµν ] = [ḡµν ] = [gµν ] = −2, and likewise for the fluctuations: [ĥµν ] =
[hµν ] =−2.
5We use the notation c[ϕ ]ϕ ≡ {c[ϕi]ϕi} for the set in which each field is rescaled according to its individual
canonical dimension.
364 Chapter 11. A generalized C-theorem
Note that the quantity (11.27), in whatever way we write it, is dimensionless. When we insert
the dimensionless fie[lds rathe]r than ϕ and Φ̄ into the basis functionals, the latter loose their
nonzero dimension: Iα [ϕ̃ ;Φ
˜̄ ] = 0.
Exploiting (11.27) in (11.25) we obtain the following representation of the EAA which is
entirely in terms of dimensionless quantities6
Γk[ϕ ;Φ̄] = ∑uα(k) Iα [ϕ̃;Φ˜̄ ] ≡ A ˜̄k[ϕ̃;Φ] (11.30)
α
Alternatively, one might wish to make its k-dependence explicit, writing,
Γ [ϕ ;Φ̄] = ∑u (k) I [k−[ϕ ]ϕ ;k−[Φ̄]k α α Φ̄] (11.31)
α
In the second equality of (11.30) we introduced the new functional Ak which, by definition,
is numerically equal to Γk, but its independent variables (arguments) are the dimensionless
fields ϕ̃ and Φ˜̄ . Hence the k-derivative of Ak[ϕ̃ ,Φ
˜̄ ] is to be performed at fixed (ϕ̃ ,Φ˜̄ ), while
the analogous derivative of Γk[ϕ ;Φ̄] refers to fixed dimensionful arguments. This leads to the
following two trivial but momentous equations:
k∂ ˜̄kAk[ϕ̃;Φ] = ∑k∂kuα(k) Iα [ϕ̃ ;Φ˜̄ ] (11.32a)
α { }
k∂kΓk[ϕ ;Φ̄] = ∑ k∂kuα(k)+d u dαα α(k) k Iα [ϕ ;Φ̄] (11.32b)
α
The extra term ∝ dαuα(k) in eq. (11.32b) arises by differentiating the factor kdα in (11.25).
It leads to the well-known canonical scaling term in the β -functions of the dimensionless cou-
plings.
The (dimensionless) theory space. For the following it is crucial to recall that it is the
dimensionless couplings u ≡ (uα) that serve as local coordinates on theory space, henceforth
denoted T. Its points are functionals Awhich depend on dimensionless arguments: A[ϕ̃;Φ˜̄ ] =
∑α uα Iα [ϕ̃ ;Φ
˜̄ ]. The RG trajectories are curves k 7→ Ak = ∑α uα(k) Iα ∈ T that are everywhere
tangent to
k∂kAk = ∑βα(u(k)) Iα (11.33)
α
The functions βα , components of a vector field ~β on T, are obtained by translating k∂kūα(k) =
b̄α(ū(k);k) into the dimensionless language. This leads to the autonomous system of differential
equations
k∂kuα(k)≡ βα(u(k)) =−dαuα(k)+bα(u(k)) (11.34)
Here bα , contrary to its dimensionful precursor b̄α , has no explicit k-dependence, thus defining
an RG-time independent vector field, the ‘RG flow’ (T,~β ).
If it has a fixed point at some u∗ then βα(u∗) = 0, and the ‘velocity’ of any trajectory passing
this point vanishes there7, k∂kuα = 0. Hence by (11.33) the redefined functional Ak becomes
stationary there, that is, its scale derivative vanishes pointwise,
k∂kA
˜̄
k[ϕ̃;Φ] = 0 for all fixed ϕ̃ , Φ˜̄ . (11.35)
6Here one should also switch from k to the manifestly dimensionless ‘RG time’ t ≡ ln(k)+ const, but we shall
not indicate this notationally.
7To keep the notation simple, we assume here that among the uα ’s there are no ‘inessential’, aka ‘redundant’,
couplings that would not have to approach fixed point values.
11.2 The effective average action as a ‘C-function’ 365
So the entire functional Ak approaches a limit, A∗ =∑ u∗α α Iα . The standard EAA instead keeps
running in the fixed point regime:
Γ [ϕ ;Φ̄] = ∑u∗ kdαk α Iα [ϕ ;Φ̄] when uα(k) = u∗α . (11.36)
α
The monotonicity of Ak This brings us back to the ‘defect’ of Γk we wanted to repair:
While Γk[ϕ ;Φ̄] was explicitly seen to decrease monotonically along RG trajectories, it does not
come to a halt at fixed points in general. The redefined functional Ak, instead, approaches a
finite limit A∗ at fixed points, but can we argue that it is monotone along trajectories?
Unfortunately this is not the case, and the culprit is quite obvious, namely the dαuα -terms
present in the scale derivative of Γk, but absent forAk: The positivity of the RHS of eq. (11.32b)
does not imply the positivity of the RHS of eq. (11.32a), and there is no obvious structural
reason for k∂kAk[ϕ̃ ;Φ˜̄ ]≥ 0 at fixed ϕ̃ , Φ˜̄ . The best we can get is the bound
k∂ ˜̄kAk[ϕ̃;Φ]≥−∑dαuα(k) Iα [ϕ̃;Φ˜̄ ] (11.37)
α
which follows by subtracting the two eqs. (11.32) and making use of (11.27).
11.2.6 The proposal
The complementary virtues of Ak and Γk with respect to monotonicity along trajectories and
stationarity at critical points suggest the following strategy for finding a C-type function with
better properties: Rather than considering the functionals pointwise, i.e. with fixed configura-
tions of either the dimensionless or dimensionful fields inserted, one should evaluate them at
?
explicitly scale dependent arguments: Ck = Γk[ϕk;Φ̄k]≡ Ak[ϕ̃k;Φ˜̄ k].
The hope is that the respective arguments ϕ ≡ k[ϕ ]k ϕ̃k, and Φ̄ [Φ̄] ˜̄k ≡ k Φk can be given a k-
dependence which is intermediate between the two extreme cases (ϕ ,Φ̄) = const and (ϕ̃ ,Φ˜̄ ) =
const, respectively, so as to preserve as much as possible of the monotonicity properties of Γk,
while rendering Ck stationary at fixed points of the RG flow.
The most promising candidate of this kind which we could find is
s.c.
Ck = Γk[0;Φ̄k ] = Ak[0;Φ
˜̄ s.c.
k ] (11.38)
Here the fluctuation argument is set to zero, ϕk ≡ 0, and for the background we choose a self-
consistent one, Φ̄s.c.k , a solution to the tadpole equation (11.14), or equivalently its dimensionless
variant
δ ˜̄ ∣∣Ak[ϕ̃;Φ] ˜̄ ˜̄ s.c. = 0 (11.39)
δ ϕ̃(x) ϕ̃=0,Φ=Φk
The function k→7 Ck defined by eq. (11.38) has a number of interesting properties to which we
turn next.
Stationarity at critical points
When the RG trajectory approaches a fixed point, A ˜̄k[ϕ̃ ;Φ] approaches A∗[ϕ̃;Φ˜̄ ] pointwise.
Furthermore, the tadpole equation (11.39) becomes (δA∗/δ ϕ̃)[0;Φ˜̄ ∗] = 0. It is completely k-
independent, and so is its solution, Φ˜̄ ∗. Thus Ck approaches a well defined, finite constant:
Ck −F→P C∗ = A∗[0;Φ˜̄ ∗] (11.40)
Of course we can write this number also as ˜̄C∗ = Γ [0;k[Φ̄]k Φ∗] wherein the explicit and the
implicit scale dependence of the EAA cancel exactly when a fixed point is approached.
366 Chapter 11. A generalized C-theorem
Stationarity at classicality
In a classical regime (‘CR’), by definition, b̄α → 0, so that it is now the dimensionful cou-
plings whose running stops: ūα(k) → ūCRα = const. Thus, by (11.23), Γk approaches ΓCR =
∑ ūCRα α Iα pointwise. Hence the dimensionful version of the tadpole equation, (11.14), becomes
k-independent, and the same is true for its solution, Φ̄s.c.CR . So, when the RG trajectory approaches
a classical regime, Ck looses its k-dependences and approaches a constant:
−CCk −→R CCR = ΓCR[0;Φ̄s.c.CR ] (11.41)
Alternatively we can write C = A [0;k−[Φ̄]Φ̄s.c.CR k CR ] where it is now the explicit and implicit k-
dependence of Ak which cancel mutually.
We observe that there is a certain analogy between ‘criticality’ and ‘classicality’, in the sense
that dimensionful and dimensionless couplings exchange their roles. The difference is that the
former situation is related to special points of theory space, while the latter concerns extended
regions in T. In those regions, A keeps moving as A [ · ] = ∑ ūCR k−dαk k α α Iα [ · ]. Nevertheless it
is thus plausible, and of particular interest in quantum gravity, to apply a (putative) C-function
not only to crossover trajectories in the usual sense which connect two fixed points, but also to
generalized crossover transitions where one of the fixed points, or even both, get replaced by a
classical regime.
Monotonicity at exact split-symmetry
If split-symmetry is exact in the sense that Γk[ϕ ;Φ̄] depends on the single independent field
variable Φ̄+ϕ ≡ Φ only, and the theory is such that pointwise monotonicity (11.22) holds true,
then k 7→ Ck is a monotonically increasing function of k. In fact, differentiating (11.38) and
using the chain rule yields
∫ ( )( )δ δ ∣
∂ s.c. d s.c. ∣kCk = (∂kΓk) [0;Φ̄k ]+ d x ∂kΦ̄k (x) − Γk[ϕ ;Φ̄] (11.42)δ Φ̄(x) δϕ(x) (0,Φ̄s.c.k )
In the first term on the RHS of (11.42) the derivative ∂k hits only the explicit k-dependence
of the EAA. By eq. (11.22) we know that this contribution is non-negative. The last term, the
δ/δϕ-derivative, is actually zero by the tadpole equation (11.14). Including it here it becomes
manifest that the integral term in (11.42) vanishes when Γk depends on ϕ and Φ̄ only via the
combination ϕ + Φ̄. Thus we have shown that
∂kCk ≥ 0 at exact split-symmetry (11.43)
Note that the result for ∂kCk in (11.42) is much closer to what one needs to prove in order to
rightfully call Ck a ‘C-function’ than the inequality (11.37). In theories that require no breaking
of split-symmetry, for instance, the integral term in (11.42) is identically zero and we know that
∂kCk ≥ 0 holds true.
The degree of split-symmetry violation varies over theory space in general. Split-symmetry
is unbroken at points u= (uα ) where at most those coordinates uα are non-zero that belong to
basis functionals Iα [ϕ ;Φ̄] which happen to depend on Φ̄+ϕ only.
The breaking of split-symmetry is best discussed in terms of the functional Γk[Φ,Φ̄] ≡
Γk[Φ− Φ̄;Φ̄] for which perfect symmetry amounts to independence of the second argument:
δ Γk[Φ,Φ̄] = 0. In this language, C is written as C = Γ [Φ̄s.c.k k k k ,Φ̄
s.c.
k ], and its scale derivativeδ Φ̄
assumes the form
∫ ( ) ∣∣
∂ = (∂ Γ ) [Φ̄s.c.,Φ̄s.c.]+ ddx ∂ Φ̄s.c.
δΓk[Φ,Φ̄]
kCk k k k k k k (x)
∣ (11.44)
δ Φ̄(x) ∣Φ=Φ̄=Φ̄s.c.k
11.2 The effective average action as a ‘C-function’ 367
Whether or not ∂kCk is always non-negative depends on the size of the split-symmetry
breaking the EAA suffers from. To prove monotonicity of Ck one would have to show on a
case-by-case basis that the second term on the RHS of (11.44) never can override the first one,
known to be non-negative, so as to render their sum negative.
In fact, for the derivative δΓk/δ Φ̄ in (11.44) we have an exact formal identity at our disposal,
the WISS of eq. (11.21). The equations (11.44) and (11.21) together with the FRGE for the
(∂kΓk)-term could be the starting point of future work on exact estimates.
In the next section of the present paper we shall investigate the monotonicity properties of
Ck in certain truncations of pure Quantum Einstein Gravity using a different strategy. In this
case it is easier to work directly with the definition of Ck, eq. (11.38), rather then using the
WISS.
11.2.7 The mode counting property revisited
Equipped with the EAA machinery, we now return to the heuristic argument about the mode
‘counting’ property of Zk that was presented at the end of subsection 11.2.1. Trying to make it
more precise, we shall now demonstrate that our candidate Ck ≡ Γk[0;Φ̄s.c.k ] is really a measure
for the ‘number’ of field modes, since it is closely related to a spectral density.
In fact, after our preparations in the previous subsections this should not be too much of
a surprise. By virtue of the general identity (11.19), a special case of the FIDE satisfied by
the EAA, the exponential e−Γk[0,Φ̄
s.c.
k ] equals precisely the partition function considered Zk in
subsection 11.2.1, provided the latter is specialized for a self-consistent background.
In order to present a picture which is as clear as possible let us make a number of specializa-
tions and approximations. In particular we consider purely bosonic theories again, and invoke
the idealization of perfect split-symmetry. Then, by eq. (11.42), we have ∂ C = (∂ Γ ) [0;Φ̄s.c.k k k k k ],
and upon expressing (∂kΓk) via the FRGE we arrive at8
[( ) ]1 (2) −1
k∂kCk = Tr Γk [0;Φ̄
s.c.
k ]+Rk(L) k∂kRk(L) (11.45)2
Now we consider a situation where the Hessian appearing in this equation itself qualifies as a
(2)
cutoff operator. When we choose L= Γ s.c.k [0;Φ̄k ] we obtain
∫
1 [ ]−1 1 k∂kRk(Ω2)
k∂kCk = Tr (L+Rk(L)) k∂kRk(L) = ∑ (11.46)2 2 Ω2+Rk(Ω2)
Ω2
∫
Here Ω2 denotes the eigenvalues of L, and ∑ indicates the summation and/or integration over
its spectrum, leaving the corresponding spectral density implicit. To proceed, we opt for a
particularly convenient cutoff function R , namely the sharp cutoff9k :
R (Ω2) = lim R̂ Θ(k2−Ω2k ) (11.47)
R̂→∞
The limit R̂→ ∞ in (11.47) is to be understood in the distributional sense. It should be taken
only after the integration over Ω2 h∫as been performed. If we fo[rmally use](11.47) in the equation
(11.46) this leads us to k∂kCk = 2∑Ω2 δ (1−Ω2/k2) = 2k2Tr δ (k2−L) , or equivalently,
d [ ( )](2)
Ck = Tr δ k
2−Γ [0;Φ̄s.c.] ≥ 0 (11.48)
dk2 k k
8Note that the equation (11.45) is of course insufficient to determine the function k 7→ Ck. We rather use it to
interpret a given Ck which was derived from a known solution to the full-fledged FRGE.
9See ref. [191] for a detailed discussion of the sharp cutoff. It is often used in quantum gravity since it allows for
an easy closed-form evaluation of the threshold functions pΦn and
p
Φ̃n that frequently appear in QEG beta-functions
[122].
368 Chapter 11. A generalized C-theorem
The equation (11.48) is quite remarkable and sheds some light on the interpretation of Ck:
Its derivative equals exactly the spectral density of the Hessian operator evaluated at the s.c.-
(2)
background field configuration and for vanishing fluctuations, Γk [0;Φ̄
s.c.
k ]. It is a manifestly
non-decreasing function of k therefore.
Let us integrate (11.48) over k2. Provided the RG effects are weak and Γk runs only very
slowly, the k-dependence of the resulting field Φ̄s.c.k is weak, too, so that it may be a sensible
(2)
approximation to neglect the k-dependence of Γk [0;Φ̄
s.c.
k ] in the δ -function of (11.48) relative
to the explicit k2. Under these special circumstances, the integrated version of (11.48) reads:
[ ( )]
2− (2)Ck = Tr Θ k Γk [0;Φ̄s.c.k ] + const (11.49)
Thus, our conclusion is that, at least under the conditions described, the function Ck indeed
counts field modes, in the almost literal sense of the word, namely the eigenfunctions of the
Hessian operator which have eigenvalues not exceeding k2.
Regardless of the present approximation we define in general
Nk1 ,k2 ≡ Ck2 −Ck1 (11.50)
Then, in the cases when the above assumptions apply and (11.49) is valid, Nk1 ,k2 has a simple
interpretation: it equals the number of eigenvalues between k21 and k
2
2 > k
2
1 of the Hessian
operator (2)Γk [0;Φ̄
s.c.
k ], when the spectrum is discrete. When the assumptions leading to (11.49)
are not satisfied, the interpretation of Nk1,k2 , and Ck in the first place, is less intuitive, but these
functions are well defined nevertheless.
As an aside let us also mention that the function (11.49) is closely related to the Chamseddine-
Connes spectral action [240–243] in Noncommutative Geometry, where the squared Dirac op-
erator plays the same role as the Hessian operator above.
11.3 Asymptotically safe quantum gravity
In this section we make the above ideas concrete and apply them to an appropriately truncated
form of Quantum Einstein Gravity (QEG) which is asymptotically safe, that is, all physically
relevant RG trajectories start out in the UV, for k ‘=’∞, at a point infinitesimally close to a non-
Gaußian fixed point (NGFP). When k is lowered they run towards the IR, always staying within
the fixed point’s UV critical manifold, and ultimately approach the (dimensionless) ordinary
effective action [99, 100, 121, 123, 126, 130–135, 138, 139, 141, 142, 146–171].
11.3.1 The single- and bi-metric Einstein-Hilbert truncations
This subsection presents a short summary of the general truncation setting and the differences
between the three employed calculations. For more details on the explicit evaluation steps of
the FRGE we refer to part II.
Truncated theory space. In the following we study the C-function properties of Ck in
pure, metric-based quantum gravity in an arbitrary spacetime dimension. We rely on results
obtained with the so called single- and bi-metric Einstein-H∫ i√lbert tru∫nc√ations where the consid-
ered subspace of theory space is spanned by the invariants g and gR only, with gµν - and
ḡµν -contributions disentangled in the bi-metric case.
In either case the ansatz for the EAA in this subspace is given by
Γ [ ,ξ , ξ̄ , ] = Γgravk g ḡ k [g, ḡ]+Γ
gf
k [g, ḡ]+Γ
gh
k [g,ξ , ξ̄ , ḡ]. (11.51)
11.3 Asymptotically safe quantum gravity 369
It consists of a purely gravitationa(l part, Γ
grav
k [g, ḡ], and )an essentially classical gauge sector
10
based on the coordinate condition βδµ ḡ
αγD̄γ −ϖ ḡαβ D̄µ hµν = ∫0 from which the gauge fixing
term Γgf[ ghk g, ḡ] and the corresponding ghost action Γk [g,ξ , ξ̄ , ḡ]∝ ξ̄ M[g, ḡ]ξ are derived. Here
ξ µ and ξ̄µ denote the diffeomorphism ghosts, and M[g, ḡ] is the Faddeev-Popov operator [122].
We will mostly focus in the following on the Einstein-Hilbert truncation in a bi-metric set-
ting. In this case, Γgravk comprises two separate Einstein-Hilbert terms built from the dynamical
metric gµν and its background analog, ḡµν , respectively (cf. part II):
∫
1 √ ( )
Γgravk [g, ḡ] =− ddx g R(g)−2λ̄D16πGDk ∫ ( )
− 1 √ddx ḡ R(ḡ)−2λ̄ B (11.52)
16πGBk
The couplings, GDk , λ̄
D and GBk , λ̄
B represent k-dependent generalizations of the classical New-
ton or cosmological constant in the dynamical (‘D’) and the background (‘B’) sector, respec-
tively. Expanding eq. (11.52) in terms of the fluctuation field hµν = gµν − ḡµν yields the
level-expansion of the EAA:
∫
1 √ ( )
Γgrav[h; ḡ] =− dd ( )−2λ̄ (0)k x ḡ R ḡ16π (0) kGk ∫ √ [ ]
− 1 dd − µν − λ̄ (1)x ḡ Ḡ ḡµνk hµν16π (1)G
∫ k
− 1 √d µν grav(2)
ρσ
d x ḡ h Γk [ḡ, ḡ]µν hρσ +O(h
3) (11.53)
2
In the level-description, the background and dynamical couplings appear in certain combina-
tions in front of invariants that have a definite level, i.e. order in hµν . The two sets of coupling
constants are related by
1 1 1 λ̄ (0)k λ̄
B λ̄D
= + , = k + k , (11.54a)
(0) GB GD (0)G B Dk k k G G Gk k k
(p)
1 1 D
= for p≥ λ̄1, k λ̄= k for p≥ 1. (11.54b)
(p) (p)
G GDk G G
D
k k k
Notice that the level-(0) couplings, (0)Gk and λ̄
(0)
k , multiply pure background invariants and thus
do not contribute to the dynamical field equations. They are, however, relevant to the statistical
mechanics of black holes, for instance [166] or chapter 13 of this thesis.
In the present ansatz all couplings of higher level, p≥ 1, are identical and agree with the dy-
namical (‘D’) ones. However, the level-(1) Newton and cosmological constants, (1)Gk ≡ GDk and
λ̄ (1) ≡ λ̄Dk k , which enter the effective field equations and the tadpole equation, differ in general
from the level-(0) couplings.
Single-metric approximation. When the distinction of the different levels is artificially
suppressed in the truncation ansatz by hypothesizing perfect split-symmetry along the entire
RG trajectory, i.e. if we set (0) = (p) ≡ sm and λ̄ (0) = λ̄ (p)G smk Gk Gk k k ≡ λ̄k for all p and k, then the
gravitational action Γgravk [g, ḡ] reduces to a functional of a single metric:
∫ ( )
Γgrav − 1 √k [g, ḡ] = ddx g R(g)−2λ̄ smk (11.55)16πGsmk
10For a single-metric extension of the ghost sector, see refs. [132–134].
370 Chapter 11. A generalized C-theorem
The second argument of the action functional, ḡ, has actually disappeared from the RHS of
(11.55). This approximation, for obvious reasons, is referred to as the single-metric Einstein-
Hilbert truncation. In general split-symmetry is violated during the RG evolution, and a detailed
comparison with the more advanced bi-metric ansatz (11.53) has revealed that there are even
qualitative differences in the respective flows, especially in the crossover regime, see chapter
10.
Gauge sector. In the sequel we will analyze the RG flows obtained in three different RG
studies. One is based upon the single-metric ansatz (11.55), while the other two are bi-metric
calculations which employ the same, more general 4-parameter ansatz (11.53), but differ in
various details of the computational setting, the gauge choice in particular.
All three calculations use a gauge fixing action of the form
∫
1 √ [ ( )][αβ ρσ ( )]
Γgfk [g, ḡ] = d
dx ḡ ḡµν Fµ [ḡ] gαβ − ḡαβ FSg fν [ḡ] gD/sm ρσ − ḡρσ32πα Gk
(11.56)
It depends on (the gauge parameter α) and the coefficient ϖ occurring in the gauge conditionαβ
F [ḡ]h ≡ βδ ḡαγD̄ −ϖ ḡαβµ µν µ γ D̄µ hµν . Both α and ϖ are k-independent by assumption.
The single-metric results obtained in [122] are based on the choice (ϖ = 1/2, α = 1), whereas
the bi-metric calculations performed in [172], henceforth denoted [I], and in part II of this thesis,
in the following referred to as [II], use (ϖ = 1/d, α → 0), and (ϖ = 1/2, α = 1), respectively.
In this chapter, the investigation of Ck will be based upon the beta-functions derived in refs.
[122], [I], and [II], respectively.
RG flow and phase diagram. The beta-functions describing the flow on theory space per-
tain to the dimensionless couplings gI ≡ kd−2GIk k and λ I −2 Ik ≡ k λ̄k for I ∈ {B,D,sm,(0),(1), · · · }.
In the bi-metric case, the 4 independent RG differential equations are partially decoupled, dis-
playing the hierarchical structure [173]:
( )
D/(p), λD/(p) → B/(0)gk k gk → λ
B/(0)
k , for p≥ 1 (11.57)
In order to solve the system of differential equations one starts by finding solutions of the D or
p≥ 1-sector, and then substitutes them successively into the decoupled RG equations of the B-
or level-(0) couplings, depending on which ‘language’ one uses.
In [II] it was shown that the beta-functions obtained for the different gauge choices in [I]
and [II] yield the same qualitative results, with only minor numerical differences. In particular,
a UV-fixed point was found in both cases with remarkably stable properties under the change
of gauge. Quite surprisingly, its ‘D’-coordinates agree quite well with the results from the
single-metric approximation; in fact the latter turned out to be unusually reliable within this
regime.
Since the RG flows in the single- and bi-metric truncations are qualitatively similar their
(projected) phase portraits in the gD-λD and gsm-λ sm plane, respectively, share the same overall
structure, as depicted in Fig. 11.1. The integral curves in the upper half plane (gD/sm > 0) are
classified as type Ia, type IIa, or type IIIa trajectories, depending on whether the cosmological
constant λD/sm approaches −∞, 0, or +∞ in the IR,11 respectively [146]. The type IIa trajectory
11The Einstein-Hilbert truncation is known to be inapplicable to type IIIa trajectories when λD/sm approaches
values of order unity. Here, we assume that their classical regime (CR) (having λD/sm ≪ 1) represents their true
k → 0 limit. Even if ultimately this should turn out not to be the case, our treatment of the NGFP→CR crossover
will remain valid.
11.3 Asymptotically safe quantum gravity 371
g
Type Ia
Type IIIa
Type IIa
λ
Figure 11.1: The schematic structure of the phase portrait on gsm-λ sm or the projected gD-λD
plane as predicted by all three truncations considered. Here and in the following the arrows
always point in the direction of decreasing k.
is a separatrix: it separates solutions with an ultimately positive cosmological constant from
those with a negative one at k = 0. Likewise the trajectory gD/sm = 0 separates the upper and
lower half plane, indicating that once the Newton coupling is chosen positive, it remains so on
all scales.
The type IIIa trajectories display a generalized crossover of the kind mentioned in section
11.2.6. It connects a fixed point in the UV to a classical regime in the IR. The latter is located
on its lower, almost horizontal branch where g, λ ≪ 1 [209, 212].
11.3.2 Gravitational instantons
General self-consistent background solutions. Let us now set up the tadpole equations
which result from the truncation ansatz Γk = Γ
grav
k +Γ
gf
k +Γ
gh
k . To be consistent with the con-
ventions in eq. (11.14) we must introduce background fields also for the ghosts, at least for a
moment. We decompose them as ξ µ = Ξµ +η µ and ξ̄µ = Ξ̄µ + η̄µ where (Ξ, Ξ̄) and (η , η̄)
denote their backgrounds and fluctuations, respectively. Then the tadpole condition (11.14)
amounts to the following three coupled equations for ϕ ∈ {h, η , η̄}:
( )∣
δ Γgrav+Γgf+Γgh ∣
0= k k k ∣∣ (11.58)
δϕ(x) ∣
h=0,η=0,η̄=0; ḡ=ḡs.c.,Ξ=Ξs.c., Ξ̄=Ξ̄s.c.k k k
If we begin∫by solving the equations involving δ/δη and δ/δ η̄ and take advantage of the
fact that Γghk ∝ (Ξ̄+ η̄)M(Ξ+η) is bilinear in the ghosts we conclude immediately that the
only self-consistent background they admit for a non-∣degenerate M is the trivial one, Ξs.c.k = 0=
Ξ̄s.c.k . As a consequence, the third equation, δΓ /δh
∣
k ··· = 0, receives no contribution from Γ
gh
k
since its h-derivative vanishes upon inserting the vanishing background ∫ghosts and η = 0 = η̄ .
Furthermore, the gauge fixing action, too, does not contribute since Γgfk ∝ (Fh)
2, being bilinear
in h, has a vanishing derivative at h= 0.
As a result, for every truncation ansatz of the above fo(rm, that is, fo)r an(y choice)of the
‘gravitational’ piece Γgravk , the tadpole condition for Φ̄
s.c.
k ≡ ḡs.c.,Ξs.c., Ξ̄s.c. = ḡs.c.k k k k ,0,0 boils
down to a single non-trivial equation, namely
∣
δ ∣
Γgravk [h; ḡ]
∣∣ = 0 (11.59)δhµν(x) h=0;ḡ=ḡs.c.k
372 Chapter 11. A generalized C-theorem
This equation determines the self-consistent metrics which can ‘live’ on a given spacetime man-
ifold, M, without being modified by the agitation of the quantum fluctuations.
For pure metric gravity, in truncations of the type Γ = Γgrav gf ghk k +Γk +Γk , the C-function
candidate Ck ≡ Γk[∣ϕ = 0;Φ̄s.c.k ] which we motivated above for a generic theory now becomes
concretely Ck = Γ ∣k s.c. . It involves only the ‘grav’-part of the ansatz:h=η=η̄=Ξ=Ξ̄=0,ḡ=ḡk
grav s.c.
Ck = Γk [h= 0; ḡk ] (11.60)
The Einstein-Hilbert case. In the special case of the Einstein Hilbert truncation the tad-
pole equation (11.59) happens to have the same mathematical structure as the classical vacuum
Einstein equation in presence of a cosmological constant. The hµν -derivative of the bi-metric
ansatz (11.53) for Γgravk yields at hµν = 0:
G s.c.µν(ḡk ) =−λ̄ (1) ḡs.c.k kµν , or Rµν(ḡs.c.) = 2
(1) s.c.
k d−2 λ̄k ḡkµν (11.61)
In the single-metric approximation the tadpole equation is the same, except that λ̄ (1)k is replaced
with λ̄ sm then. Thus ḡs.c.k k is always an Einstein metric, and upon contraction we get from eq.
(11.61):
2d
( s.c.) = λ̄ (1)R ḡk (d−2) k (11.62)
Inserting this expression for the curvature scalar into Γgravk yields the following representation
of the EAA, evaluated for hµν = 0 and a self-consistent background geometry:
∫ √ { }∣
= Γgrav[0; ḡs.c. − 1] = dd (0) ∣Ck k k x ḡ R(ḡ)−2λ̄( ) k ∣16π 0G ḡ=ḡs.c.k M k
1 [( ) ]
=− d−2 λ̄
(1) − λ̄ (0)k k Vol(M, ḡs.c.k ) (11.63)8π (0)G dk
∫ √
Here Vol(M,g) ≡ dM d x g denotes the Euclidean volume of the manifold M measured with
the metric written in the argument, gµν .
Note that Γgravk evaluated at (h; ḡ) = (0; ḡ
s.c.
k ) depends on both the level-(0) and the level-(1)
couplings in a non-trivial way: the former enter via the action Γk|h=0 which has a level-(0)
component only, the latter via the tadpole equation which is entirely ‘level-(1)’.
Assuming the running dimensionful couplings are regular at the scale k considered, eq.
(11.63) shows that Ck is finite if, and only if, the spacetime manifold has finite volume. The
self-consistent background being an Einstein metric, its curvature structure and other details
play no role for the value of the action, it is only the volume that matters.
Instanton solutions. Trying to find solutions to (11.61) that exist for all scales from ‘k =
∞’ down to k = 0 the simplest situation arises when all metrics ḡs.c.k , k ∈ [0,∞) can be put on
the same smooth manifold M, leading in particular to the same spacetime topology at all scales,
thus avoiding the delicate issue of a topological change. This situation is realized, for example,
if the level-(1) cosmological constant is positive on all scales, which is indeed the case along
the type (IIIa) trajectories: λ̄ (1)k > 0, k ∈ [0,∞).
In the following we focus on precisely this situation. The requirement of a finite action is
then met by a well studied class of Einstein spaces which exist for an arbitrary positive value
of the cosmological constant, namely certain 4-dimensional gravitational instantons [244–246],
see Table 11.1 for some examples.
11.3 Asymptotically safe quantum gravity 373
Metric M V
Eucl. de Sitter S4 3π
Page P2+ P̄2 1.8π
S2×S2 S2×S2 2π
Fubini-Study P2(C) 9π/4
Table 11.1: Various 4-dimensional gravitational instantons and the related normalized volumes
V(M, g̊). (See [244] for a detailed account.)
Let g̊µν be the metric of one such instanton, corresponding to a fixed reference value of the
cosmological constant, λ˚̄ , say. Then the tadpole equation (11.61), at any k, is solved by the
following rescaled metric [210, 211]:
˚̄
ḡs.c.
λ
kµν = g̊µν (11.64)
λ̄ (1)k
As a result, the k-dependence of the total volume behaves as, for arbitrary d,
[ ]
(1) −d/2
Vol(M, ḡs.c.k ) = 8π λ̄k · V(M, g̊) (11.65)
Here we introduced the dimensionless constant
≡ 1V(M, g̊) λ˚̄ d/2Vol(M, g̊) (11.66)
8π
which is characteristic of the instanton under consideration.12 The number V is manifestly
independent of k, and it is easy to see that it is also independent of λ˚̄ . The reason is that g̊
depends on λ˚̄ via the equation Rµν(g̊) =
2
−2 λ
˚̄
g̊µν . This implies that upon rescaling λ
˚̄ by a
d
constant factor, λ˚̄ → c2λ˚̄ , the metric responds according to g̊ −2µν → c g̊µν , and so the volume
behaves as Vol(M, g̊) → c−dVol(M, g̊). In the definition of V, eq. (11.66), the factor c−d
coming from the volume is therefore precisely canceled by a corresponding factor c+d which is
produced by its pre-factor, λ˚̄ d/2 → ˚̄cd λ d/2.
Thus the value of V is a universal number which depends only on the type of the instanton
considered13 . For the round metric on Sd we find, for instance,
(d−1)
d [(d−1)(d−2)]d/2V(S ) = π 2 (11.67)
2(d+4)/2Γ(d+12 )
Table 11.1 contains the corresponding Vvalues for some more examples in d = 4.
The Ck-function for the Einstein-Hilbert truncation. Using (11.65) in (11.63) we ob-
tain the following two equivalent representations of Ck:
1 [( ) ]
=− [ ] d λ̄ (1) (0)Ck
(0) (1) d/2 d−2 k − λ̄k V(M, g̊)
Gk λ̄k
[( ) ]
=− 1[ ] d (1) (0)−2 λk −λk V(M, g̊) (11.68)(0) (1) d/2 d
gk λk
12It is closely related to the normalized volume ṽ(M,g) defined in the mathematical literature [245, 247, 248].
13For a discussion of the topological properties of the normalized volume see [245, 247, 248].
374 Chapter 11. A generalized C-theorem
In the second line of (11.68) we eliminated the dimensionful quantities (p)Gk and λ̄
(p)
k in favor of
their dimensionless analogs whereby all explicit factors of k dropped out.
We observe that the result (11.68) for the function k 7→ Ck has the general structure
≡ ( (0),λ (0) (1)Ck C gk k ,λk ) = (
(0) (0) (1)
Y gk ,λk ,λk )V(M, g̊) (11.69)
Here Y( ·)≡ C ( ·)/V stands for the following function over theory space:
[( ) ]
d λ (1) −λ (0)
Y(g(0),λ (0),λ (1)) =− d−2 (11.70)
g(0) [λ (1)]d/2
Equation (11.70) represents our main result. We shall study its properties below. In 4 dimen-
sions we have in particular
2λ (1) −λ (0)
Y(g(0),λ (0),λ (1)) =− (d = 4) (11.71)
g(0) (λ (1))2
Several comments are in order here.
A. The function C depends on both the RG trajectory and on the solution to the running
self-consistency condition, along this very trajectory, that has been picked. In eq. (11.69)
those two dependencies factorize: the former enters via the function Y, the latter via the
constant factor V(M, g̊) that characterizes the gravitational insta(nton. )
B. The dependence on the RG trajectory, parametrized as k →7 (0,1) (0,1)gk ,λk , is obtained
(by evaluati)ng a scalar function on theory space along this curve, namely Y : T→ R,
g(0,1),λ (0,1) 7→Y(g(0),λ (0),λ (1)). It is defined at all points of Twhere g(0) =6 0 and λ (1) 6= 0,
and turns out to be actually independent of g(1).
We shall refer to ≡ ( (0), λ (0)C. Yk Y gk k , λ
(1)
k )≡ Ck/V(M, g̊) and Y( ·)≡ C ( ·)/V(M, g̊) as the
reduced Ck and C ( ·) functions, respectively.
D. Denoting the collection o(f running)couplings by u(k), we may write the scale derivative of
Ck as k∂kCk = V(M, g̊) β
∂
α ∂ Y (u(k)) which involves the directional derivative
~β ·~∇
uα
acting upon scalar functions on theory space. This motivates defining the subset T+ of
Ton which Y : T→ R has a positive directional derivative in the direction of ~β :
{ }
T+ ≡ u ∈ T|βα(u) ∂∂ Y(u)> 0 . (11.72)uα
The interpretation is that Ck increases monotonically with k along those (parts of) RG
trajectories that lie entirely inside T+, i.e. k∂kCk > 0 at all points of T+.
E. Invoking the idealization of exact split-symmetry, i.e. assuming that the Newton and cos-
mological constants of different levels are all equal ( (p)g ≡ gsm, λ (p)k k k ≡ λ smk , p= 0,1,2, · · · ),
we obtain Ck in the single-metric approximation.
It reads C = C (gsm,λ sm) = Ysm(gsm,λ smk k k k k ) · V(M, g̊) with
( )
Ysm(gsm
2 1
,λ sm) =− − (11.73)d 2 gsm (λ sm)d/2−1
Note that it depends only on the dimensionless combination Gsm(λ̄ sm)d/2−1k k = g
sm
k (λ
sm)d/2−1k
whose robustness properties under changes of the cutoff and the gauge fixing has often
been used to check the reliability of single-metric truncations [123, 135, 151, 191].
F. In general, the beta-functions which govern the RG evolution of Γk may depend on the
topology of M, see [236] for an example. Within the truncation considered here this is
not the case, however, the reason being the universality of the heat-kernel asymptotics
which is exploited in the computation of the beta-functions, see section 7.5 of part II of
this thesis and ref. [172].
11.3 Asymptotically safe quantum gravity 375
G. Switching from the level language, which employs the couplings g(p) and λ (p), to the B-D
language, based upon the couplings {gD, λD, gBk , λ B}, with
1 1 1 λ (0) λ B λD
= + , = + and g(1) = gD, λ (1) = λD ,
g(0) gB gD g(0) gB gD
the function Y : T→ R assumes the following form, for d = 4,
1 1 [ λ B ]
Y(gD,λD,gBk ,λ
B) =− − 2− (11.74)
gDλD gBλD λD
This representation is particularly convenient for part of the numerical analyses to which
we turn in the next section.
11.3.3 Numerical results
Above we investigated the general properties of Ck and we derived its explicit form for the
Einstein-Hilbert truncation. In this subsection we will study the monotonicity properties of Ck
in 4 spacetime dimensions for solutions of the FRGE in both the single-metric [122] and the
two bi-metric approaches [I], [II]. We will focus on type IIIa trajectories (see Fig.11.1) which
exhibit the aforementioned NGFP→CR crossover. Possibly those solutions are relevant to real
Nature even [209, 212]. For the corresponding discussion of type Ia and type IIa trajectories we
refer to appendix 11.B.
A. Returning to Ck for the bi-metric truncation given in eq. (11.69), the entire information
on the RG trajectory is contained in the factor
2λ (1) −λ (0)
Yk ≡ Y( (0)gk ,λ
(0),λ (1) k kk k ) =− (11.75)(0) [λ (1)g 2k k ]
We are going to evaluate this function of k for a number of RG trajectories on the 4-
dimensional theory space which we generate numerically.
B. Likewise we compute the reduced Ck-function predicted by the single-metric truncation,
that is, we evaluate
Ysm ≡ Ysm(gsm sm − 1k k ,λk ) = (11.76)gsm smk λk
for trajectories on the corresponding 2-dimensional theory space. We obtain them numer-
ically by solving a system consisting of 2 differential equations only.
C. We are also going to perform a hybrid calculation which is intermediate between the 2-
and 4-dimensional treatment, in the following sense.
As a rule, the single-metric approximation to a bi-metric truncation is a valid description
of the flow if split-symmetry is only weakly broken, i.e. there is no significant difference
between couplings at different levels: (0) = (1)uα uα =
(2)
uα = · · · . If we make the correspond-
ing identifications λ (0) = λ (1) = · · · ≡ λ and (0) = (1)k k k gk gk = · · · ≡ gk in (11.75) we obtain
1
Ysplit-symk ≡ Y (gk,λk,λk) =− (11.77)gkλk
Here (gk, λk) stands for (
(0)
gk , λ
(0)
k ) or, what should be the same, (
(1), λ (1)gk k ) ≡ (gD Dk , λk )
as obtained from the bi-metric RG equations. If the trajectory respects split-symmetry it
does not matter from which level we take the couplings. If split-symmetry is not perfect, it
376 Chapter 11. A generalized C-theorem
does however matter from which level they come, and we obtain two, in general different
functions:
split-sym,(0) ≡− 1 split-sym,(1) ≡− 1Yk , Y (11.78)(0)λ (0) k (1)λ (1)gk k gk k
It will be instructive to compare the two functions (11.78) for various representative tra-
jectories on 4-dimensional theory space. This will provide us with some insights about
what is more important in an approximate calculation of Ck: good control over the details
of the underlying RG trajectory, or precise (analytic) knowledge about how Γ [0;Φ̄s.c.k k ]
depends on the couplings from the various levels when split-symmetry is broken.
Note that while (11.77) and (11.78) have the same structure as the single-metric result (11.76),
there is a crucial difference: the former Yk-functions involve running couplings obtained from
the 4-dimensional bi-metric system of RG equations, whereas the latter, Ysmk , has the solutions
to the 2-dimensional single-metric flow equations as its input.
Note also that by virtue of (11.74) the full-fledged bi-metric Yk may be written as
[ ]
B
split-sym,(1) 1 λ
Y kk = Yk +∆Yk with ∆Yk ≡− 2− (11.79)gB λD λDk k k
The magnitude of the ∆Yk-term is a measure for the degree of split-symmetry violation as
split-sym,(p)
∆Yk = 0 when the symmetry is exact14 and Yk ≡ Yk for all p= 0,1,2, · · · .
Single-metric truncation
We begin the computation of the reduced single-metric Ck-function by numerically calculating
a number of type IIIa trajectories on the 2-dimensional gsm-λ sm theory space, and then evaluate
Ysmk for them. We find that the reduced Ck-functions thus obtained always have the same qual-
itative properties: they become stationary (approach plateaus) for k → ∞ and k → 0, but they
are not on all scales monotonically increasing with k. For all trajectories there exists a regime
of scales where ∂kCk < 0. This negative derivative typically occurs while the trajectory crosses
over from the NGFP to its turning point close to the GFP.
In Fig. 11.2 we display a representative single-metric example. For reasons of a clearer
presentation we plot here, and in all analogous diagrams that will follow, the inverse of the
reduced Ck function, along with its derivative. Furthermore, here and in the following, the scale
k√is always measured in units of the Planck mass defined by the classical regime, mPlanck ≡ 1/
GCR. The example of Fig. 11.2 shows the ‘wrong sign’ of the scale derivative (∂kYsmk < 0) for
k in an interval between about 3 and 5 Planck masses, which is the typical order of magnitude.
Bi-metric Einstein-Hilbert truncation
Turning now to the bi-metric Einstein-Hilbert truncation with its 4-dimensional theory space
we employ the two sets of RG equations from [I] and [II], respectively, and compare the results
they imply.
Furthermore, we must distinguish two fundamentally different cases with respect to the RG
trajectories, namely trajectories which restore split-symmetry in the IR, and trajectories which
do not.
In either case we begin by numerically computing a type IIIa trajectory of the decoupled
gD-λD subsystem. Then, when we ‘lift’ this 2D trajectory to a 4D one, we must pick initial
14Of course,[this can] also be seen directly. Reinstating dimensionful couplings, the two contributions to
λ̄B
∆ 1Y = − 2− kk B D D are proportional to 1/GBk and λ̄ Bk /GBk , respectively. Those quantities are the pre-factorsGk λ̄k λ̄k
of the monomials responsible for the extra background dependence of Γk, and so they must vanish to achieve split-
symmetry.
11.3 Asymptotically safe quantum gravity 377
0.00
0.05
0.00
-0.05
-0.05
-0.10
-0.15
1 sm -0.20/Yk
0.10 0 2 4 6 8 10-
k/mPlanck
-0.15
0 2 4 6 8 10
k/mPlanck
Figure 11.2: The inverse of Ysmk for a typical single-metric type IIIa trajectory. The inset shows
its k-derivative whose positive values indicate a violation of monotonicity.
conditions for gB and λ B, and it is at this point that we must decide about restoring, or not
restoring the symmetry. As we showed in detail in ref. [173], the requirement of split-symmetry
implies uniquely fixed values for the couplings gBk and λ
B
k in the limit kց 0, namely precisely
the coordinates (gB•(k), λ
B
• (k)) of the running UV-attractor. We now discuss the cases with and
without symmetry restoration in turn.
Split-symmetry restoring trajectories. Opting for the symmetry restoring IR values of
the B-couplings, what remains free to vary is the underlying type IIIa trajectory in the D sector.
We find that the qualitative properties of the resulting functions Yk are the same for all trajecto-
ries of this type, and that these properties do not depend on whether we use the RG equations
from [I] or from [II]. The picture is always the following: The reduced Ck-function Yk ap-
0.00
0.00
-0.05 0.4
-0.10
-0.05
-0.15 0.2
-0.20
-0.25 0.0
-0.10 -0.30
1 2 3 4 -0.2
k/m
-0.15 Planck
-0.4
-0.6
-0.20
-0.8
1 2 3 4 0.5 1.0 1.5 2.0 2.5 3.0
k/mPlanck k/mPlanck
Figure 11.3: The left plot shows 1/Yk and its scale derivative for a typical bi-metric type IIIa
trajectory that restores split-symmetry in the IR. It is based on the RG equations of [I]. For
these trajectories, Ck is always found to be perfectly monotone. The inset in the left plot shows
k∂k1/Yk, which is decomposed in the right plot into the derivative of the split-symmetric com-
ponent 1/ split-sym,(1)Yk (dashed, gray curve) and of ∆(1/Yk) (solid, gray curve). Neither of the two
contributions is negative definite separately, but their sum is (solid, dark red curve).
proaches plateau values in the fixed point and in the classical regime, i.e. it becomes stationary
there, and it is a strictly monotone function of k on all scales in between, k∂kYk > 0.
Clearly the latter property is in marked contrast with the single-metric results. In Fig. 11.3
1/Yk
k∂k1/Yk
k∂k1/Y
sm
k
k∂k1/Yk
378 Chapter 11. A generalized C-theorem
0.3
0.00
0.2
-0.02 0.00
0.1
-0.05
-0.04
-0.10 0.0
-0.06 -0.15 -0.1
-0.20
0.08 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.2-
k/mPlanck
-0.3
-0.10
-0.4
-0.12
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
k/mPlanck k/mPlanck
Figure 11.4: The function 1/Yk as in Fig. 11.3, but now based on the RG equations of [II].
we show Yk for a representative IIIa-trajectory. The plots were obtained with the RG equations
derived in [I]. Their analogs based on the equations from [II] are displayed in Fig. 11.4. It is
gratifying to see that there is hardly any difference between the results from the two calcula-
tional schemes.
In eq. (11.79) we decomposed Yk as =
split-sym,(1)
Yk Yk + ∆Yk in order to make its split-
symmetry violating part explicit. For 1/Yk we have correspondingly 1/ = 1/
split-sym,(1)
Yk Yk +
∆(1/ ) with ∆(1/ ) = −∆ /( split-sym,(1)Yk Yk Yk Yk Yk) and exact split-symmetry (∆Yk = 0) amounts
to ∆(1/Yk) = 0, of course.
In the Figs. 11.3 and 11.4 we show how the scale derivative of 1/Yk decomposes into the
derivative of 1/ split-sym,(1)Yk and of the symmetry violation term ∆(1/Yk). For all trajectories of
the class considered, and with the RG equations from both [I] and [II], we always find that the
scale derivative of neither 1/ split-sym,(1)Yk , nor of ∆(1/Yk) is negative definite separately, but their
sum is!
When split-symmetry is intact, ∆Yk = 0, (violation of) monotonicity for 1/
split-sym,(1)
Yk is
equivalent to a (non-) monotone function 1/Yk. Now, for generic RG trajectories from the bi-
metric calculations [I] and [II] this condition is known to be approximately satisfied only for
k → ∞, i.e. in the vicinity of the NGFP. The trajectories considered in the present paragraph
are fine-tuned to fulfill the requirement of split-symmetry restoration in the IR, so ∆Yk vanishes
also there. But on all intermediate scales split-symmetry is broken, the monotonicity of Ck is
not guaranteed by any general argument, and in general 6= split-sym,(1)Yk Yk . And indeed Figs. 11.3
and 11.4 show a strong violation of this equality. In fact, the part split-sym,(1)Yk is seen to be non-
monotone exactly in the regime where the split-symmetry of the RG trajectory is known to be
significantly broken.
The status of split-symmetry violation displayed by an RG trajectory is thus also reflected
by the deviation of the pertinent Yk from its ‘split-symmetry enforced’ version
split-sym,(1)
Yk . Quite
remarkably, for the present class of trajectories a perfect compensation of the split-symmetry vi-
olation the RG trajectories suffer from, and a nonzero correction term ∆Yk takes place. Mirac-
ulously, the term ∆ split-sym,(1)Yk modifies the non-monotone Yk in precisely such a way that the
total Yk is monotone.
This perfect compensation for all eligible trajectories strongly supports our hope that the
candidate Ck-function really qualifies as a ‘C-function’ since the structure of ∆Yk, i.e. the way
how it depends on the couplings, is a direct consequence of having set C = Γ [0;Φ̄s.c.k k k ]. It is
indeed surprising to see that a function as simples as the ∆Yk of eq. (11.79) can do the job of
rendering Ck monotone for all physically relevant trajectories at once.
1/Yk
k∂k1/Yk
k∂k1/Yk
11.3 Asymptotically safe quantum gravity 379
Split-symmetry violating trajectories. We continue to use the bi-metric RG equations
from [I] and [II], but now we deliberately break split-symmetry by selecting a generic trajectory
in the gB-λ B-subspace, on(e that w)ould not hit the running UV-attractor for kց 0. After having
generated solutions k 7→ gD Dk , λk of the two decoupled D-equations, again corresponding to
a type IIIa trajectory, we solve the resulting B-equations with initial values for (gB Bk , λk ) that
explicitly break split-symmetry even in the IR.
In Figs. 11.5 and 11.6 the numerical results for 1/Yk are displayed for the RG equations
of [I] and [II], respectively. They show the same qualitative behavior: While the function Yk
becomes stationary towards the NGFP-regime in the UV, the second plateau in the IR, which
we had found for trajectories restoring split-symmetry, is now destroyed by the appearance of
extrema in the function Yk, rendering it non-monotone. In the right panels of Figs. 11.5 and
11.6 this is reflected by the changing sign of the derivative k∂kYk plotted there.
In order to visualize how sign flips of k∂kYk can come about it is helpful to define, and to
determine numerically, the following subset of the gB-λ B-plane:
{ }
TB (k)≡ (gB, λ B) ∈ R2+ | (gDk , λDk , gB, λ B) ∈ T+ ⊂ T (11.80)
The RG time-dependent set TB+(k) consists of all those points of the 4D theory space at which
the directional derivative is positive, βα ∂Y/∂uα > 0, and which have (gD, λD( )-coo)rdinates that
agree with the current position of the selected ‘D’ trajectory at time k, i.e. gD, λDk k .
0.4
0.05 0.2
0.1
0.2
0.0
0.00
-0.1
-0.2 0.0
0.5 1.0 1.5 2.0
-0.05
k/mPlanck
-0.2
-0.10
-0.4
0.0 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 3.0
k/mPlanck k/mPlanck
Figure 11.5: The left plot shows the function 1/Yk for a bi-metric trajectory of type IIIa that
does not restore split-symmetry in the IR. It is based on the RG equations of [I]. We observe
a sign-change of k∂k(1/Yk) at moderate values of k, indicating a violation of monotonicity. In
the decomposed form of k∂k(1/Yk), shown in the right plot, we see that the contribution 1/
split-sym,(1)
Yk is in fact monotone, but the correction term ∆(1/Yk) is not, and neither is their sum.
Not restoring split-symmetry in the IR results in a violation of the monotonicity of Ck along the
trajectory considered.
Using the same typical IIIa trajectory in the dynamical sector as above in the split-symmetry
restoring case, we now study the RG evolution in the gB-λ B-plane, see Fig. 11.7. To this end,
we subdivide this plane into TB+(k), the shaded regions in the diagrams of Fig. 11.7, and its
complement, the white regions in Fig. 11.7. This subdivision is d(ifferent at each)instant of RG
time. We are particularly interested in those 4D trajectories k 7→ gD, λDk k , gBk , λ Bk that give rise
to a monotonically increasi(ng Ck-f)unction, or in other words, in trajectories whose projection
on the B-plane is such that gBk , λ
B
k ∈ TB+(k) holds true for all k.
From Fig. 11.7 it is now clear why the trajectory restoring split-symmetry in the IR, starting
at the UV-attractor located at P2, is so special: As the scale k changes, so does the region defined
by TB+(k). In the IR, the domain T
B
+(k) defines a narrow band around the running UV-attractor
(P2), and this results in a monotonically increasing Ck. While the distinguished split-symmetry
1/Yk
k∂k1/Yk
k∂k1/Yk
380 Chapter 11. A generalized C-theorem
0.4
0.15
0.2
0.10 0.1 0.2
0.0
0.05
-0.1 0.0
0.00 -0.2
0.5 1.0 1.5 2.0
-0.05 k/m -0.2Planck
-0.10
-0.4
-0.15
0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5
k/mPlanck k/mPlanck
Figure 11.6: The function 1/Yk as in Fig. 11.5, but now based on the RG equations of [II].
restoring trajectory is safely within this band, most of the split-symmetry breaking trajectories
lie well outside TB+(k) at low k. Increasing k, we move towards the UV, and T
B
+(k) extends
especially to regions with negative λ B, while its boundary approaches, and ultimately touches
the asymptotic position of the NGFP.
The crucial fact to notice is the following. At all scales, the symmetry restoring trajectory is
seen to stay within TB+(k), and this is in agreement with the results obtained in paragraph (A).
Since when k is increased sooner or later all trajectories converge to this particular one15, they
are necessarily all pulled towards a regime TB+(k), if they are not yet inside already. This can be
observed in Fig. 11.7 by following the trajectory that passes through the point P1 at some low
scale. As P1 lies outside TB+(k) this implies that ∂kCk < 0 in the IR. Increasing k the trajectory
is pulled towards the running UV-attractor and between the first and second snapshot of Fig.
11.7 it crosses the boundary of TB+(k). At this moment the derivative of Ck crosses zero and
from this point onward we have ∂kCk > 0. In the third snapshot the trajectory is already well
inside TB+ and it approaches the symmetry-restoring one. Once close to this ‘guiding trajectory’
it remains in its vicinity and together, for k→ ∞, they approach the boundary of TB+(k) from its
interior. This is as it should be since we know that k∂kCk = 0 at the NGFP.
The hybrid calculation. In the hybrid calculation, we retain only the Ysplit-symk -part of
Yk, omitting the correction term ∆Yk. In Fig. 11.8 we show (the inverse of) its two variants
split-sym,(0) split-sym,(1)
Yk and Yk which are obtained by extracting the couplings from, respectively, the
0th and the 1st level of a bi-metric type IIIa trajectory. This particular trajectory restores split-
split-sym,(0)
symmetry in the IR. Fig. 11.8 shows that the graphs of the resulting functions Yk and
split-sym,(1)
Yk are quite different, the former function is monotone, the latter is not.
This observation once more tells us that the correction term ∆Yk is needed in order to com-
pensate for the split-symmetry violation that goes into Ck via the trajectories. In fact, at inter-
mediate scales, all trajectories suffer from this disease, both the symmetry restoring and the
non-restoring ones; the unmistakable symptoms are the substantial differences among the lev-
els.
This confirms our earlier findings: The correction term ∆Yk is indispensable. It is needed
in order to protect the sum Ysplit-symk +∆Yk against the otherwise unavoidable infection with the
symmetry violation the trajectories must live with. This protection is successful, i.e. Ck =
(Ysplit-symk +∆Yk)Vhas a monotone dependence on k, provided we do not break split-symmetry
by hand, that is, by selecting inappropriate initial conditions for the background couplings.
15See ref. [173] for a detailed demonstration of this behavior.
1/Yk
k∂k1/Yk
k∂k1/Yk
11.3 Asymptotically safe quantum gravity 381
10 10
8 8
6 6
gB gB
4 4
2 2
P2
P1
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
10 10
8 8
6 6
gB gB
4 4
2 2
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
λB λB
Figure 11.7: This series of snapshots represents the gB-λ B-plane at four RG times which in-
crease from the upper left to the lower right diagram. They are given by the maximum k-value
of the incomplete dynamical trajectory k 7→ (gD Dk , λk ) shown in the respective inset. The shaded
regions correspond to TB+(k) at that particular time; hence every trajectory in the shaded (white)
region will give rise to a positive (negative) value of k∂kCk at the instant of time k. Furthermore,
two different B-trajectories that are evolved upward (towards increasing scales k) are shown
at the corresponding moments. The one passing the point P1 (P2) is split-symmetry violating
(restoring). The symmetry restoring trajectory starts its upward evolution close to P2, the po-
sition of the running UV attractor [173]; we see that this trajectory never leaves the shaded
area, and thus its Ck-function is strictly monotone. This is different for the trajectory through
P1: Attracted by the running UV-attractor, it is pulled into the shaded region, thus unavoidably
crossing the boundary of TB+(k), which causes a sign flip of ∂kCk, rendering Ck non-monotone.
382 Chapter 11. A generalized C-theorem
0.00
-0.05
-0.10 level-(0)
-0.15
level-(1)
sm
-0.20
0 1 2 3 4 5
k/mPlanck
split-sym,(0)
Figure 11.8: The inverse of Yk and
split-sym,(1)
Yk is shown for a generic bi-metric type IIIa
trajectory that restores split-symmetry in the IR. On intermediate scales, the RG trajectory is
not split-symmetric, as is evident from the different graphs of the level-(1) (dark, solid) and the
level-(0) (light, solid) variants of Ysplit-symk . For comparison, the single-metric result Y
sm
k is also
included (dashed curve).
Testing pointwise monotonicity. We have seen in eq. (11.42) that for exact RG trajecto-
ries k→7 Γk the only source of obstructions for Ck to become a monotone function is the second
term on its RHS, which measures to what extent Γk breaks split-symmetry. In the case of exact
RG solutions, we know that the first term on the RHS is positive, (∂ s.c.kΓk) [0;Φ̄k ]≥ 0, since this
is a special case of the pointwise monotonicity, (∂kΓk) [ϕ ;Φ̄] ≥ 0 ∀(ϕ ,Φ̄), ∀k. However, the
latter property might not always be true for approximate solutions to the flow equation, those
obtained by using truncations, for instance. Testing pointwise monotonicity, (∂kΓk) [ϕ ;Φ̄] ≥ 0,
may therefore serve as a device to judge the validity of a truncation. In future work we will come
back to this method for arbitrary arguments (ϕ ,Φ̄). We focus here only on (∂kΓk) evaluated at
the special arguments (ϕ ,Φ̄) = (0,Φ̄s.c.k ).
(∂kΓk)[0;Φ̄
s.c.
k ] (∂kΓk)[0;Φ̄
s.c.]
20 k
600
15 400
200
10
0.4 0.5 0.6 0.7 0.8
5
-200 k/mPlanck
2 4 6 8 10 -400
k/mPlanck
Figure 11.9: The first term on the RHS of eq. (11.42), i.e. (∂ Γ ) [0;Φ̄s.c.k k k ] is evaluated for
a typical single-metric (left) and split-symmetry violating bi-metric (right) trajectory. In both
cases, it is seen to be negative for certain scales. This indicates a severe failure of the underlying
approximation since, at the exact level, (∂kΓk) is known to be positive at all field arguments and
for any k.
It turns out that the single-metric and the ‘unphysical’ bi-metric RG trajectories (those
without split-symmetry restoration) actually fail this pointwise monotonicity test. As shown
in Fig. 11.9, there are k-intervals on which (∂ s.c.kΓk) [0;Φ̄k ] is negative. On the other hand, for
the split-symmetry restoring bi-metric trajectories this quantity is positive throughout, as it is
at the exact level, see Fig. 11.10. These findings make it very clear that the non-monotonicity
displayed by our Ck-function candidate, when applied to single-metric and symmetry violating
1/ split-symYk
11.3 Asymptotically safe quantum gravity 383
(∂kΓ )[0;Φ̄
s.c.
k k ]
(∂kΓk)[0;Φ̄
s.c.
k ]
5000
1.5 4000
3000
2000
1.0
1000
0 k/mPlanck
0.2 0.4 0.6 0.8 1.0
0.5
k/mPlanck
2 4 6 8 10
Figure 11.10: The quantity (∂ Γ ) [0;Φ̄s.c.k k k ] is now evaluated for a split-symmetry restoring bi-
metric trajectory. It always stays non-negative, even in those regimes where the single-metric
or the split-symmetry violating bi-metric (see inset) trajectories fail the pointwise monotonicity
test.
bi-metric truncations, is not due to a defect of the proposed form of Ck but rather originates
in insufficient approximations. Only the symmetry-restoring, bi-metric trajectories are close
enough to the exact ones to render both (∂kΓk) [0;Φ̄s.c.k ] and the full ∂kCk positive.
Summary and Conclusion
To sum it up we can say that the expected monotonicity of Ck arises under the following con-
ditions: First, the bi-metric version of the Einstein-Hilbert truncation is used, and second, the
underlying RG trajectory is split-symmetry restoring. Violating either of these conditions may
destroy the monotonically increasing behavior of Ck. We saw that there are cases where the
split-symmetry violation of Γk is sufficiently small to leave the monotonicity of Ck intact, the
main example being the bi-metric type IIIa trajectories approaching the UV attractor for k→ 0.
We have seen that down-grading the bi-metric truncation ansatz to the level of a single-
metric approximation is paid by loosing the monotonicity property of Ck. As for its dependence
on the running couplings, the reduced Ck-function in the bi-metric truncation, Yk, differs from
its single-metric counterpart Ysmk by the correction term ∆Yk, which vanishes when Γ
grav
k is
exactly split-symmetric. (In the single-metric approximation, this is always the case, by decree.)
We found that there is a numerically highly non-trivially conspiracy and compensation between
∆Yk and those properties of the RG trajectories which stem from the split-symmetry violation
in the flow equation, and which could easily destroy the monotonicity of Ck. The fact that this
does not happen for any of the physically relevant trajectories is directly linked to the specific
properties of our candidate function, C =Γ [0;Φ̄s.c.k k k ], its scale dependent argument in particular,
since it determines the structure of ∆Yk.
Taken together these findings strongly support the following conjecture: In the full theory,
QEG in 4 dimensions, or in a sufficiently general truncation thereof, the proposed candidate for
a generalized C-function is a monotonically increasing function of k along all RG trajectories
that restore split-symmetry in the IR and thus comply with the fundamental requirement of
Background Independence.
If the conjecture can be established we will have a particularly easy to apply diagnostic tool
for testing the reliability of truncations. Since then solutions to the untruncated flow equation
for sure have a monotone Ck, any truncation that violates the monotonicity misses qualitatively
important features of the RG flow and would therefore be judged an insufficient approximation
384 Chapter 11. A generalized C-theorem
to the full flow. In this light we provisionally conclude that the single-metric approximation
is not fully reliable, while the bi-metric Einstein-Hilbert truncation is superior as it keeps the
monotonicity of Ck intact at least. Of course this conclusion is fully consistent with all other
results available on the bi-metric Einstein-Hilbert truncation, ref. [172] as well as the chapters
10 and 12 of this thesis.
As split-symmetry is essential in this context, it might be helpful to recall its physical con-
tents. Split-symmetry and the corresponding Ward Identities for split-symmetry (WISS) are
the technical device by means of which Background Independence in the physical sector (‘on-
shell’) is imposed on the effective action and similar ‘off-shell’ quantities16 . The prototypical
example of a Background Independent theory is classical General Relativity [234]. Now, even
though we describe it by an effective action Γ[g, ḡ], we would like QEG to enjoy Background
Independence exactly at the same level as General Relativity.
Let us contrast QEG with genuine bi-metric theories, in the original sense of the word,
that is, extensions of General Relativity employing two physically distinct metrics, (1)gµν and
(2)
gµν , say. Depending on the structure of their action S[
(1)
gµν ,
(2)
gµν , · · · ] they could differ, for in-
stance, in their coupling to matter, or their propagation properties. In an appropriate limit,
matter particles of a certain species could, for example, follow the geodesics of (1)gµν , or of
(2)
gµν ;
but it also can happen that trajectories are no geodesics at all and have no geometric interpre-
tation. In a genuine bi-metric theory, these different cases are experimentally distinguishable.
The metrics have equal status in that both of them, independently, can make their way into
observables. In canonical quantization both (1) (2)gµν and gµν are turned into operators. This is
fundamentally different when one applies the background field technique to the quantization
of a system with a bare action S[ĝ] depending on one metric only, and introduces ḡµν only
as a technical convenience, for coarse-graining and gauge-fixing purposes in particular. Set-
ting ĝµν = ḡµν + ĥµν we transfer the physical degrees of fre∫edom entirely∫ from ĝµν to ĥµν
which is made the new dynamical quantum field by replacing [dĝµν ] with [dĥµν ]. From the
functional perspective, ḡµν is merely an arbitrary shift on which no observable consequence
of the theory may depend. In canonical quantization, ĝµν and ĥµν are operators, while ḡµν
continues to be a classical c-number field. The logical dissimilarity between dynamical and
background metric gets slightly obscured at the level of the expectation values hµν ≡ 〈ĥµν〉
and gµν ≡ 〈ĝµν〉 = ḡµν +hµν since Γk[h; ḡ] ≡ Γk[g, ḡ] depends on two independent fields, two
metrics in fact, if one uses the EAA in the ‘comma notation’, Γk[g, ḡ]. Now, the role of the
split-symmetry as encoded in the WISS of eq. (11.21), is to express the requirement that there
is only one physical metric and that no observable quantity may depend on how ḡµν was cho-
sen. Setting for example k= 0 and ignoring gauge fixing issues for a moment, invariance of the
bare action under {δ ĥµν = εµν , δ ḡµν = −εµν} ⇐⇒ {δ ĝµν = 0, δ ḡµν = −εµν} implies that
Γ0[h; ḡ] can depend on the sum ḡ+ h ≡ g only, while Γ0[g, ḡ] ≡ Γ0[g] simply does not depend
on its second argument. In reality, because we use a ‘background-type’ gauge fixing condition,
Γ0[g, ḡ] does have a certain ḡ-dependence, again dictated by the WISS, but it disappears upon
going on-shell and cannot be seen in any experiment therefore.
11.3.4 Crossover trajectories and their mode count
Stationarity of Ck. In subsection 11.2.6 we proved that the exact Ck is stationary at fixed
points as well as in classical regimes. The explicit Ck functions obtained from both the single-
and the bi-metric truncation indeed display this behavior. In fact, looking at the two alternative
formulas for Ck in (11.68) it is obvious that Ck becomes stationary when the dimensionless
couplings are at a fixed point of the flow, and when the dimensionful ones become scale inde-
pendent; this is the case in a classical regime (‘CR’) where by definition no physical RG effects
16For them, ‘Background Independence’ is not naively ‘independence of ḡµν ’.
11.3 Asymptotically safe quantum gravity 385
occur. If λ̄ ICR and G
I
CR are the constant values of the cosmological and Newton constants there,
this regime amounts to the trivial canonical scaling λ I = k−2λ̄ I and gI = kd−2GIk CR k CR.
Generalized crossovers and limits of Ck. In subsection 11.2.6 we mentioned already the
possibility of generalized crossover transitions, not only in the standard way from one fixed
point to another, but rather from a fixed point to a classical regime or vice versa. Thereby Ck
will always approach well defined stationary values C∗ and CCR in the respective fixed point or
classical regime. (See Fig. 11.11 for a schematic sketch.)
FP
T FP1 T
CR
FP2
(a) Crossover: FP → FP. (b) Crossover: FP → CR.
Figure 11.11: Crossover trajectories in theory space: from one fixed point to another, (a), and
from a UV fixed point to a classical regime, (b).
In the case of an asymptotically safe RG trajectory, the initial point in the UV is a non-
Gaußian fixed point, by definition. For the corresponding limit C UV ≡ limk→∞ Ck the bi-metric
calculation yields C UV = C∗, with
( )
d
−2 λ
(1)−λ (0)
C∗ =− d
∗ ∗
[ ] V(M, g̊) (11.81)
(0) λ (1)
d/2
g∗ ∗
This result simplifies to
( )
sm − 2 V(M, g̊)C∗ = − (11.82)d 2 gs∗m [λ∗sm]d/2−1
in the single-metric approximation. Note that C∗ diverges at the trivial (‘Gaußian’) fixed point
at which all dimensionless couplings vanish.
If the trajectory ends in an IR fixed point the corresponding limit C IR ≡ limk→0Ck, if it
exists, is again given by the formula (11.81), C IR ≡ C∗, but for different fixed point coordinates.
If the trajectory is instead destined to enter a classical regime and to approach k = 0 by an
infinitely ‘long’, and boring, since purely canonical running on T, then the value at the end
point equals C IR = CCR where
( )
− 2 V(M, g̊)CCR =
d− (11.83)2 [ ]d/2−1GCR λ̄CR
In writing down eq. (11.83) we assumed that the same values of GCR and λ̄CR apply at all levels,
as required by split-symmetry.
Thus, for any of the above crossover types we expect a finite value of
N≡N ≡ UV− IR0,∞ C C (11.84)
386 Chapter 11. A generalized C-theorem
Physical interpretation of Ck. Beside monotonicity and stationarity, Ck has another es-
sential property in common with a C-function: The limiting value C∗ has a genuine inherent
interpretation at the fixed point itself. It is a number characteristic of the NGFP which does not
depend on the direction it is approached, and in this role it is analogous to the central charge.
The interpretation of C∗ is best known for d = 4 and the single-metric approximation where,
apart from inessential constants, it is precisely the inverse of the dimensionless combination
g∗λ∗ =Gk→∞λ̄k→∞. Its physical interpretation is that of an ‘intrinsic’ measure for the size of the
cosmological constant at the fixed point, namely the limit of the running cosmological constant
in units of the running Planck mass ( −1/2Gk ). In numerous single-metric studies the product
g∗λ∗ has been investigated, and it was always found that g∗λ∗ is a universal quantity, i.e. it
is independent of the cutoff scheme and the gauge fixing, within the accuracy permitted by
the approximation. In fact, typically the universality properties of g∗λ∗ were even much better
than those of the critical exponents. Completely analogous remarks apply to d 6= 4, and to the
quantity (11.81) in the bi-metric generalization.
We interpret this number, with all due care, as a measure for the total ‘number of field
modes’ integrated out along the entire trajectory. Clearly C UV and C IR play a role analogous to
the central charges of the 2D conformal field theories sitting at the end points of the trajectory
in the case of Zamolodchikov’s theorem.
Within the Einstein-Hilbert truncation, our numerical results for C∗ at the NGFP in d = 4
are as follows for the three calculations we compared:

7.3 single-metric
C∗ =−V(M, g̊)×4.3 bi-metric I (11.85)
8.1 bi-metric II
Obviously, in all calculations C∗ is a negative number of order unity.
Let us now focus on the case depicted in Fig. 11.11b, which can be seen as a simple
caricature of the real Universe. We consider a family of de Sitter spaces along a type IIIa
trajectory which is known to possess a classical regime with λ̄CR > 0. Assuming that this regime
represents the true final state of the evolution, we obtain
IR
C =− 3π (11.86)
GCRλ̄CR
Note that this C IR is negative, too, and that −C IR equals precisely the well known semi-classical
Bekenstein-Hawking entropy of de Sitter space [79].
Thus, combining (11.85) and (11.86) for C UV and C IR, respectively, we arrive at the follow-
ing important conclusion: In an asymptotically safe theory of quantum gravity which is built
upon a generalized crossover trajectory from criticality (the NGFP) to classicality the total num-
ber of modes integrated out, N= C UV −C IR, is finite according to the natural counting device
provided by the EAA itself.
In the special situation when GCRλ̄CR ≪ 1, like in the real world, we have |C IR| ≫ 1, while
|C UV| = O(1) according to the NGFP data of (11.85). As a consequence, the number N is
completely dominated by the IR part of the trajectory, N= C UV −C IR ≈ −C IR, and so we
obtain
≈ 3πN + ≫ 1 (11.87)
GCRλ̄CR
It is tempting to identify λ̄CR and GCR with the corresponding values measured in the real
Universe. Because of their extremely tiny product GCRλ̄CR we find a tremendous number of
modes then: N≈ 10120. Nevertheless, in sharp contradistinction to what standard perturbative
field theory would predict, this number is finite.
11.4 Discussion and outlook 387
Different types of trajectories. Concerning the finiteness of N, the situation changes if
we try to define the function k 7→Ck along trajectories of the type Ia, those heading for a negative
cosmological constant λD after leaving the NGFP regime, and of type IIa, the single trajectory
which crosses over from the NGFP to the Gaußian fixed point (GFP) at which all 4 couplings
vanish.
In the type IIa case, eq. (11.81) yields ‘C∗ = −∞’ at the IR end point of the trajectory
so that the total number of modes diverges, ‘N= +∞’. Clearly this behavior can be seen as
the limit λ̄CR → 0 of eq. (11.87) since the GFP has a vanishing cosmological constant. The
divergent value of N is the signal of a ‘topology change’ that occurs at λ̄ (1) = 0: While the
self-consistent backgrounds (of maximal symmetry, say) are spheres Sd for λ̄ (1) > 0, it is flat
space (Rd) if λ̄ (1) = 0. The Euclidean volume of the former is always finite, but that of Rd is
infinite.
While along the type IIa trajectory the divergence of Ck occurs only at the very end of the
RG evolution, i.e. in the limit k→ 0, for type Ia trajectories Ck becomes singular already at a
finite scale k = ksing > 0. All trajectories of this type cross the hyperplane λD = 0 at a nonzero
scale, ksing. However, as eq. (11.74) shows, Y( ·) and C ( ·) are singular on this plane17, so
that Ck diverges in the limit k ց ksing. The number of modes, Nk ,∞ is infinite then, whichsing
however by no means implies that all modes have been integrated out already. In fact, there is
a non-trivial RG evolution also between ksing and k = 0.
Along a type Ia trajectory, the tadpole equation has qualitatively different solutions for
k > ksing, k = ksing, and k < ksing, namely spherical, flat, and hyperbolic spaces, respectively
(Sd , Rd, and Hd, say). This topology change prevents us from smoothly continuing the mode
count across the λD = 0 plane. This is the reason why in this paper we mostly focused on type
IIIa trajectories. Some further details for the type Ia and IIa cases can be found in appendix
11.B, however.
11.4 Discussion and outlook
The proposed Ck-function
The effective average action is a variant of the standard effective action which has an IR cutoff
built-in at a sliding scale k. As such, it possesses a natural ‘mode counting’ and monotonicity
property which is strongly reminiscent of Zamolodchikov’s C-function in 2 dimensions, at least
at a heuristic level. For a broad class of systems, this property (‘pointwise monotonicity’) is
easy to demonstrate, the essential input being that in every system with a well-defined RG flow
the action Γk +∆Sk is a strictly convex functional on all scales, that is, the Hessian operator
(2)
satisfies the positivity constraint Γk +Rk > 0, k ∈ (0,∞). Motivated by this observation, and
taking advantage of the structures and tools that are naturally provided by the manifestly non-
perturbative EAA framework, we tried to find a map from the functional Γk[Φ,Φ̄] to a single
real valued function Ck that shares two main properties with the C-function in 2 dimensions,
namely monotonicity along RG trajectories and stationarity at RG fixed points.
We do not expect such a map to exist in full generality. In fact, an essential part of the
research program we are proposing consists in finding suitable restrictions on, or specializations
of the admissible trajectories (restoring split-, or other symmetries, etc.), the theory space (with
respect to field contents and symmetries), the underlying space of fields (boundary conditions,
regularity requirements, etc.), and the coarse graining methodology (choice of cutoff, treatment
of gauge modes, etc.) that will guarantee its existence.
17One might be worried about the hyperplanes gD = 0 and gB = 0 on which Y( ·) is singular too, see eq. (11.74).
However, within the truncation considered there exist no trajectories that would ever cross or touch those planes.
388 Chapter 11. A generalized C-theorem
In this chapter we motivated and analyzed a specific candidate for a map of this kind, namely
C = Γ [Φ̄s.c.,Φ̄s.c.] where Φ̄s.c.k k k k k is a running self-consistent background, a solution to the tadpole
equation implied by Γk. We showed that the function Ck is stationary at fixed points, and a non-
decreasing function of kwhen the breaking of the split-symmetry which relates fluctuation fields
and backgrounds is sufficiently weak. Thus, for a concrete system the task is to identify the
precise conditions under which the split-symmetry violation does not destroy the monotonicity
property of Ck, and to give a corresponding proof then.
It would be interesting to work out the properties of this Ck-function for further concrete
examples, but also to explore and test structurally different maps from Γk to Ck. A new kind of
map should in particular be devised if one wants to count the modes of fermionic fields with the
same weight as those of bosons. Because of the sign factors produced by the super-trace in the
flow equation, Ck = Γk[Φ̄s.c. s.c.k ,Φ̄k ] rather counts bosons and fermions with opposite signs, and
so N equals the total number of bosonic modes integrated out minus the number of fermionic
ones. (The different count for bosons and fermion is reminiscent of, but not precisely identical
to the functional integral representing the Witten index instead of the partition function. There,
fermionic states of fermion number F contribute with the weight (−1)F .) While the ‘boson
minus fermion’ counting is at variance with the standard c-theorem, the properties and potential
applications of the present Ck with fermions should be explored in more detail before dismissing
it prematurely. As we stressed already, rather then reproducing known results, our main goal
consists in finding maps Γk →Ck that are simple and ‘geometrically natural’ in the theory space
and functional RG context.
Einstein-Hilbert truncations
By means of a particularly relevant example, QEG in d > 2 dimensions, we demonstrated that
our approach is viable in principle and can indeed lead to interesting candidates for ‘C-functions’
under conditions which are not covered by the known c- and a-theorems. As exact proofs are
not within reach for the time being, the practical problem is of course the same as in all non-
perturbative functional RG studies, namely the necessity to truncate the theory space. Here, for
asymptotically safe quantum gravity we computed Ck directly from the RG trajectories obtained
with both the single- and bi-metric Einstein-Hilbert truncation, respectively.
It is one of our main results that in the bi-metric truncation the function Ck has the de-
sired properties of monotonicity and stationarity, while the single-metric truncation is too poor
an approximation to correctly reproduce the monotonicity which we expect at the exact (un-
truncated) level. We demonstrated explicitly that the monotonicity property obtains only for the
RG trajectories which are physically meaningful, that is, those which lead to a restoration of
split-symmetry once all field modes are integrated out.
We studied generalized crossover trajectories from a fixed point in the UV to a classical
regime in the IR, in which by definition the dimensionful cosmological and Newton constants
loose their k-dependence. In many ways they are analogous to a standard (fixed point → fixed
point) crossover. In quantum gravity they are of special importance as one of the main chal-
lenges consists in explaining the emergence of a classical spacetime from the quantum regime.
For the trajectories with positive cosmological constant (‘type IIIa’) the self-consistent back-
ground configurations needed are gravitational instantons. The resulting Ck depends on the in-
stanton type via the normalized volume, a quantity of topological significance. For the example
of Euclidean de Sitter space, the sphere S4, for instance, we obtained the ‘integrated C -theorem’
N= UVC − IRC ≈ 3π/GCRλ̄CR (11.88)
This result is intriguing for several reasons. First of all, N, and also the values of C UV and
C IR separately, are well defined finite numbers, in marked contrast to expectations based on the
11.4 Discussion and outlook 389
counting in perturbative field theory. The quantity N can be interpreted as a measure for the
‘number of modes’ which are integrated out while the cutoff is decreased from k ‘=’∞ to k = 0.
Hereby the notion of ‘counting’ and the precise meaning of a ‘number of field modes’ is
defined by the EAA itself, namely via the identification C = Γ [Φ̄s.c.k k k ,Φ̄
s.c.
k ]. Under special
conditions it reduces to a literal counting of the (2)Γk -eigenvalues in a given interval. Generically
we are dealing with a non-trivial generalization thereof which, strictly speaking, amounts to a
definition of ‘counting’. As such it is the most natural one from the EAA perspective, however.
The approximate equalityN≈ 3π/GCRλ̄CR is valid if GCRλ̄CR ≪ 1 in the classical regime. In
this limiting case, N≈ |C IR| equals exactly the Bekenstein-Hawking entropy of de Sitter space,
and the contribution from the UV fixed point is negligible, |C UV| ≪ |C IR|. Asymptotic Safety is
crucial for this result, making C UV finite.
So we are led to the following interpretation of the entropy of de Sitter space: it equals the
number N of metric and ghost fluctuation modes that are integrated out between the NGFP in
the UV and the classical regime in the IR. Asymptotic Safety is ‘taming’ the ultraviolet and
renders this number perfectly finite. (In the real Universe, N≈ 10120.)
Outlook
Future work along these lines will be in various different directions. Clearly one of the goals
will be to corroborate our result based on the bi-metric Einstein-Hilbert truncation on larger
theory spaces. This should also lead to a better understanding and to a physics interpretation of
the stationary values C∗ and CCR which replace the central charge of 2D conformal field theory.
We found that CCR coincides with the familiar Bekenstein-Hawking entropy18, but this is likely
to change beyond the Einstein-Hilbert truncation.
Ultimately one could hope to establish, perhaps even at some level of rigor, the existence
of aC-function in 4D asymptotically safe Quantum Einstein Gravity by using a combination of
WISS and FRGE in order to derive bounds for the ‘disturbing’ term in the equation (11.44) for
∂kCk.
Besides its obvious relevance to the global structure of the RG flow, this will also allow
us to use the monotonicity of Ck as a powerful criterion and easy to apply practical test for
assessing the reliability of truncations or other approximations.
Furthermore, it would be interesting to find further examples, based on other theory spaces
which admit a simple map from Γk to some Ck. It also remains to clarify the precise relationship
between the existing c- and a-theorems on the one side, and the present framework on the other.
We will address this question in future work.
In the existing work on the known (generalized) c-theorems in 2, 3, and 4 dimensions,
usually no reference is made to a (bare) action and fields it depends on; only the existence of a
local energy momentum tensor is assumed. In the present approach the emphasis is instead on
(effective) action functionals depending on a set of fields that is fixed from the start. However,
recalling the discussion (of the ‘reconstruction problem’) in ref. [121] it becomes clear that this
clash is much less profound than it seems: In the EAA approach to Asymptotic Safety, Γk[ · ]
should primarily be seen as a generating function (or functional) for a set of n-point functions.
Hereby the field arguments of the EAA serve a purely technical purpose, and in general the
relationship between those field arguments and the fundamental physical degrees of freedom
(DOF) whose quantization would result in a given RG trajectory is at best a highly indirect one.
The main reason is that in the case of an asymptotically safe theory Γk→∞, i.e. the fixed point
action is extremely complicated, highly non-linear, contains higher derivatives, and is nonlocal
18It is nevertheless intriguing that the Bekenstein-Hawking entropy appears here in a role analogous to the central
charge in conformal field theory. In fact the thermodynamics of 2-dimensional black holes, or correspondingly
dimensionally reduced ones, is closely related tot the Virasoro algebra and its central charge [249].
390 Chapter 11. A generalized C-theorem
probably. Furthermore, Γk→∞ is a gauge-fixed action, but it will not be of the familiar form
S+ Sgf + Sgh in general, with some invariant action S and a quadratic, second derivative ghost
action Sgh, as it is the case when one applies the Faddeev-Popov trick. This is another issue that
complicates the identification of the physical contents of the fixed point theory. In perturbation
theory, Γk→∞ which is essentially the same as the bare action S, contains only a few relevant
field monomials and only second derivative terms. As a result, there is basically a one-to-one
relation between degrees of freedom and fields. In Asymptotic Safety, rather than an ad hoc
input, the theory’s bare action, or what comes closest to it, Γk→∞, is the result of a complicated
non-perturbative evaluation of the fixed point condition. As the structure of propagating modes
is crucially affected by higher derivative and nonlocal terms, it is the fixed point condition that
decides about the na∫ture ∫of the unde∫rlying DOF’s. To identify them, a phase-space functional
integral of the form dx dπ exp(i π j ẋ j−H[π,x]) must be found which reproduces the RG
trajectory obtained from the FRGE. We can then read off canonically conjugate pairs x j, π j
and the (local) Hamiltoni∫an H[π,x] which governs their bare dynamics. To bring the original
functional integral [121] [dΦ̂]e−Γk→∞ to this form, field redefinitions and the introduction of
further, or different fields will be necessary in order to remove nonlocal and higher-derivative
terms. It is quite conceivable that there is more than one set {π j, x j} and Hamiltonian H that
reproduces a given RG trajectory. In this case we would say that the quantum theory defined by
the latter has ‘dual’ descriptions employing different (bare) actions and fields. As yet, not much
work has been devoted to this ‘reconstruction problem’, see however ref. [121] for a first step.
Relation of Ck to the N-bound
To close with, we mention another intriguing aspect of the integrated C -theorem (11.88) which
deserves being investigated further, namely its connection to the hypothesis of the ‘N-bound’
which is due to Banks [250] and, in a stronger form, to Bousso [251]. In Bousso’s formulation,
the claim is that in any universe with a positive cosmological constant, containing arbitrary
matter that even may dominate at all times, the observable entropy Sobs is bounded by Sobs ≤
3π/Gλ̄ ≡ N.
Here Sobs includes both matter and horizon entropy, but excludes entropy that cannot be
observed in a causal experiment. As for the notion of an ‘observable entropy’, it is identified
[251] with the entropy contained in the causal diamond of an observer, i.e. the spacetime region
which can be both influenced and seen by the observer. It is bounded by the past and future
light cones based at the endpoints of the observer’s world line.
Remarkably, while the number N equals the Bekenstein-Hawking entropy of empty de Sitter
space, the bound is believed to apply in presence of arbitrary matter, and for arbitrary spacetimes
with λ̄ > 0, which not even asymptotically need to be de Sitter.19
Given the methods developed in this chapter the intriguing possibility arises to check whether
the N-bound holds in asymptotically safe field theories and to tentatively identify N with N≡
C UV −C IR. In principle we have all tools available for a fully non-perturbative test that treats
gravity at a level well beyond the semi-classical approximation. We would have to add matter
fields to the truncation ansatz [130, 152, 153] and include for all types of fields the corre-
sponding (Gibbons-Hawking-York, etc.) surface terms that are needed on spacetimes with a
non-empty boundary [166, 252, 253].
Originally the N-bound grew out of string theory based arguments which hinted at the pos-
sibility of a ‘λ̄ -N-connection’ [250, 251]. It would be such that all universes with a positive
cosmological constant are described by a fundamental quantum theory which has only a finite
number of degrees of freedom, and that this number is determined by λ̄ .20
19In [251] the original requirement [250] of spacetimes that are asymptotically de Sitter has been dropped.
20For a similar discussion in Loop Quantum Gravity see ref. [254].
11.A The special status of Faddeev-Popov ghosts 391
Is there a corresponding ‘λ̄ -N-connection’ in asymptotically safe field theory? For pure
gravity we can answer this question in the affirmative already now: The fundamental quantum
field theory is defined by the Asymptotic Safety construction with an RG trajectory of the type
IIIa, we get the required positive cosmological constant in the IR, λ̄CR, which in turn fixes the
number of degrees of freedom, here to be interpreted as C UV −C IR, by N= 3π/GCRλ̄CR < ∞.
Recalling our discussion of the Ia and IIa trajectories in section 11.3.3 and appendix 11.B we
can now easily understand what is special about a strictly positive λ̄ , and why the connection
fails for a negative or vanishing classical cosmological constant: in the latter cases, we found
that N is not finite.
11.A The special status of Faddeev-Popov ghosts
In subsection 11.2.4 we argued that Faddeev-Popov ghosts, even though they contribute with a
negative sign to the supertrace on the RHS of the flow equation, do not destroy the pointwise
monotonicity of the EAA when they are the only fields present with odd Grassmann parity. The
reason was that the ghosts are merely a way of rep∫resenting the Faddeev-Popov determinant,
det(M), the functional integral actually being Z = [dg] det(M)e−Sgfe−S, wherein the gauge
fixing term and the determinant effectively restrict the integration over all metrics to an integral
over the gauge orbit space of metrics modulo diffeomorphisms. If we had parametrized the latter
directly we were dealing with a purely Grassmann-even integral [62], which when modified
by an IR cutoff, obviously leads to a pointwise monotone EAA as the gauge orbit space is
independent of k. In this appendix, we briefly indicate how this general argument can be made
concrete.
We start out from the functional integral that has been gauge-fixed à la Faddeev-Popov, but
without IR cutoff yet. Then, after the usual background split, we perform a partial21 transverse-
traceless (TT)-decomposition of the fluctuation field,
( )
h = hT
2 1
µν µν + D̄µVν + D̄νVµ − ḡµν D̄αVα + ḡµνh (11.89)d d
with D̄µhT = 0, ḡµνµν h
T
µν = 0, h ≡ ḡµνhµν . Henceforth we interpret [dgµν ] as [dhTµν ][dh][dVµ ].
Furthermore, we write the Faddeev-Popov determinant as det (M[g, ḡ]) ≡ det (M)e−S1 with
M≡ M[ḡ, ḡ]. Diagrammatically speaking the action S1 contains the ghost-antighost-graviton
vertices and M−1 is the ‘free’ ghost propagator. Thus
∫
Z[ḡ] = [dhTµν ] [dh] [dVµ ] det(M)e
−Sgfe−S̃ (11.90)
∫ ∫
with S̃≡ S+S1 and the representation det(M) = [dξ ][dξ̄ ] exp ξ̄Mξ .
Next, consider the family of gauge fixing functions
Fµ [ḡ](h) = D̄
νhµν −ϖD̄ h( µ ) ( )
= D̄2
2 1
V + 1− D̄ D̄ν νµ µ Vν + R̄
d µ
Vν + −ϖ D̄µh (11.91)
d
As for the parameter ϖ , we choose the ‘un-harmonic’ gauge [124, 255, 256], setting ϖ = 1/d.
(The harmonic gauge has ϖ =∫1/2 i√nstead.) This gauge leads to a remarkable conspiracy of the
gauge fixing action, S = 1gf 2α d
dx ḡ ḡµνFµ [ḡ](h)Fν [ḡ](h), and the ghost action at hµν = 0,
21It is ‘partial’ in that the vector field Vµ is not decomposed further here as this is usually done, setting Vµ =
VT µν Tµ + D̄µ σ with ḡ D̄µVν = 0.
392 Chapter 11. A generalized C-theorem
∫ √
Sgf = ddx ḡ ξ̄ Mµ ξ νµ ν . One finds that Sgf is a bilinear form in Vµ (and only Vµ !) whose
kernel is precisely the square of the inverse ghost propagator22 :
∫
1 √
Sgf = d
dx ḡVµ M
µ α
αM νV
ν (11.92)
2α
In this gauge, Mµ ν ≡ M[ḡ, ḡ]µ ν = D̄2δ µ ν − (1− 2/d)D̄µD̄ µν − R̄ ν . As a consequence, the
integral over the ghosts, producing the determinant det(M), when combined with e−Sgf , yields
a Dirac δ -functional in the limit α → 0:
1 ∫ 2
det(M)e− 2α VM V → δ [V ] (11.93)
∫
It satisfies [d∫V ]δ [V ]→∫1. That δ [V ] is indeed correctly normalized, up to a constant, follows− 1 VM2from det(M) [dV ]e 2 V = det(M)det−1/2α (M2) = det(M)det−1(M) = 1.
As a result, the limit α → 0 simplifies the integral for Z quite considerably: after integrating
over Vµ and using (11.93) we are left with
∫ ( )
Z = [dhTµν ]Dh exp −S̃[hT +d−1µν ḡµνh; ḡµν ] (11.94)
This functional integral is manifestly over fields of even Grassmann parity only. So when we go
through the usual procedure and define the associated EAA, the derivation of ∂kΓk ≥ 0 in the
main part of this paper applies to it, provided the above exact compensation of the ghost and Vµ
contributions persists in presence of an IR cutoff. While this is not the case for a generic cutoff,
it has been shown [126] that if the cutoff operators Rk of the ghost and metric fluctuations,
respectively, are appropriately related, which always can be achieved, the compensation does
indeed persist. For further details the reader is referred to [126].
Thus we have shown that (at the very least) when the ghosts are the only Grassmann-odd
fields it is in principle always possible to set up the gauge fixing and ghost sector of the EAA
and its FRGE in such a way that ∂kΓk ≥ 0 holds true pointwise.
The various sets of beta-functions studied in this chapter were not obtained using this very
special set-up for the gauge-fixing and ghost sector. However, as the truncations considered
here anyhow neglect all RG effects in this sector we have the freedom to use any gauge at this
level of accuracy since this should not lead to an extra error. A similar remark applies to the
choice of the cutoff operators.
11.B The trajectory types Ia and IIa
In this appendix we evaluate gravC = Γ [0; ḡs.c.k k k ] along type Ia and IIa trajectories in d = 4 for the
bi-metric Einstein-Hilbert truncation. The analysis parallels to some extent the one in subsec-
tion 11.3.3 for the type IIIa case. Since in the Ia and IIa cases the cosmological constant λ (1)
turns zero or even negative at some scale the corresponding self-consistent background under-
goes a topological change, from S4 to flat Euclidean space R4 and to H4, respectively. While for
the separatrix, the type IIa trajectory, the change from S4 to R4 happens in the limit k→ 0 only,
the type Ia solutions have a negative λ (1) at finite scales k< ksing already. At the transition point
k = ksing equation (11.75) is no longer valid and Yk diverges: limkցk Yk = ∞. For k < ksing sing
a different formula for Ck, employing a new background configuration, could be derived. We
shall not do this here and rather restrict our attention to the subspace of Twith λ (1) > 0.
The Figs. 11.12 and 11.13 depict the explicit k-dependence of 1/Yk for a representative
type Ia trajectory and the unique IIa trajectory, respectively, both for the case of restored split-
symmetry in the IR. The plots are based on the RG equations of [II]; the results with those
22This ‘magic’ property has been discovered by F. Saueressig et al. [126, 257].
11.B The trajectory types Ia and IIa 393
from [I] are quite similar. For the type Ia trajectory in Fig. 11.12 the singularity of Yk occurs at
about ksing ≈ 0.5mPlanck , and Yk is seen to be perfectly monotone above this scale.
Along this trajectory, we integrated out Nk ,∞ = ∞ modes already before the end of thesing
trajectory. But as there is a nontrivial RG evolution also below ksing, there are still further
modes left to be integrated out. Using a hyperbolic background we could count how many there
are in some interval [k1,k2] with k1 < k2 < ksing. But clearly there is no meaningful way of
associating a finite number N0,∞ to the complete trajectory as this was possible in the IIIa case.
For the symmetry-restoring bi-metric separatrix, the plot of 1/Yk is very similar to the
case of the IIIa-trajectories discussed in the main part of the paper, see Fig. 11.13. The only
difference is that 1/Yk vanishes exactly at k = 0, while 1/Yk was always nonzero for the type
IIIa solutions. So the separatrix is the marginal case where N= ∞ is reached precisely at the
IR-end point of the trajectory.
0.4
0.3 0.2
0.5
0.0
0.2
-0.2
-0.4
0.1 0.0
0.5 1.0 1.5 2.0
k/m
0.0 Planck
-0.5
-0.1
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
k/mPlanck k/mPlanck
Figure 11.12: The function 1/Yk and its derivative are shown for a typical bi-metric type Ia
trajectory which restores split-symmetry in the IR. Due to the sign flip of λ (1)k near ksing ≈
0.5mPlanck, the function 1/Yk has a zero there and Ck diverges. As long as (11.75) is still valid,
to k > ksing, the function Yk is seen to be monotone.
0.3
0.00
0.2
-0.02 0.00
0.1
-0.05
-0.04
0.10 0.0-
0.06 -0.15- -0.1
-0.20
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.2
-0.08
k/mPlanck
-0.3
-0.10
-0.4
-0.12
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
k/mPlanck k/mPlanck
Figure 11.13: The function 1/Yk for the bi-metric type Ia trajectory (separatrix) and its deriva-
tive. Its properties are similar to the type IIIa results obtained in subsection 11.3.3. In particular
Yk is seen to be monotone. The asymptotic topology change of the self-consistent background
in the limit k→ 0 is not directly visible in these plots. It can be checked though that Yk diverges
for k→ 0 (and that it does not in the IIIa case).
In a series of snapshots, Figs. 11.14 and 11.15 show the evolution of the RG trajectories
in the background sector, from the IR to the UV, on the basis of typical Ia and IIa dynamical
trajectories, respectively. The shaded (white) regions partition the gB-λ B-plane into subsets of
positive (negative) slope k∂kYk. A crossing of the corresponding boundary indicates a violation
1/Yk 1/Yk
k∂k1/Yk k∂k1/Yk
k∂k1/Yk k∂k1/Yk
394 Chapter 11. A generalized C-theorem
10 10
8 8
6 6
gB gB
4 4
2 2
P1
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
λB λB
10 10
8 8
6 6
gB gB
4 4
2 2
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
λB λB
Figure 11.14: The gB-λ B-plane is shown in a series of subsequent ‘snapshots’ at different RG-
times which increase from the upper left to the lower right diagram. They are given by the
maximum k-value of the incomplete dynamical type Ia trajectory k 7→ (gDk , λDk ) shown in the
respective inset. The shaded regions corresponds to TB+(k) at that particular time, so that every
trajectory in the shaded (white) region will give rise to a positive (negative) value of k∂kCk at the
instant of time k. Furthermore, two different B-trajectories that are evolved upward (towards
increasing scales k) are shown at the corresponding moments. The one passing the point P1
(P2) is split-symmetry violating (restoring). The symmetry restoring trajectory starts its upward
evolution close to P2, the position of the running UV attractor, see chapter 10. As long as
k > ksing, which is assumed here to avoid a topology change, this trajectory never leaves the
shaded area, and thus its Ck-function is strictly monotone. This is different for the trajectory
through P1: Attracted by the running UV-attractor, it is pulled into the shaded regime, thus
unavoidably crossing the boundary of TB+(k), which causes a sign flip of ∂kCk, rendering Ck
non-monotone.
11.B The trajectory types Ia and IIa 395
10 10
8 8
6 6
gB gB
4 4
2 2
P2 P1
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
λB λB
10 10
8 8
6 6
gB gB
4 4
2 2
0 0
-0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4
λB λB
Figure 11.15: A series of snapshots as in Fig. 11.14, but for the type IIa trajectory. The results
are similar to those in subsection 11.3.3 for the type IIIa trajectories.
396 Chapter 11. A generalized C-theorem
of the monotonicity of Ck. While in the case of the separatrix the requirement of split-symmetry
restoration in the IR is sufficient to assure this condition, a more careful study is needed for
type Ia trajectories. There λ (1) turns negative at finite scales k, and thus makes eq. (11.75)
inapplicable. In any case, trajectories that break split-symmetry at k= 0 are more vulnerable to
monotonicity violation than those restoring it.
12.1 Introduction
12.2 Anomalous dimension in single- and bi-
metric truncations
Single-metric truncation
The (TT-based) bi-metric calculation
The (‘Ω-deformed’) bi-metric calculation
Significance of the cosmological constant
From anti-screening to screening and back
12.3 Interpretation and Applications
The dark matter interpretation
Primordial density perturbations from the
NGFP regime
12.4 Summary
12. PROPAGATING MODES IN QUANTUM GRAVITY
C This chapter follows closely ref. [192].
Within the Asymptotic Safety scenario, we discuss whether Quantum Einstein Gravity (QEG)
can give rise to a semi-classical regime of propagating physical gravitons (gravitational waves)
governed by an effective theory which complies with the standard rules of local quantum field
theory. According to earlier investigations based on single-metric truncations there is a ten-
sion between this requirement and the condition of Asymptotic Safety since the former (latter)
requires a positive (negative) anomalous dimension of Newton’s constant. In this chapter we
show that the problem disappears using the bi-metric renormalization group flows that became
available recently: They admit an asymptotically safe UV limit and, at the same time, a genuine
semi-classical regime with a positive anomalous dimension. This brings the gravitons of QEG
on a par with arbitrary (standard model, etc.) particles which exist as asymptotic states. We
also argue that metric perturbations on almost Planckian scales might not be propagating, and
we propose an interpretation as a form of ‘dark matter’.
12.1 Introduction
One of the indispensable requirements an acceptable fundamental quantum gravity theory must
satisfy is the emergence of a classical regime where in particular small perturbations, i.e. grav-
itational waves, can propagate on an almost flat background spacetime. This regime should
be well described by classical General Relativity or, if one pushes its boundary towards the
quantum domain a bit further, by the effective quantum field theory approach pioneered by
Donoghue [258, 259].
In the present context we consider the scenario where the ultraviolet (UV) completion of
quantized gravity is described by an asymptotically safe quantum field theory [145]. In a for-
mulation based upon the gravitational average action [122], this quantum field theory is defined
by a specific Renormalization Group (RG) trajectory k 7→ Γk which lies entirely within the UV-
critical hypersurface of a non-Gaußian fixed point (NGFP). Here Γk ≡ Γk[hµν ; ḡαβ ] denotes
398 Chapter 12. Propagating modes in Quantum Gravity
the Effective Average Action, a ‘running’ action functional which, besides the scale k, depends
on the (expectation value of the) metric fluctuations, hµν , and the metric of the background
spacetime on which they are quantized, ḡαβ , see chapter 4 and part II of this thesis.
To recover classical General Relativity in this setting it would be most natural if the asymp-
totically safe RG trajectory of the fundamental theory, emanating from the NGFP in the UV
(k → ∞), contains a segment in the low energy domain (k → 0) where the full fledged descrip-
tion in terms of the effective average action, valid for all scales and all backgrounds, smoothly
goes over into the effective field theory of spin-2 quanta propagating on a rigid background
Minkowski spacetime. The simplest picture would then be that the approximating low energy
theory which is implied by the fundamental asymptotically safe one is ‘standard’ in the sense
that it complies with the usual axiomatics of local quantum field theory on Minkowski space
which underlies all of particle physics, for instance.
However, almost all existing RG studies of the Asymptotic Safety scenario, using functional
RG methods, indicate that there is a severe tension, if not a clash, between their predictions
and the picture of a conventional Minkowski space theory describing propagating gravitons or
gravitational waves at low energies [99, 100, 121, 123, 126, 130–135, 138, 139, 141, 142, 146–
171].
In the following we try to describe this tension as precisely as possible. It is necessary to
distinguish the real question of (non-)existing propagating gravitational waves in the classical
regime from certain objections against Asymptotic Safety in general that were raised occasion-
ally but were based on misconceptions and are unfounded therefore. One of these misconcep-
tions is the believe that the anomalous dimensions of quantum fields must be positive, always.
In fact, for asymptotically safe Quantum Einstein Gravity (QEG) it is crucial that the
anomalous dimension of the metric fluctuations, ηN, is negative, at least in the vicinity of the
NGFP. There, by the very construction of the theory’s UV completion, it assumes the value
η∗N =−(d−2), in d spacetime dimensions.1 And indeed, the RG equations obtained within the
special class of non-perturbative approximations that have been considered in the past almost
exclusively, the so called ‘single-metric’ truncations of theory space, had always given rise to
a negative anomalous dimension [168–171]. Moreover, ηN < 0 was found not only near the
NGFP but even everywhere on the truncated theory space considered.
In these ∫trunca√tions the ansatz for the Effective Average Action (EAA) always included a
term ∝ G−1k d
dx gR(g) from which ηN was obtained as the scale derivative of the running
Newton constant: ηN = k∂k lnGk. Since in this term the metric gµν is to be interpreted as
gµν = ḡµν + hµν , the running Newton constant fixes the normalization of the fluctuation field,
hµν . While extremely tiny in magnitude, ηN turned out negative with this entire class of trun-
cations even in the ‘classical regime’ displayed by the special (Einstein-Hilbert truncated, Type
IIIa) trajectory which matches the observed values of Newton’s constant and the cosmological
constant [191, 209, 212].
To see why the sign of the anomalous dimension is important let us consider an arbitrary
field in d spacetime dimensions with an inverse propagator ∝ Z(k2)p2 which depends on an
RG scale k. In abse[nce of oth]er relevant scales we may identify k
2 = p2, obtaining the dressed
propagator G̃(p) ∝ Z(p2
−1
)p2 . For example in a regime where Z(k2) ∝ k−η with a constant
exponent η we have, in momentum space, G̃(p) ∝ 1/(p2)1−η/2. If this propagator pertains to
an Euclidean field theory on flat space it is natural to perform a Fourier transformation with
respect to all d coordinates, whence
1
GE(x− y) ∝ | − | − (12.1)x y d+η 2
1We assume d > 2 throughout.
12.1 Introduction 399
For field theories on Minkowski space the static limit of the propagator is particularly inter-
esting; setting the time component of pµ to zero and taking the (d− 1) dimensional Fourier
transform of G̃(p) we get, with x≡ (x0,~x) and y≡ (x0,~y) at equal times,
1
GM(0,~x−~y) ∝ | − (12.2)~x ~y|d+η−3
Eqs. (12.1) and (12.2) confirm that the exponent η which comes into play via the scale depen-
dent field normalization Z(k2) ∝ k−η indeed deserves the name of an ‘anomalous dimension’:
the renormalization effects changed the effective dimensionality of spacetime, which manifests
itself by the fall-off behavior of the 2-point function, from d to d+η . In d = 3+1, for instance,
we obtain the modified Coulomb potential
− 1GM(0,~x ~y) ∝ | (12.3)~x−~y|1+η
The point to be noted here is that, as compared to the classical Coulomb Green’s function, a
positive value of the anomalous dimensions renders the propagator more short ranged, while it
becomes more long ranged when η is negative.
Thus we conclude that the anomalous dimension ηN < 0 found by the single-metric trunca-
tions of QEG corresponds to a graviton propagator on flat space which falls off for increasing
distance more slowly than 1/|~x|. Also notice that, strictly speaking, eq. (12.1) holds only when
d+η − 2 6= 0. If d+η − 2 = 0 one has an increasing behavior even, GE(x− y) ∝ ln(x− y)2.
This is precisely the case relevant at the NGFP of quantum gravity where η∗N = −(d− 2). In
the fixed point regime the momentum dependence is G̃(p) ∝ 1/pd . Note that at the NGFP the
function (12.2) becomes linear: GM(0,~x−~y) ∝ |~x−~y|.
The fall-off properties of the propagator have occasionally been adduced as a difficulty
for the Asymptotic Safety idea. We emphasize that in reality there is no such difficulty. It is
nevertheless instructive to go through the argument, and to see where it fails. For this purpose,
consider an arbitrary bosonic quantum field Φ on 4D Minkowski space. Under very weak
conditions one can derive a Källén-Lehmann spectral representation [260, 261] for its dressed
propagator:
∫ ∞
∆′F(x− y) = dµ2 ρ(µ2) ∆F(x− y;µ2) (12.4)
0
Here
∫ 4
− 2 − d p e
ıp(x−y)
∆F(x y;µ ) = (12.5)
(2π)4 p2−µ2+ ıε
is the free Feynman propagator (with possible tensorial structures suppressed), and the spectral
weight function
ρ(q2) = (2π)3∑δ 4(pα −q) |〈0|Φ(0)|α〉|2 (12.6)
α
contains a sum over all states |α〉 with momenta pα where p2α ≥ 0, pα 0 ≥ 0 (the one-particle
contribution included). It is assumed that the states are elements of a vector space which is
equipped with a positive-definite inner product. Therefore it follows directly from its definition
(12.6) that ρ(µ2) is a non-negative function. The Källén-Lehmann representation itself follows
from only a few, very basic additional assumptions: (a) completeness of the momentum eigen-
states, in particular completeness of the asymptotic states, (b) the spectral condition p2 ≥ 0,
p0 ≥ 0 for the states, (c) Poincaré covariance, in particular invariance of the vacuum state.
400 Chapter 12. Propagating modes in Quantum Gravity
If a dressed propagator ∆′F possess a Källén-Lehmann representation it follows that its
Fourier transform behaves as 1/p2 for p2 → ∞ limit, exactly as for the free one, ∆F. Conversely,
for |~x−~y| → ∞ at equal times, ∆′F cannot decay more slowly than( ∝ 1/|~x−~y|). Indeed, the free
massive Feynman propagator behaves as ∆F(0,~x−~y;µ2) ∝ exp − µ |~x−~y| in this limit, so
that the µ2-integral in (12.4) amounts to a superposition of decaying exponentials with non-
negative weight, since ρ(µ2)≥ 0. The best that can happen is that ρ(µ2) has support at µ2 = 0,
in which case the free propagator behaves Coulomb-like ∝ 1/|~x−~y|, and, as a consequence, the
dressed one as well, ∆′F(0,~x−~y) ∝ 1/|~x−~y|. Obviously this is the behavior corresponding to an
anomalous dimension η = 0. If a Källén-Lehmann representation exists, ∆′F may fall off faster,
so η > 0 is possible, but not more slowly.
As a consequence, under the conditions implying the existence of a Källén-Lehmann rep-
resentation negative anomalous dimensions η < 0 cannot occur. This entails that, conversely,
whenever an anomalous dimension is found to be negative one or several of those conditions
must be violated.
In the case of asymptotically safe gravity, described by the EAA, we can easily identify
at least one of the above necessary conditions which is not satisfied: The functional integral
related to Γk[h µµν ,ξµ , ξ̄ ; ḡµν ] is a modified version (containing an IR regulator term) of the
standard Faddeev-Popov gauge-fixed and BRST invariant functional integral which quantizes
hµν in some background gauge, usually the de Donder-Weyl gauge [122]. However, the opera-
torial reformulation of this quantization scheme is well-known to involve a state space with an
indefinite metric [262]. Therefore, ρ(q2) has no reason to be positive, and the short distance
behavior of the dressed hµν propagator may well be different from 1/p2 in momentum space.
In fact, Asymptotic Safety makes essential use of this possibility: For p2 → ∞, and in d = 4, the
propagator must be proportional to 1/p4 as a consequence of the UV fixed point.
A well-known example with similar properties is the Lorentz-covariant quantization of
Yang-Mills theories on flat space, Quantum chromodynamics (QCD), for instance. Here the
anomalous dimension related to the gluon, η ≡ ηF, is negative too, and its negative sign is pre-
cisely the one responsible for asymptotic freedom. Analogous to the computation done for the
Newton constant, one can obtain ηF in the EAA∫approach by using a (covariant) background
type gauge and reading off ηF from the term
1 F22 µν in Γk as the logarithmic scale derivative4g
k
of the gauge coupling gk, see ref. [110–114] for details. A long ranged gluon propagator due to
η < 0 could be indicative of gluon confinement, at least in certain gauges. Again the pertinent
state space is not positive-definite, and so even propagators increasing with distance are not
excluded by general principles.
It is actually quite intriguing that a linear confinement potential ∝ |~x−~y| for static color
charges, corresponding to a 1/p4 behavior in the IR, is precisely what in gravity is realized in
the UV. While the fixed point regime of QEG is realized at small rather than large distances,
the graviton carries a large negative anomalous dimension there.
Up to now we exploited only a rather technical, non-dynamical property of the quantization
scheme used, namely the indefinite metric on state space, in order to reject the implications
of a Källén-Lehmann representation with a positive spectral density. This was sufficient to
demonstrate that within the setting of the (background gauge invariant) gravitational EAA of
ref. [122] the exact anomalous dimension derived from the running Newton constant is not
bound to be positive for any general reason. Therefore there is nothing obviously wrong with
the negative ηN’s that were found in concrete QEG calculations on truncated theory spaces, and
a similar statement is true for Yang-Mills theory.
However, the previous argument has not yet much to do with the dynamical properties of
the respective theory. Taking QCD as an example again, we can solve the BRST cohomology
problem which underlies its perturbative quantization, and in this way we learn how to reduce
12.1 Introduction 401
the indefinite-metric state space to a subspace of ‘physical’ states which carries a positive def-
inite inner product. One finds that, in this sense, transverse gluons and quarks are ‘physical’,
while longitudinal and temporal gluons, as well as Faddeev-Popov ghosts are ‘unphysical’.
Now, it is a highly non-trivial question whether the dynamics of the ‘physical’ states is such
that the above requirements (a), (b), (c) are satisfied so that a Källén-Lehmann representation of
the transverse gluon propagator could exist. The general believe is that the answer is negative
since gluons, being confined, do not form a complete system of asymptotic states. So here we
have a deep dynamical rather than merely kinematical reason to reject the implications of the
Källén-Lehmann representation concerning the propagator’s fall-off behavior. This opens the
door for a gluon propagator which might even increase with distance, like, for instance, the ‘IR
enhanced’ propagator proportional to 1/p4 for p2 → 0.2
Even though the gluon propagator is gauge dependent there is a direct connection to the
gauge invariant confinement criterion of an area law for Wilson loops. It has been shown [267]
that if the gluon propagator possesses the singular 1/p4 behavior for p2 → 0 in just one gauge
then QCD is confining in the Wilson loop sense; in any other gauge it need not show this
singular behavior. In covariantly gauge fixed QCD, it is of interest to know the properties of
the gluon, ghost, and quark propagators also because they contain information about the non-
perturbative dynamical mechanism by means of which the theory cuts down the indefinite state
space to a positive-definite subspace, containing ‘physical’ states only.
In gravity, the analogous question concerns the status of the transverse gravitons, that is, the
hµν modes which are not ‘pure gauge’ but rather ‘physical’ in the BRST sense. Let us envisage
a universe which, on all its vastly different scales, from the Planck regime to cosmological
distances, is governed by QEG, and let us ask whether a transverse graviton which it may
contain is more similar to a photon (unconfined, freely propagating, exists as an asymptotic
state3) or to a gluon (confined, no asymptotic state, no Källén-Lehmann representation with
positive ρ)?
In its full generality this is a very hard question. The attempt at an answer on the basis of
existing single-metric computations would be that the graviton is more similar to the gluon than
to the photon, a claim that might appear surprising, in particular if one thinks of astrophysical
gravitational waves.
In quantum gravity, where Background Independence adds to the standard principles of
quantum field theory, a particular convenient way to ensure a covariant formalism in presence
of a gauge fixing and a cutoff term is the background field method. Thereby one introduces
a generic background metric ḡµν at intermediate stages of the quantization in addition to the
usual dynamical metric gµν . Consequently, the most general ansatz for Γk (possibly including
matter fields) has to be of ‘bi-metric’ type and thus contains all possible field monomials that
can be constructed from gµν , ḡµν , and the matter fields which respect all relevant symmetries
(diffeomorphisms, (gauge-) symmetries in the matter sector, etc.)[136, 137]. On the one hand,
this affects for instance the mathematical description of the UV-completion of the theory, where
a suitable UV fixed point defines the relation between the various invariants of background and
physical metric. However, on the other hand, the background is a purely technical artifice that
does not affect the observables. Those are obtained from the physical sector of Γk=0 ≡ Γ that
must respect Background Independence, i.e. which is independent of the auxiliary background
field in the IR limit. Hence, ḡµν – even though crucial in the construction of an exact RG flow
2Please note that by no means we are saying here that this behavior must occur, rather only that it can occur
without violating any of the general principles discussed. In fact, detailed analyses of the IR properties of QCD, em-
ploying various independent non-perturbative techniques, indicate that in reality the picture is far more complex. For
recent results concerning the gluon propagator, the properties of the spectral densities and the positivity properties
of Yang-Mills theory we must refer to the literature [263–266].
3to the extent this can make sense as an approximate notion in curved spacetime
402 Chapter 12. Propagating modes in Quantum Gravity
at all intermediate scales k – becomes redundant when invoking fully intact split-symmetry
in the IR (k = 0). A technical simplification in the RG computations consists of neglecting
the background invariants in the ansatz for the non-gauge part of the EAA, giving rise to the
aforementioned single-metric approximation. Since the gauge fixing and the cutoff action still
rely explicitly on ḡµν , the RHS of the FRGE generates invariants depending on the background
field and thus remain unresolved. Hence, the reliability of the single-metric approximation has
to be tested by comparison of the class of solutions, in particular the UV completion with
its more general bi-metric counterparts. Thus, from a physical perspective, the ultimate theory
(physical sector of Γk=0, on-shell S-matrix elements, for example) is not ‘bi-metric’ in the sense
that two independent metric tensors would play a role individually. There is only one physical
metric; the background metric at intermediate stages of the quantization is only a technical
artifice. Fully intact split-symmetry in the physical sector, for vanishing IR cutoff, is precisely
the statement that ḡµν has become redundant and no observable depends on it.
The purpose of this chapter is to go beyond the single-metric approximation and investigate
the crucial sign of the anomalous dimension ηN using differently truncated functional RG flows
of asymptotically safe metric gravity, i.e. QEG. In particular we explore the corresponding
predictions of two ‘bi-metric truncations’ of theory space. They have been studied in ref. [172],
henceforth denoted [I], and in ref. [173] or part II of this thesis (with boundary terms omitted)
which in the sequel is referred to as [II], respectively. Both employ a similar truncation ansatz
for Γk[g, ḡ], namely two separate Einstein-Hilbert terms for the dynamical and the background
metric gµν and ḡµν , respectively. The calculations in [I] and [II] differ, however, with respect
to the gauge fixing-conditions and -parameters they use, as well as the field parameterization
they employ. In [I] the ‘geometric’ or ‘anharmonic’ gauge fixing [62, 124, 172, 255, 256] is
used, with gauge fixing parameter α = 0, while [II] relies on the harmonic gauge and α = 1.
Furthermore, in [I], the functional flow equation and in particular its mode suppression operator
Rk was formulated in terms of a transverse-traceless (TT) decomposed field basis for hµν , while
in part II we have seen that no such decomposition is necessary to obtain the results of [II]. It is
to be expected that these differences of the coarse graining schemes employed should have only
a minor impact on the RG flow and leave its essential qualitative features unchanged.
This chapter is organized as follows. In section 12.2 we present a detailed analysis of the
two bi-metric calculations [I], [II] and a comparison of their respective RG flows with the well-
known one based on the single-metric Einstein-Hilbert truncation. We demonstrate that the
former imply a positive anomalous dimension, hence a ‘photon-like’ behavior of gravitons in
the semi-classical regime. There is no obvious physical reason or qualitative argument that
would explain the sign flip of η in going from the single- to the bi-metric truncation. Therefore,
detailed quantitatively precise calculations are particularly important here.
Section 12.3 is devoted to metric fluctuations outside this regime. Their precise propagation
properties near, but close to the Planck scale remain unknown for the time being. We argue that,
in this range of covariant momenta, they behave as a form of gravitating, but non-propagating
‘dark matter’. Possible implications for the early Universe are also discussed. Finally section
12.4 contains a brief summary.
12.2 Anomalous dimension in single- and bi-metric truncations
As a short reminder, our approach to the quantization of gravity assumes that the fundamen-
tal degrees of freedom mediating the gravitational interaction are carried by the spacetime
metric. It heavily relies upon the Effective Average Action (EAA), a k-dependent functional
Γ µk[gµν , ḡµν ,ξ , ξ̄µ ] which, in the case of QEG, depends on the dynamical metric gµν , the
background metric ḡµν , and the diffeomorphism ghost ξ µ and anti-ghost ξ̄µ , respectively. We
employ the background field method to deal with the key requirement of Background Indepen-
12.2 Anomalous dimension in single- and bi-metric truncations 403
dence, and are thus led to the task of quantizing the metric fluctuations hµν ≡ gµν − ḡµν in all
fixed but arbitrary backgrounds simultaneously.4
For all truncations of theory space studied in this chapter the corresponding ansatz for the
EAA has the same general structure, namely
Γk[g,
grav gf gh
ḡ,ξ , ξ̄ ] = Γk [g, ḡ]+Γk [g, ḡ]+Γk [g, ḡ,ξ , ξ̄ ] (12.7)
Concretely we consider the Einstein-Hilbert truncation, both in its familiar single-metric form
[122, 191] and a more advanced bi-metric variant thereof [172, 173]. In the single-metric
truncation the gravitational (‘grav’) part of the ansatz has the form
∫ ( )
Γgrav
1 √
[g, ḡ] =− ddx g R(g)−2λ̄ smk (12.8)16πGsmk
It contains two running coupling constants, Newton’s constant Gsmk and the cosmological con-
stant λ̄ smk . (The superscript ’sm’ stands for single-metric.)
For the most general bi-metric refinement of this truncation one should in principle include
the infinitely many invariants which one can construct from the metrics gµν and ḡµν that reduce
to (12.8) when both metrics are identified, g = ḡ. Here, we follow [I] and [II] retaining for
technical simplicity only four such invariants, namely two independent Einstein-Hilbert actions
for g and ḡ, respectively:
∫ ( )
Γgravk [g, ḡ] =−
1 √
ddx g R(g)−2λ̄D
16πGDk ∫ √ ( )
− 1 ddx ḡ R(ḡ)−2λ̄ B (12.9)
16πGBk
This family of actions comprises 4 running coupling constants, the dynamical (‘D’) Newton and
cosmological constants as well as their background (‘B’) counterparts.
An equivalent and sometimes more useful description of the action (12.9) is obtained by
expanding Γgravk [g, ḡ] in powers of the fluctuation field hµν = gµν − ḡµν . We have, up to terms
of second order in hµν :
∫
1 √ ( )
Γgrav dk [h; ḡ] =− d x ḡ R(ḡ)−2λ̄
(0)
16π (0) kGk ∫ √ [ ]
− 1 ddx ḡ − Ḡµν − λ̄ (1)ḡµν hµν
16π (1) kG
∫ k
− 1 √d grav(2)d x ḡ hµν Γk [ḡ, ḡ] hρσ +O(h3) (12.10)2
This expansion in powers of hµν is referred to as the ‘level representation’ of the EAA, and a
term is said to belong to level-(p) if it contains p factors of hµν , for p = 0,1,2, · · · . The level-
(p) couplings (p)Gk , λ̄
(p)
k , by definition, correspond to invariants that are of order (h
p
µν) . Their
relation to the ‘D’ and ‘B’ couplings that were used in eq. (12.9) is given by, for p= 0,
1 1 1 λ̄ (0) Bk λ̄ λ̄
D
= + , = k + k , (12.11)
(0)
G GB GD
(0)
G GB Dk k k Gk k k
4In this chapter we are dealing with pure gravity. If one includes matter fields a general truncation ansatz for
Γk contains all possible field monomials that can be constructed from gµν , ḡµν , and the matter fields that respect
the full set of imposed symmetries. At the fundamental level (k → ∞) the fixed point condition will fix the precise
combination in which gµν and ḡµν occur; in the final theory (k → 0) instead it is, again, split-symmetry that forces
one of the two metrics to become irrelevant or more precisely, ‘invisible’ by the physical observables.
404 Chapter 12. Propagating modes in Quantum Gravity
and (p) = D, λ̄ (p)Gk Gk k = λ̄
D
k at all higher levels p≥ 1.
Note that the couplings at level-(1) are precisely those which enter the field equation for
self-consistent backgrounds, δΓk/δhµν |h=0 = 0, while those at level-(2) and levels-(3,4, · · · ) de-
termine the propagator and the vertices of the hµν -self-interactions, respectively. In the present
(2)
truncation the latter roles are played by the same coupling namely (1)Gk = Gk = · · · ≡ GDk , and
(2)
likewise λ̄ (1) Dk = λ̄k = · · · ≡ λ̄k . However, it goes beyond a single-metric truncation as it re-
solves the differences between level-(0) and level-(1).5 Single-metric calculations retain only
terms of order (h 0µν) , i.e. of level-(0), and then postulate that the RG running of the couplings
at the higher levels is well approximated by that at level-(0). (See section 10.3 for a detailed
discussion.)
The gauge fixing and the ghost terms Γgfk and Γ
gh
k in (12.7) are determined by the gauge
fixing function
αβ ( β )
F [ḡ]h ≡ δ ḡαγD̄ −ϖ ḡαβµ µν µ γ D̄µ hµν (12.12)
which involves a free parameter, ϖ , whose RG running is neglected here. Special cases include
the harmonic gauge (ϖ = 1/2) and the geometric, or ‘anharmonic’ gauge (ϖ = 1/d). In addition
there appears the gauge parameter α in the gauge fixing action whose k-dependence will be
neglected as well:
∫ √ [ ( )][ ( )]
Γgf
1 αβ ρσ
k [g, ḡ] = d
dx ḡ ḡµν Fµ [ḡ] gαβ − ḡαβ Fν [ḡ] gρσ − ḡρσ
32πα D/smGk
(12.13)
Specifically, the two gauge fixing parameters were chosen as (ϖ = 1/2, α = 1), (ϖ = 1/d, α →
0), and (ϖ = 1/2, α = 1) in the single-metric truncation of [122], the ‘TT-decomposed’6 bi-
metric calculation of [I], and the ‘Ω-deformed’7 bi-metric analysis in [II], respectively.
When the full ansatz is inserted into the Functional Renormalization Group Equation (FRGE)
we obtain a coupled system of RG differential equations which, when expressed in terms of di-
mensionless couplings8 , has the following structure:
[ ]
∂ D/sm = −2+ηD/sm( D/sm,λD/sm) D/smtgk d gk k gk (12.14a)
∂ λD/sm = βD/sm( D/sm D/smt k λ g ,λ[ k k
) (12.14b)
]
∂ (0)g = d−2+η (0)(gD,λD (0) (0)t k k k ,gk ) gk (12.14c)
∂ λ (0)t k = β
(0)( D D (0) (0)λ gk ,λk ,gk ,λk ) (12.14d)
The two equations (12.14a) and eq. (12.14b) constitute the single-metric system, while the
bi-metric system is described by the full set of all 4 differential equatio(ns. )
Since the above equations are partially decoupled, solutions k →7 gDk ,λDk ,
(0)
gk ,λ
(0)
k can be
obtained from (12.14) in a hierarchical fashion: ( D, λD)⇒ (0) ⇒ λ (0)gk k gk k . Notice that the explicit
5For structurally different calculations disentangling background and fluctuation fields see [130, 206, 268–270].
6The Hessian of Γk in the Einstein-Hilbert truncation contains uncontracted derivative operators such as D̄µ D̄ν .
In [I] a transverse-traceless (TT) decomposition of the fluctuation field hµν was employed to deal with this compli-
cation. The problematic operators act on the component fields as fully contracted Laplacian ḡµν D̄µ D̄ν then, and
heat kernel methods can be applied to evaluate the functional traces due to the various irreducible fields.
7In [II], Ω denotes a conformal parameter introduced as a tool to distinguish between dynamical and background
contributions. The freedom in choosing a gauge parameter α was exploited to reduce the functional trace on the
RHS of the FRGE to a function of the Laplacian D̄2 alone, which then could be computed using standard heat
kernels again.
8The dimensionless couplings, gIk and λ
I
k , are related to the dimensionful ones, G
I
k and λ̄
I
k , appearing in the
truncation ansatz, by GI 2−d Ik = k gk and λ̄
I
k = k
2λ Ik , respectively.
12.2 Anomalous dimension in single- and bi-metric truncations 405
form of the beta-functions to be used is different for the three truncations we are going to
consider here; they can be found in [122], [I], and [II], respectively.
In the sequel, we mostly focus on the Newton couplings GIk and their non-canonical RG
running which is described by the respective anomalous dimension k∂ lnGI ≡ η Ik k . In all trun-
cations considered here its general structure is
BII 1(λ )g
I
η = − for I ∈ {D, B, (0), sm} (12.15)1 BI2(λ )gI
The level- and background-η I ’s are related by η (0)/g(0) = ηB/gB +ηD/gD.
In the sequel we employ the language of levels and always present the couplings of the hµν -
independent invariants, denoted by a superscript (0), together with the higher level couplings
which are collectively denoted by ’D’, standing for (p), p ≥ 1. (The ‘B’ couplings could be
obtained from (12.11) if needed.)
In the following subsections we analyze the anomalous dimensions related to the various
versions of Newton’s constant. We begin with the single-metric case and then proceed to the
two bi-metric calculations [I] and [II].
Unless stated otherwise, we always assume 4 spacetime dimensions (d = 4) in the rest of
this paper, and we employ the optimized cutoff shape function [188].
12.2.1 Single-metric truncation
In the single-metric Einstein-Hilbert truncation the RG running of Newton’s constant is gov-
erned by
Bsm(λ sm)gsm
η sm(gsm,λ sm) = 1− (12.16)1 Bsm(λ sm2 )gsm
The function Bsm(λ sm1 ) in the numerator of eq. (12.16) is given by
{ }
Bsm
1
1 (λ
sm) =− −5Φ11 (−2λ sm)+18Φ2 sm 1 23π 2 (−2λ )+4Φ1 (0)+6Φ2 (0) (12.17)
and Bsm(λ sm2 ) in the denominator reads
{ }
Bsm sm
1
2 (λ ) = −5Φ̃11 (−2λ sm)+18Φ̃2 (−2λ sm2 ) (12.18)6π
Here Φ and Φ̃ are the standard threshold functions introduced in [122] which depend on the
details of the cutoff scheme, its ‘shape function’ (0)Rk in particular.
We are interested in the sign of η sm in dependence on gsm and λ sm, the two coordinates
on theory space. As can be seen from the plot in Fig. 12.1a, in the single-metric truncation,
the anomalous dimension η sm is negative in the entire physically relevant region of the gsm-λ sm
theory space. This is a well-known fact, already mentioned in the∫ √Introduc∫tio√n, and has been
confirmed also by all single-metric truncations with more than the g and gR terms in the
ansatz that were analyzed so far [123–126, 128, 129, 147, 236, 255, 256, 271–276].
In the semi-classical regime9 where 0 < gsm, λ sm ≪ 1 the term Bsm sm sm2 (λ )g in the denom-
inator on the RHS of (12.16) is negligible, hence the negative sign of η sm is entirely due to
the negative sign of Bsm(λ sm1 ) that occurs for small arguments λ
sm ≪ 1. Here it is a reliable
approximation to set η sm ≈ Bsm1 (λ sm)gsm.
9To be precise, we consider a ‘type IIIa’ trajectory here, which, by definition, has a positive cosmological constant
in the IR, see [191].
406 Chapter 12. Propagating modes in Quantum Gravity
1.2
-1.4 -1.8
1.2
1.0 -1
1.0
-2.4
-2
0.8
-2.7
0.8
-3
0.6
gsm 0.6
gsm
1.1 -2.8-
0.4
0.4
0.2
0.2
0 -0.1
0.0 0.0
0.2
0.6
-0.2 0.4 0.8-0.2
-0.1 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3
λ sm λ sm
(a) The sign of η sm. (b) The contour plot of η sm.
Figure 12.1: The phase-portrait of the single-metric Einstein-Hilbert truncation. The shaded ar-
eas in the left diagram indicate regions in the (gsm,λ sm)-plane of positive anomalous dimension
η sm. In the single-metric approximation, η sm is seen to be negative everywhere on the physi-
cally accessible part of theory space. The contour plot of the right diagram shows the lines of
constant η sm values (‘iso-η’ lines).
It is instructive to expand the function Bsm1 for small values of the (dimensionless) cosmo-
logical constant:
[ ]
Bsm(λ sm
1 26
1 ) = Φ
1
1 (0)−24Φ2 sm2 (0) − λ +O((λ sm)2) (12.19)3π 3π
This linear approximation confirms the negative values of Bsm1 in the semi-classical regime: Its
λ sm-independent term Bsm1 (0) is known to be negative for any admissible cutoff [122], and the
term linear in the cosmological constant is negative, too, when λ sm > 0.
Notice that the slope of the linear function (12.19) is universal, i.e. cutoff scheme indepen-
dent. Every choice of the shape function (0)Rk used in the threshold functions Φ and Φ̃ yields the
same slope, −26/3π , which is negative and thus favors an anomalous dimension which is neg-
ative, too. The constant term in (12.19) is cutoff scheme dependent, however its negative sign
is not. Hence, starting from Bsm1 (0)< 0, the function B
sm
1 (λ
sm), and therefore also η sm(gsm,λ sm),
decreases with increasing values of λ sm, and in fact stays negative throughout the relevant part
of theory space (λ sm < 1/2).
12.2.2 The (TT-based) bi-metric calculation [I]
Turning to truncations of bi-metric type now, let us consider the approach followed in [I] first.
In the dynamical sector the dependence of the corresponding anomalous dimension
BD(λD)gD
ηD(gD,λD) = 1− (12.20)1 BD2(λD)gD
on the cosmological constant λD is described by the numerator function,
1 { ( )
BD D1(λ ) = 25Φ
2
2 (−2λD)+Φ2 − 4λD −80Φ3 (−2λD6π 2 3 3 ) }
−3Φ22 (0)+28Φ3 (0)+72Φ43 4 (0) (12.21)
12.2 Anomalous dimension in single- and bi-metric truncations 407
1.2 1.2
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.1 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3 0.4
λD λ (0)
(a) The D-sector. (b) The level-(0) sector.
Figure 12.2: The phase portraits on the (gD,λD)- and the (g(0),λ (0))-plane, respectively, obtained
using the bi-metric results of [I]. The shaded (white) areas are regions of positive (negative)
anomalous dimension ηD and η (0), respectively. While the dynamical anomalous dimension ηD
exhibits regions of physical interest where it is positive, η (0) does not.
while the denominator contribution in eq. (12.20) contains
1 { ( ) }
BD D 22(λ ) =− 25Φ̃2 (−2λD)+ Φ̃2 − 4λD2 3 −80Φ̃
3
3 (−2λD) (12.22)12π
The beta-functions of the level-(0) and the background-sector are sensitive to the dynamical
couplings as well. In particular the sign of the anomalous dimension η (0), pertaining to the
level-(0) Newton constant g(0), is strongly dependent on the dynamical cosmological constant,
λD. Explicitly,
( ) 1
[
η 0 (gD,λD,g(0)) = 10q11 (−2λD)+2q
1 (− 4 D 2 D
12π 1 3
λ )−40q2 (−2λ )
]
−8Φ11 (0)−13Φ22 (0) g(0) (12.23)
where p p η
D p
q ( )≡ Φ (w)− Φ̃ (w) involves ηD ≡ ηD(gD,λDn w n 2 n ).
In Figs. 12.2a and 12.2b we display the (gD,λD) and (g(0),λ (0)) phase portraits for the dy-
namical and the level-(0) couplings, respectively. For the latter, the overall picture is essentially
the same as for the single-metric truncations: The anomalous dimension η (0) is negative every-
where on theory space (where g(0) > 0), in particular in the semi-classical regime. However,
the dynamical (gD,λD)-flow reveals a novel aspect of the bi-metric truncation: the anomalous
dimension ηD is positive for λD smaller than a certain critical value λDcrit > 0, and turns negative
only when λD > λDcrit.
While this conclusion is drawn on the basis of the complete formula (12.15) including the
denominator, the sign of ηD coincides with the sign of BD since BDgD1 2 ≪ 1 in the entire region
of interest, so that ηD ≈ BD(λD1 )gD is a good approximation. As a consequence, the domains
where ηD > 0 and ηD < 0, respectively, are separated by a straight line on the (gD,λD)-plane
located at λD = λDcrit with B
D(λD1 crit) = 0.
The sign flip of BD1 can be demonstrated analytically by expanding B
D
1 in powers of the
cosmological constant:
1 [ ] 43
BD(λD1 ) = 23Φ
2 4
2 (0)+72Φ4 (0)−72Φ3 D D 23 (0) − λ +O((λ ) ) (12.24)6π 9π
gD
g(0)
408 Chapter 12. Propagating modes in Quantum Gravity
1.2 0.2
0.4
1.0
-0.2 -1.2
0.3
0.8
-0.4
-1.4
0.6 0 -1.8
gD
0.4
0.1 -2
0.2
-1
λDcrit
0.0
0 1
-0.2
-0.1 0.0 0.1 0.2 0.3
λD
Figure 12.3: The contour plot shows the lines of constant anomalous dimension ηD implied by
the bi-metric calculation [I]. The shaded regions correspond to ηD > 0. Note the sign flip of ηD
at a non-zero critical value of the cosmological constant, λDcrit > 0.
Again, the slope of this linear function is found to be both universal and negative, −43/9π in
this case. The difference in comparison with the single-metric truncation lies in the constant
term BD1(0): according to the bi-metric calculation it is positive for all plausible cutoff schemes,
in sharp contradistinction to Bsm1 (0)< 0 in the single-metric case.
For the example of the optimized shape function we have BD1(0) = +35/36π , yielding the
critical cosmological constant λD D 2 Dcrit|opt.cutoff ≈ 0.2. The quadratic terms ∝ (λ ) in B1 correct
this result to about λDcrit|opt.cutoff ≈ 0.1 which is then stable under the addition of still higher
orders and coincides with the exact value.
So we may conclude that the dynamical anomalous dimension ηD is positive in the semi-
classical regime where 0< gD, λD ≪ 1 and becomes negative for λD > λDcrit ≈ 0.1. This is also
seen in Fig. 12.3 which shows the lines of constant ηD values on the (gD,λD) plane. Recall that
the NGFP, for instance, is located on the curve with ηD =−2.
To verify that the novel feature of a positive ηD in the semi-classical regime is independent
of the cutoff-scheme chosen, we have checked the corresponding condition BD1(0) > 0 for the
one-parameter family of exponential shape functions (0)Rk ( ) = [exp( )−1]
−1
y sy sy , for example
[123, 135, 191, 277]. Its threshold functions at vanishing argument can be evaluated exactly:
1 [ ( ) ]
BD1(0) = − − (7s−3)(s−1)+ 9+ s(14s−19) ln(s) (12.25)6π(s 1)3
This result for the s-dependence, plotted in Fig. 12.4, is indeed reassuring: even though the
value of BD1(0) decreases for increasing ‘shape parameter’ s, it stays always positive. This yields
a critical value λDcrit which is positive, too. Thus, the bi-metric calculation [I], very robustly,
predicts a semi-classical regime with a positive value of the dynamical anomalous dimension
ηD.
12.2.3 The (‘Ω-deformed’) bi-metric calculation [II]
A different bi-metric approach that is more closely related to the single-metric computation in
[122] was developed in [II] recently. While it employs the same truncation ansatz, namely two
separate Einstein-Hilbert actions for gµν and ḡµν , the gauge fixing and the field parametrization
12.2 Anomalous dimension in single- and bi-metric truncations 409
1
2.0
0.001 1.5
1.0
10-6
BD(0) 0.0 0.2 0.4 0.6 0.8 1.01
10-9
10-12
1 1000 106 109 1012 1015
s
Figure 12.4: The constant BD1(0) as obtained in [I] for different values of the shape parameter
s characterizing the exponential shape functions. While cutoff scheme dependent, its sign is
always positive, implying the existence of a critical cosmological constant λDcrit > 0 such that
ηD > 0 for λD < λDcrit.
chosen are different from the calculation [I]. In order to explore whether the novel properties
displayed by [I] are actually due to its bi-metric character, and to what extent gauge fixing and
field parametrization issues play a role possibly, we shall now repeat the analysis of the previous
subsection, this time using the beta-functions obtained in [II].
1.2 1.2
1.0 1.0
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
-0.2 -0.2
-0.1 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3 0.4
λD λ (0)
(a) The D-sector. (b) The level-(0) sector.
Figure 12.5: The phase portraits for the dynamical and the level-(0) sector according to the beta-
functions of [II]. As can be seen from the shading the dynamical RG flow exhibits a transition
from negative to positive anomalous dimensions at λDcrit > 0, while its level-(0) counterpart is
negative everywhere in the physically relevant part of theory space.
The property at stake is the λD-dependence of ηD. It has the structure (12.20) again, but
with seemingly rather different functions in the numerator and denominator:
1 { }
BD(λD1 ) =
23 2
3 Φ2 (−2λ
D)−24Φ33 (−2λD)− 2Φ23 2 (0)+8Φ
3
3 (0) (12.26a)π
1 { }
BD D 23 22(λ ) =− 3 Φ̃2 (−2λ
D)−24Φ̃33 (−2λD) (12.26b)2π
gD
g(0)
410 Chapter 12. Propagating modes in Quantum Gravity
The level-(0) sector is governed by the following anomalous dimension η (0):
[ ]
η (0)(gD,λD,g(0)
2 5
) = q1 (− D)−3q21 2λ D2 (−2λ )− 2Φ1 2 (0)π 6 3 1 (0)− Φ2 (0) g (12.27)
The resulting phase-portraits for the dynamical and level-(0) sectors are depicted in Fig.
12.5. Only in case of the dynamical anomalous dimension ηD do the shaded areas, indicating
regions of positive anomalous dimension, appear in the physically relevant part of the phase
diagram. For the level-(0) sector we obtain a negative value of η (0) everywhere. The contour
plot over the (gD,λD) plane showing the lines of constant ηD is displayed in Fig. 12.6.
Comparing the diagrams in Figs. 12.5 and 12.6 to their analogs of the calculation [I], in
Figs. 12.2 and 12.3, we find perfect agreement at the qualitative level between the two bi-metric
approaches [I] and [II], respectively. However, the results differ significantly from their single-
metric counterparts in Fig. 12.1.
1.2
0.4
1.0
0.2
0.8
-4
0.6
-3.6
gD -2
0.8 -2.4-
0.4
-1.2
-0.4
0
0.2
λDcrit
0.0
0.2
0 0.4 0.8 1
-0.2
-0.1 0.0 0.1 0.2 0.3
λD
Figure 12.6: The contour plot shows the lines of constant anomalous dimension ηD implied by
the bi-metric calculation [II]. The shaded regions correspond to ηD > 0. Consistent with the
calculation in [I], ηD is found to change its sign on a straight line λD = λDcrit with λ
D
crit > 0. In the
semi-classical regime, ηD is seen to be positive.
As the semi-classical regime is of special importance let us expand BD1(λ
D) in the vicinity
of λD = 0 again:
1 [ ] 26
BD1(λ
D) = 7Φ22 (0)−16Φ33 (0) − λD+O((λD)2) (12.28)π 3π
Eq. (12.28) confirms the picture implied by the previous bi-metric calculation [I]: a universal,
negative slope (which in this case happens to coincide with the single-metric value, −26/3π)
along with a universally positive constant term, BD1(0). Together they give rise to a region in
which ηD > 0.
The cutoff-scheme dependence of BD1(0) was again checked by evaluating B
D
1(0) for the
optimized and the s-family of exponential shape-functions, for instance. The optimized cutoff
yields BD1(0) = 5/6π . The linear approximation (12.28) corresponds to the critical value λ
D
crit ≈
0.096 in this case; it coincides almost perfectly with the corresponding exact value from the
full non-linear equation: λDcrit ≈ 0.064. For the exponential shape-functions the constant term in
12.2 Anomalous dimension in single- and bi-metric truncations 411
0.01 0.34
0.32
0.30
10-5 0.28
0.26
BD(0) 0.241
8 0.0 0.2 0.4 0.6 0.8 1.010-
10-11
10-14
1 1000 106 109 1012 1015
s
Figure 12.7: The value of BD1(0) according to the bi-metric calculation [II] is shown for different
choices of the family parameter s characterizing the exponential cutoff shape functions.
(12.28) evaluates to D(0) = 4−4s+ln(s)+3s ln(s)B1 − 2 which is positive for all admissible values
10 of s,
π(s 1)
as shown in Fig. 12.7.
Thus, the second set of bi-metric results fully confirms all conclusions drawn in the previous
subsection on the basis of the RG equations obtained in [I].
12.2.4 Summary: significance of the cosmological constant
We investigated the possibility of a positive anomalous dimension (ηD or η sm) in the semi-
classical regime of three different truncations. In section 12.1 we discussed already that while
at negative η I near the NGFP is the very hallmark of Asymptotic Safety, there is no general
reason that would forbid η I to be positive in other parts of theory space, the semi-classical
regime in particular. While a transition to a positive η I was not observed in any single-metric
truncation, we found that both bi-metric calculations which we analyzed do indeed show that
ηD is actually positive on a large portion of theory space, namely the half plane −∞ < λD < λDcrit.
Here λDcrit is a strictly positive critical cosmological constant, necessarily smaller than the NGFP
coordinate λD∗ .
The region in theory space with a negative ηD, which is indispensable for a non-Gaußian
fixed point and the non-perturbative renormalizability of QEG, crucially owes its existence to
the negative, universal slope of BI1(λ ) at λ = 0. It occurs in all three truncations, including the
single-metric one, and indicates an anti-screening component in the beta-function of gD. In the
‘sm’ case the intercept Bsm1 (0) is negative as well, and so B
sm
1 (λ ) is negative for all λ . In both
bi-metric truncations BD1(0) is positive, however, and this gives rise to a window λ
D ∈ (−∞,λDcrit)
with a certain λD Dcrit > 0 in which B1(λ ) is positive.
In the semi-classical regime, the linear (in gD) relationship ηD ≈ BD1(λ )gD always turned
out to be an excellent approximation. Hence, for a positive Newton constant (which we always
assume) the anomalous dimension is positive in the window λD ∈ (−∞,λDcrit). The precise value
of λDcrit depends on the cutoff shape function; generically it is of the order 10
−1 or 10−2, say.
The main message is summarized in Fig. 12.8 which depicts the exact (i.e., all-order) λ -
dependence of BI1. The single- and bi-metric functions all decrease with increasing λ . But
while the ‘sm’ function Bsm1 is negative everywhere, both of the dynamical bi-metric functions
are non-negative in the vicinity of λ = 0, implying a positive dynamical anomalous dimension
10It is known that for quantitative reliability the shape parameter s∈ (0,∞) should not be chosen too small, s& 0.5,
say [123, 135, 191].
412 Chapter 12. Propagating modes in Quantum Gravity
BI1(λ)
[I]
λ
-0.10 -0.05 0.05 0.10 0.15
-0.5 [II]
-1.0
sm
-1.5
Figure 12.8: The λ -dependence of BI1(λ ) for the two bi-metric truncations [I] and [II], as well as
the single-metric approximation (sm). Notice that the latter has not only a negative slope but also
a negative intercept Bsm1 (0)< 0, while both bi-metric functions are positive in the semi-classical
regime of not too large dimensionless cosmological constant.
there: ηD(gD,λD)> 0 for all gD > 0 and −∞ < λD < λDcrit.
12.2.5 From anti-screening to screening and back
Recalling the definition ηD ≡ k∂ lnGDk k , it follows from the above that along every RG trajectory
running on the half space λD < λDcrit the dynamical Newton constant G
D
k increaseswith increasing
scale k. Stated differently, the gravitational interaction shows a screening behavior there. This
is in stark contrast to its anti-screening character in the NGFP regime.
gD
UV
gD k∗ crit
gDT T
kIRcrit
O λD DT λcrit λ
D
∗ λD
Figure 12.9: Schematic behavior of a bi-metric type IIIa trajectory on the (gD,λD)-projection
of theory space. The dashed line separates the half spaces with ηD > 0 and ηD < 0, respec-
tively. The part of the trajectory located above (below) the turning point T is referred to as the
trajectory’s UV (IR) branch.
For the example of a bi-metric trajectory which is of type IIIa in the ‘D’ projection [173]
the situation is depicted schematically in Fig. 12.9. The trajectory k 7→ (gD Dk ,λk ) emanates from
the NGFP at ‘k = ∞’, then leaves the asymptotic scaling regime for k ≈ mPlanck, but stays in
the half-space with ηD > 0 as long as k is larger than a certain critical scale kUVcrit at which the
running cosmological constant λDk drops below λ
D
crit. As k decreases further below k
UV
crit, the
cosmological constant continues to decrease until the turning point T is reached, beyond which
12.3 Interpretation and Applications 413
the (dimensionless!) λDk now increases for decreasing k. Ultimately, it will re-enter the half-
space with ηD < 0, namely at a second critical scale, kIRcrit. So, by definition,
λD| D D IR UVk k=kIR = λcrit = λk |k=kUV with kcrit < kcrit . (12.29)
crit crit
As it is already well-known for the type IIIa trajectories in the single-metric truncation [191,
209, 212], the bi-metric trajectories of this type, too, can have a long classical regime where the
(dimensionful!) Newton- and cosmological constant are approximately constant. This requires
tuning the turning point T very close to the Gaußian fixed point, the origin (0,0) in Fig. 12.9.
The point T is passed at k = kT with kIRcrit ≪ kT ≪ kUVcrit where the two critical scales are far apart
then.
For example, the ‘RG trajectory realized in Nature’, that is, the specific single-metric
(gsm,λ smk k )- or bi-metric (g
D
k ,λ
D
k )-trajectory whose parameters are matched against the measured
values of G and λ̄ [209, 212] is well-known to be highly fine-tuned, with turning point coordi-
nates as tiny as g ≈ λ ≈ 10−60T T . Following the discussion in [209, 212] it is easy to see that, for
this trajectory, and for a λD −2 UVcrit value of, say, 10 , the UV critical scale is about kcrit ≈ mPlanck/10,
while the one in the IR is slightly above the present Hubble parameter, kIRcrit ≈ 10H0. Newton’s
constant reaches its maximum at k = kUVcrit; it is about 2% larger there than at laboratory scales.
12.3 Interpretation and Applications
The ‘dynamical’ anomalous dimension ηD governs the running of that particular version of New-
ton’s constant which controls the strength of the gravitational self-interaction and the coupling
of gravity to matter. We found gravitational screening (rather than anti-screening, as predicted
by the single-metric truncations) in the semi-classical regime, that is, GDk grows with k as long as
λDk < λ
D
crit. The strong renormalization effects associated with Asymptotic Safety, the formation
of a fixed point, anti-screening, and large negative values of ηD, are confined to the half-space
with λD > λDcrit instead.
In the following two subsections we discuss a number of possible implications of these
findings. In subsection 12.3.1 we interpret the sign change of ηD in terms of a dark matter
description, and in subsection 12.3.2 we briefly comment on an application in cosmology.
12.3.1 The dark matter interpretation
Physical significance of the dimensionless cosmological constant
For the interpretation of the above results it is helpful to recall that, upon going on-shell, the
value of the dimensionful cosmological constant λ̄D ≡ k2λDk determines the curvature of space-
time when it is explored with an experiment, or a ‘micro(scop)e’ of resolving power
11 ℓ ∝ 1/k.
D −1/2The radius of curvature of spacetime is of the order rc ∝ λ̄ then, and the dimensionless
cosmological constant is approximately the (squared) ratio of the two distance scales involved:
( )
≈ ℓ
2
λDk (12.30)rc
Thus we see that the sign-flip of ηD is controlled by the background curvature: on self-consistent
backgrounds [166] which are only weakly curved on the scale of the microscope, ℓ ≪ rc, we
have λDk ≪ 1, therefore ηD > 0, and so we observe a screening behavior of the gravitational
interaction. Conversely, when the spacetime is strongly curved on the scale of the microscope
(i.e. the scale set by the modes just being integrated out at this k) the ratio ℓ/rc approaches unity,
implying λDk > λ
D
crit and, as a result, strong anti-screening effects.
11This estimate could be made more precise using the method of the ‘cutoff modes’, see refs. [210, 211].
414 Chapter 12. Propagating modes in Quantum Gravity
Propagating gravitons in the semi-classical regime
The positive ηD in the semi-classical regime resolves the puzzle raised in the Introduction:
On a nearly flat background spacetime the dynamics of the hµν fluctuations is such that the
interactions get weaker at large distance, and the corresponding Green’s function is short ranged.
The positive ηD causes no conflict with the existence of a Källén-Lehmann representation with
a positive spectral density, and the EAA may be seen as describing an effective field theory
very similar to those on Minkowski space. It describes weakly interacting gravitons and, in
the classical limit, gravitational waves. In the opposite extreme when the curvature is large on
the scale set by k there is no description of the hµν -dynamics in terms of a Minkowski space-( )
− 2 1−η
D/2
like effective field theory. The propagator ∝ 1/ D̄ is very different from the one on
flat space then, both because of the background curvature and of the large negative ηD which
renders it long ranged. In this regime the hµν -dynamics is anti-screening and results in the
formation of a non-trivial RG fixed point.
This general picture points in a similar direction as the mechanism of the ‘paramagnetic
dominance’ found in [165] which likewise emphasizes the importance of the background curva-
ture for Asymptotic Safety.
The positive sign of ηD near the Gaußian fixed point is furthermore consistent with the
perturbative calculations on a flat background12 performed by Bjerrum-Bohr, Donoghue, and
Holstein [278].
The screening behavior in the semi-classical regime is also consistent with the first analyses
of the ‘lines of constant physics’ [279, 280] found by numerical simulations within the CDT
approach [98].
Strong curvature regime: ‘physical’, gravitating, and (non-)propagating hµν modes
An important issue about which we can only speculate at this point is the properties of the
metric fluctuations in the regime where λD ' λDcrit. There, the field hµν still carries ‘physical’,
in the sense of ‘non-gauge’ excitations which, however, admit no description as ‘particles’ ap-
proximately governed by an effective field theory similar to those on Minkowski space. This
would not be surprising from an on-shell perspective as now the background is curved on a scale
comparable to the physics considered. However, it is not completely trivial that the quantum
fluctuations driving the RG flow13 reflect this transition, too, since those are far off-shell in
general.
All we can say about the hµν quantum field in this regime is that it is likely to carry ‘physical’
excitations which, due to the non-linearity of the theory, interact gravitationally. We do not
know the precise propagation properties of those excitations, however. They might, or might
not behave like a curved space version of the graviton, as propagating little ripples on a strongly
curved background.
What comes to mind here is the analogy to transverse gluons in QCD, at the transition
from the asymptotic freedom to the confinement regime. In either regime they are ‘physical’,
i.e. ‘non-gauge’ excitations, but only in the former regime they behave similar to propagating
particles, while they are confined in the latter.
Also the unparticles which were proposed by Georgi in a different context [281, 282] are
examples of such perfectly ‘physical’ field excitations which admit no particle interpretation,
not even on flat space.
12Provided the latter are restricted to the vacuum-polarization diagrams, i.e. those related to ηD.
13By contributing to the functional trace on the RHS of the FRGE.
12.3 Interpretation and Applications 415
The hµν propagator by RG improvement
The physics of the hµν excitations in the strong curvature regime could be explored by comput-
ing their n-point functions δ nΓ [h; ḡ]/δhn0 |h=0 from the standard effective action Γ0 = limk→0 Γk
on a self-consistent, in general curved background ḡ ≡ ḡs.c.. Particularly important is the in-
verse propagator G−1 ∝ δ 2Γ [h; ḡs.c.]/δh20 |h=0. It describes the properties of both the ‘radia-
tive’ modes carried by hµν , and the ‘Coulombic’ modes. The latter determine in particular
the response of the hµν field to an externally prescribed (static) source Tµν , the source-field
relationship having the symbolic structure G−1h= T.
The calculation of G is a very hard problem, not only because of the much more general
truncation ansatz it requires, but also because we do not yet know any realistic candidate for a
consistent background ḡs.c.k in the domain of interest [283, 284]. Clearly a technically simple
background like ḡµν = δµν is excluded here since a flat background is far from consistent when
λD is large.
Despite these difficulties we can try to get a rough first impression of this domain if we
restrict our attention to the h propagator in a regime of covariant momenta in which ηDµν ≡ η
is approximately k-independent. Then, by a standard argument [285], RG (impro)vement of1−η/2
the 2-point function suggests that the inverse propagator in Γ0 equals G−1 ∝ −D̄2 . In
general this is a complicated operator with a non-local integral kernel. Let us consider the
corresponding source-field relation,
( )
L−η − 2 1−η/2D̄ φ =−4πGρ (12.31)
with a now scale-independent Newton constant G, and a length parameter L included for dimen-
sional reasons. Here we suppress the tensor structure and employ a notation reminiscent of the
Newtonian limit which we shall take later on only; the following argument is fully relativistic
still.
Non-locality mimics dark matter
For a generic real, i.e. non-integer value of η the LHS of eq. (12.31) involves a highly non-
local operator acting on φ . In order to understan(d how th)e solutions of this equation differ fromη/2
the classical ones, let us act with the operator −L2D̄2 on both sides of (12.31). Leaving
domain issues aside this yields an equation similar to (12.31), but nowwith η = 0 and amodified
source instead:
D̄2φ = 4πGρ̃ (12.32a)
( )
≡ − 2 2 η/2ρ̃ L D̄ ρ (12.32b)
We see that the modifications caused by a non-zero anomalous dimension can be shifted from
the differential operator acting on the gravitational field to the source function. In the Newtonian
limit, for instance, eq. (12.32a) has the interpretation of the classical Poisson equation for the
gravitational potential φ generated by the mass density ρ̃ . However, the density function ρ̃ does
not coincide with the mass distribution that has actually been externally prescribed, namely ρ .
The RG effects are encoded in the way the ‘bare’ mass distribution ρ gets ‘dressed’ by quantum
effects which turn it into the ‘renormalized’ ρ̃ .
Being more explicit, the operator application in (12.32b) amounts to the convolution of ρ
416 Chapter 12. Propagating modes in Quantum Gravity
with a non-local integral kernel:14
∫
ρ̃(x) = ddx′
√
ḡ(x′)Kη(x,x
′)ρ(x′) (12.33a)
( )
K (x,x′)≡ 〈x| −L2 2 η/2η D̄ |x′〉 (12.33b)
Note that the kernel Kη , and therefore ρ̃ , still depend on the background ḡµν .
While in general x and x′ are 4-dimensional coordinates they reduce to 3D space coordinates
if we invoke the Newtonian limit where ρ , ρ̃ , and φ are time independent. In fact, to gain a
rough, but qualitatively correct intuition for the ‘dressing’ ρ 7→ ρ̃ , it suffices to consider the
Newtonian limit, an approximately flat background in particular, but to maintain a non-zero
value of η . Then, with ḡµν = ηµν , eq. (12.32a) boils down to the time independent Poisson
equation ∇2φ = 4πGρ̃ , and the kernel K (~x,~x′) ≡ K (|~x−~η η x′|) is easily evaluated in the plane
wave eigenbasis of the Laplacian on flat space, ∇2:
∫
d3p ( )2 2 η/2K (r) = L ~p ei~p·(~x−~x′)η , r ≡ |~x−~x′| (12.34)
(2π)3
Focusing on the simplest case, η ∈ [−2,−1], this integral yields15, at r 6= 0,
[ ]− η1 L
Kη(r) =− 4πΓ(−1−η)cos(π2 η) (12.35)r3+η
Now, even if the ‘bare’ ρ(~x) is due to a point mass, for example, ρ(~x) =Mδ (~x), the ‘renormal-
ized’ or ‘dressed’ mass distribution amounts to an extended, smeared out cloud with a density
profile ρ̃(~x) = MKη(|~x|). If (12.35) applies, ρ̃ has support also away from ~x = 0, falling off
according to the power law
ρ̃(r) ∝ 1/r3+η (r > 0) (12.36)
If η is negative, the ρ̃ distribution is the more extended the larger is |η |.
While strictly speaking (12.36) is valid only for η ∈ [−2,−1], it highlights the main impact
a negative η has on gravity, also beyond the Newtonian limit: If one sticks to the classical form
of the field equation (here: Poisson’s equation) the gravitational field is sourced not only by
the energy momentum tensor of the true matter (here: ρ) but in addition by a fictitious energy-
momentum-, and in particular mass-distribution (ρ̃) which is obtained by a non-local integral
transformation applied to the true, or ‘bare’, source.
In the simplest case the integral transformation is linear and assumes the form (12.33a).
Where it applies, the ‘fictitious’ matter traces the ‘genuine’ one, the latter sources the former.
Hence it seems indeed appropriate to regard the transition from ρ to ρ̃ as due to the ‘dressing’
of the bare source by quantum effects, similar to the dressing of electrons in Quantum Elec-
trodynamics (QED) by clouds of virtual particles surrounding them. It is quite clear then, in
particular in a massless theory, that the dressing of point sources results in spatially extended,
non-local structures.
14If needed, the non-integer power of D̄2 can be expressed by an appropriate integral representation. For a general
discussion of fractional powers of the Laplacian and d’Alembertian and their Green’s functions, see [286, 287].
15For other values of η we must introduce explicit distan(ce o)r momentum cutoffs into the integral (12.34) in order1−η/2
to take account of the fact that the approximation G−1 ∝ D̄2 with a constant value of η is valid only in a
restricted regime. Being interested in qualitative effects only we shall not do this here. One also has to be careful
about delta-function singularities at the origin; in particular we have K0(~x,~x′) = δ (~x−~x′), as it should be.
12.3 Interpretation and Applications 417
Modified gravity in astrophysics: a digression
Applying this discussion to the realm of astrophysics, to galaxies or clusters of galaxies, one is
tempted to interpret the fictitious matter contained in ρ̃ , over and above the true one, as the long
sought-for dark matter, and to identify ρ with the actually observed ‘luminous’ matter.
To avoid any misunderstanding we emphasize that the presently available RG flows do not
(yet?) reliably predict large negative anomalous dimension (η ≈ −1, say) on astrophysical
scales16. All we can say for the time being is that the mathematical structure of the field equa-
tions we encounter here is potentially relevant to the astrophysical dark matter problem, but
clearly much more work will be needed to settle the issue.
The much more direct reason why the mechanism of non-local gravity mimicking dark
matter is relevant to Asymptotic Safety is that on a type IIIa trajectory large negative η’s occur
in two regimes: not only at astrophysical or cosmological scales, k . kIRcrit, but also near the
Planck regime, k & kUVcrit.
As it is shown schematically in Fig. 12.9, the trajectories of type IIIa, like the one that
could perhaps apply to the real Universe, have two sections with a sufficiently large λD to make
ηD negative, one on the UV-, the other on the IR-branch. The main difference between the
branches is their typical value of gD: it is much smaller on the IR-branch than on the UV-
branch. As a result, on the IR-branch |ηD| = |BD1(λD)gD| assumes values of order unity, say,
only when λD is increased much further beyond λDcrit than this would be necessary on the UV-
branch. This distinction is best seen in the contour plots (‘iso-η-lines’) of Figs. 12.3 and 12.6.
Since the Einstein-Hilbert truncation becomes unreliable near λD = 1/2, it can deal with the
large negative η’s on the UV-branch only.
After the above precautionary remark it is nevertheless interesting to note that on the as-
trophysical side an integral transform like (12.33a), connecting luminous to dark matter in real
galaxies, has indeed been proposed long ago on a purely phenomenological basis: It is at the
heart of the Tohline-Kuhn modified-gravity approach [288–290]. Recently this approach has
attracted attention also because it was found to emerge naturally from a certain classical, fully
relativistic, and non-local extension of General Relativity [291–293].
Above we saw that quantum gravity effects can modify Einstein’s equations in precisely the
Tohline-Kuhn style. The similarity between the two theories becomes most explicit for η =−1
which leads to the integral kernel
1 1
K− (~x,~x′1 ) = (12.37)
2π2L |~x−~x′|2
This is exactly the one which appears also in the Tohline-Kuhn framework.
Using this kernel in eq. (12.33a), a point mass with ρ(~x) = Mδ (~x) is seen to surround
itself with a spherical ‘dark matter halo’ whose radial density profile is given by ρ̃(r) = M/
(2π2Lr2). By virtue of ∇2φ = 4πGρ̃ , this dark matter distribution generates the logarithmic
potential φ(r) = (2GM/πL) ln(r). In the Newtonian limit, it is well known to yield a perfectly
flat rotation curve, that is, a test particle on a circular orbit has a velocity which is independent
of its radius17, v2 = 2GM/πL.
Non-local constitutive relations as a QEG vacuum effect
Recently the Tohline-Kuhn framework turned out to describe the Newtonian limit of a fully
relativistic generalization of General Relativity which allows the incorporation of non-locality
16See, however, ref. [212].
17From the idealized case of a point particle where v2 ∝ M it might appear that this approach has difficulties
reproducing the Tully-Fisher law for spiral galaxies [294] which favors v4 ∝ M. However, the detailed studies on
the basis of fits to realistic galaxy data reported in [295, 296] seem not to encounter such difficulties.
418 Chapter 12. Propagating modes in Quantum Gravity
at a phenomenological, purely classical level [291–293, 297]. This theory, proposed by Hehl
and Mashhoon, relies on the observation that the teleparallel equivalent of General Relativity, a
special gauge theory of the translation group, is amenable to generalization through the intro-
duction of a non-trivial ‘constitutive relation’ similar to the constitutive relations between (~E,~B)
and (~D,H~ ) in electrodynamics.
Because of memory effects, such relations are non-local typically. They make their appear-
ance both in the classical electrodynamics of matter, and in vacuum Quantum Electrodynamics
where loop effects are well known to give rise to a complicated relationship between ~E and ~D,
say, which is both non-linear and non-local [298]. As for quantum gravity, it was pointed out
[165, 195] that QEG, like QED, has a non-trivial vacuum structure with a non-linear relation-
ship between the gravitational analogs of the E~ and ~D fields. From this perspective it is quite
natural that the source-field relation of quantum gravity, in a regime with large negative η , turns
out not only non-linear, but also non-local.
In this sense, a phenomenological theory like the one in [291–293], as far as its general
structure is concerned, may well be regarded as an effective field theory description of the
QEG vacuum in the large-η regime.
In fact, in QEG and the theory of ref. [291–293] the size of the new effects is determined by
essentially the same control parameter. In [291–293] the degree of non-locality is governed by
the ratio ρ ≡ Lacc/Lphen, where Lphen denotes the length scale of the phenomeno(n under con)sid-2
eration, and Lacc is the acceleration length of the observer. Interestingly, ρ−2 ≡ Lphen/Lacc is
basically the same as the dimensionless cosmological constant λk = λ̄k/k2 which controls the
size of η and the non-local effects in QEG. There, ℓ ≈ k−1 characterizes the length scale of
the physical process under consideration and so it takes the place of Lphen, while the radius of
curvature, rc, may be identified with L 18acc .
Planck scale non-locality as ‘dark matter’
At this point of the discussion we switch back from the IR to the UV regime. As we emphasized
already the beta-functions considered in the present chapter, where they are reliable, yield only
tiny values for η on astrophysical scales. So here we focus on the dark matter interpretation
which applies to the UV branch of the ‘RG trajectory realized in Nature’, see Fig 12.9. Of
course, the UV-branch exists not only for the trajectories of type IIIa but for all asymptotically
safe ones. Along any of them, for k near the Planck scale, but still above kUVcrit, the anomalous
dimension is large and negative since the trajectory just left the NGFP regime where ηD ≈ηD∗ =
−2.
It is thus plausible to re-apply the above discussion of astrophysical dark matter which is
mimicked by non-locality in the ultraviolet. The situation would then be as follows. When
we approach the UV regime, above a certain scale kUVcrit located about one or two orders of
magnitude below the Planck scale, non-local effects start becoming essential. Now, the regime
in question, kUVcrit . k . mPlanck, is exactly the one for which we concluded already that the hµν
excitations cannot be described there by an effective field theory of the conventional local form;
in particular their propagation properties are not easily established, and we conjectured that
there are indeed no propagating gravitons above kUVcrit.
Assuming this picture is correct it suggests the interpretation of the physical, but non-
propagating hµν modes as a type of Planckian dark matter that admits an effective description
in terms of a (fully relativistic!) Hehl-Mashoon-type theory [291–293]. In this scenario the
modes of the metric fluctuations with covariant momenta above kUVcrit do not propagate, but are
still physical (in the sense of ‘non-gauge’). They interact gravitationally with matter and among
themselves, they can condense to form spatially extended structures, and they dress ordinary
18In cosmology, for instance, one has indeed Lacc ∼ H−1 ∼ λ̄−1/2 ∼ rc.
12.4 Summary 419
localized energy-momentum distributions by ‘dark matter halos’ which are approximately de-
scribed by a Tohline-Kuhn-type integral transform.
This reflects an antagonism between gravitons and dark matter: The semi-classical modes
of the fluctuation field have a particle interpretation, describe massless gravitons or essentially
classical gravitational waves, while those with larger momenta are equally physical, gravitate,
but do not propagate presumably.
To visualize this situation it helps to recall the example of the transverse gluon modes in
QCD: Those with momenta well above the confinement scale propagate approximately particle-
like, the others are confined, and they form the homogeneous gluon condensate characteristic
of the QCD vacuum state.
12.3.2 Primordial density perturbations from the NGFP regime
The conjectured absence of propagating gravitons in a certain range of momenta can also be
relevant to cosmology presumably, for example in the context of the Cosmic Microwave Back-
ground (CMB). In refs. [189, 190, 209] an Asymptotic Safety-based alternative to the standard
inflationary paradigm has been proposed in which the source of the primordial density per-
turbations, responsible for later structure formation, are the quantum fluctuations of geometry
itself which occur during the Planck epoch.19 Within QEG the fluctuations in this regime are
governed by the NGFP, and so they could provide a perfect window to the very physics of
Asymptotic Safety.
It has been argued that when the Universe was in the Planck-, or NGFP-regime the scale-
free form of the hµν -propagator ∝ 1/D4 gave rise to a kind of cosmic ‘critical phenomenon’
which displays metric fluctuations on all length scales [189, 190, 209, 216]. The scale-free
nature of all physics at the fixed point renders the fluctuation spectrum scale-free automatically.
Towards the end of the Planck era, the RG trajectory leaves the asymptotic scaling regime of
the NGFP, the fluctuations ‘freeze out’, and thus prepare the initial state for the subsequent
classical evolution. They lead to a Harrison-Zeldovich like CMB spectrum with a spectral
index of ns = 1 plus small corrections [189, 190, 209, 216].
Here the absence of propagating gravitational waves at high scales could come into play
as follows. At the end of the Planck epoch the geometry fluctuations get imprinted on the (by
then essentially classical) spacetime metric and the matter fields. The imprints then evolve
classically, and ultimately, at decoupling, get encoded in the CMB. Now, a priori the frozen-in
geometry perturbations present at the end of the Planck era (k ≈ mPlanck) would affect the scalar
and the radiative (‘tensor’) parts of the metric alike. If, however, there do not yet exist physical
radiative excitations at this scale, or they are suppressed, then one has a natural reason to expect
that in real Nature the CMB tensor-to-scalar-ratio should be smaller than unity. The power in
the tensor modes is suppressed relative to the scalar ones since by the time the Universe leaves
the fixed point regime gravitational waves cannot propagate yet, the relevant scales being in the
range m UVPlanck > k > kcrit.
12.4 Summary
Since the early investigations of the Einstein-Hilbert truncation it was clear that a subset of its
RG trajectories contain a long classical regime at low scales in which Gk and λ̄k are constant
to a very good approximation; from these single-metric calculations it appeared, however, that
in the adjacent semi-classical regime at slightly larger scales the Newton constant decreases
immediately, thus rendering the anomalous dimension η ≡ k∂k lnGk negative. Even though at
the endpoint of the separatrix, for example, we have λ̄ = 0 and so the effective field equations
19For a different approach to asymptotically safe inflation see [299] and [233].
420 Chapter 12. Propagating modes in Quantum Gravity
admit Minkowski space as a solution, the quantized metric fluctuations on this background,
the gravitons, would have unexpected properties, being more similar to gluons than to photons.
However, in this chapter we provided evidence from two independent bi-metric analyses, one
based on part II, which indicate that this is actually not the case. Between the strictly classical
(η = 0) and the fixed point regime (η < 0) there exists an intermediate interval of scales with a
positive anomalous dimension. Those RG trajectories which have a positive cosmological con-
stant in the classical domain possess two regimes displaying a negative anomalous dimension,
one at Planckian, and the other on cosmological scales. At least in the former the existence of
propagating gravitons seems questionable, and we proposed a natural interpretation of the per-
tinent physical, non-propagating, but gravitating hµν excitations as a form of Planckian ‘dark
matter’.
13.1 Introduction
13.2 Global properties of the boundary sector
The running UV attractor in the ∂ (0)-sector
The UV behavior of the boundary sector
The IR behavior of the boundary sector
The semiclassical regime
13.3 Variational problems
Effective field equations and stationary
points
Variational principle in presence of a bound-
ary
13.4 Counting field modes
Split-symmetry and ∂M= 0/
Euclidean space with non-vanishing bound-
ary
Thermodynamics of the Schwarzschild black
hole
13.5 Conclusion
13. THE GIBBONS-HAWKING-YORK BOUNDARY TERM
C Section 13.4 is a condensed version of subsection 4.3 in [166]. Parts of section 13.4 and
13.5 can be found in [166].
Previous explorations of the Asymptotic Safety scenario in Quantum Einstein Gravity (QEG)
by means of the effective average action and its associated functional Renormalization Group
(RG) equation assumed spacetime manifolds which have no boundaries. In ref. [166] we took
a first step towards a generalization to non-trivial boundaries, restricting ourselves to action
functionals which are at most of second order in the derivatives acting on the metric. The
beta-functions considered in [166] were derived in a single-metric calculation as well as for a
bi-metric matter induced truncation in which the gravitational RG evolution is generated by a
coupled scalar matter sector.
In this chapter we are going to analyze truncated actions with running boundary terms for
full fledged QEG within the bi-metric Einstein-Hilbert truncation augmented by a scale depen-
dent Gibbons-Hawking-York surface term of the background metric, see part II of this thesis.
Since in the current setting the RG evolution of the boundary terms is purely induced by
the bulk contributions, the single-metric and bi-metric beta-functions associated to the boundary
invariants structurally agree, however with gsmk and λ
sm
k replaced by their dynamical counterparts
gDk and λ
D
k . It turns out that
We find that the bulk and the boundary Newton constant, pertaining to the Einstein-Hilbert
and Gibbons-Hawking-York term, respectively, show opposite RG running; proposing a scale
dependent variant of the Arnowitt-Deser-Misner (ADM) mass we argue that the running of
both couplings is consistent with gravitational anti-screening. We discuss the status of the ‘bulk-
boundary matching’ usually considered necessary for a well defined variational principle within
the functional RG framework, and we explain a number of conceptual issues related to the ‘zoo’
of (Newton-type, for instance) coupling constants, for the bulk and the boundary, which result
from the bi-metric character of the gravitational average action. In particular we describe a
simple device for counting the number of field modes integrated out between the infrared cutoff
scale and the ultraviolet. This method makes it manifest that, in an asymptotically safe theory,
422 Chapter 13. The Gibbons-Hawking-York boundary term
there are effectively no field modes integrated out while the RG trajectory stays in the scaling
regime of the underlying fixed point. As an application, we investigate how the semiclassical
theory of Black Hole Thermodynamics gets modified by quantum gravity effects and compare
the new picture to older work on ‘RG-improved black holes’ which incorporated the running
of the bulk Newton constant only. We find, for instance, that the black hole’s entropy vanishes
and its specific heat capacity turns to zero at Planckian scales, indicating a stabilization of the
thermodynamical properties when approaching the ultraviolet.
13.1 Introduction
In this chapter we conclude the analysis of the beta-functions obtained in part II of this thesis,
by focusing on the boundary sector of theory space. One of the ideas to study a bi-metric full
fledged truncation of Quantum Einstein Gravity (QEG) including surface terms, is the impor-
tance of spacetime boundaries in cosmology, in particular the evolution of the Universe [86].
We shall be thus interested in the Renormalization Group (RG) evolution of scale dependent
gravitational actions which include surface terms and extend our previous single-metric results
by the bi-metric structure of the bulk sector. Concretely we will use the results of part II relying
on the effective average action to formulate a diffeomorphism invariant and, most importantly,
background independent coarse graining flow on the ‘theory space’ of action functionals for
the metric [122]. While in the past this approach was limited to the quantization of gravity on
spacetimes M without a boundary1the bi-metric-bulk – pure-background-boundary truncation
ansatz considered in this thesis comprises a total of six independent basis invariants of which
four are defined on the bulk and two on the boundary. From the corresponding ‘running’ coef-
ficients, only the dynamical couplings of the bulk sector, i.e. the D-coefficients, enter the RHS
of the FRGE. Since it is the Hessian operator of the EAA that appears under the functional
trace on the RHS, all coefficients associated to basis invariants of less then second order in the
fluctuation fields do not affect the RG evolution at all. In the level-language in which Γk[ϕ ;Φ̄]
is expanded in powers of ϕ , this property translates into the vanishing of all level-(0) and level-
(1) couplings in the non-canonical part of the RG flow. This has a very practical implication
resulting in a differential system that exhibits a hierarchy among the couplings. For the present
truncation, the dynamical sector, gD and λD, completely decouples and significantly influences
the RG behavior of the remaining four coefficients related to level-(0) bulk and boundary invari-
ants. In detail, we have the following hierarchy structure for the six beta-functions:
λD gD
g(0) g∂ (0)
λ (0) λ ∂ (0)
(13.1)
For the present investigation another consequence of eq. (13.1) is the mutual independence
of the (g(0), λ (0)) and (g∂ (0), λ ∂ (0)). Thus any result derived in either the (0)- or ∂ (0)-sector
is not affected by the value and properties of the other one. In the previous chapters of this
thesis, we exploited this fact to study bi-metric results independently of the boundary sector.
While in general, it is very interesting to investigate split-symmetry, Background Independence,
and related issues also for the boundary terms, the present truncation is unable to disentangle
the ∂ (0)-, ∂ (1)-, ∂ (2)-, . . . , -sectors. In future work we want to incorporate this separation
extending the present truncation by a dynamical Gibbons-Hawking-York functional. Therefore,
in part II of this thesis we already paved the ground for this future project by keeping the
1See however [300] for an early perturbative calculation of the induced surface term in 2+ε dimensional gravity.
13.2 Global properties of the boundary sector 423
discussion and derivation very general. It will be very interesting to compare the results from
the previous chapters of the bulk sector with its boundary counterpart.
Our focus in this chapter is thus on the conceptual part of boundary terms within the FRGE,
which from the cosmological point of view is a very important generalization of spacetime
manifolds. Hence, we now follow the second branch of eq. (13.1) and consider the D-∂ (0) part
of the differential system, comprising the RG evolution of the four couplings gD, λD, g∂ (0), and
λ ∂ (0). For the same reasons the previous sections ignored the boundary couplings, we will now
omit the discussion on level-(0) bulk couplings, which is treated extensively in chapter 10.
We start this chapter by investigating the general (global) properties of the RG flow in the
boundary sector. We detect a similar running UV attractor for ∂ (0) that gives rise to the non-
Gaußian fixed point in the ultraviolet which is relevant for the Asymptotic Safety conjecture.
For more details and its properties, we refer to chapter 9.
In the next step, we consider the variational problem in the context of the EAA. While
the Gibbons-Hawking-York term was initially introduced to recover Einstein’s field equations
in the presence of spacetime boundaries, from the Functional Renormalization Group (FRG)
perspective there is nothing special about this invariant. In particular the classical matching
technique is now subject to the RG evolution and will in general be spoiled.
Next, we evaluate the Ck-function of chapter 11 for the average actions containing bound-
ary contributions. In particular we study the Euclidean Schwarzschild solution and derive its
thermodynamical quantities, which now become k-dependent.
Finally, we conclude this chapter by summarizing the results in section 13.5.
13.2 Global properties of the boundary sector
The beta-functions for the D-couplings are very involved, containing non-polynomial depen-
dencies of gD and λD due to their appearance in the Hessian operator under the functional trace.
On the other hand, the differential equations for the boundary couplings are quite trivial, once
some values for gD and λD are inserted. Any RG solution obeys the following two relations
∂ ∂ (0) = β ∂ (0)( ∂ (0),λ ∂ (0); ) = 2 ∂ (0) + ∂ (0)( D,λD)( ∂ (0)tgk g gk k k gk B1 gk k g )
2
k (13.2a)
∂ λ ∂ (0) = β ∂ (0)( ∂ (0),λ ∂ (0); ) =−3λ ∂ (0) + ∂ (0)( D,λD) ∂ (0) + ∂ (0) ∂ (0) ∂ (0)t k λ gk k k k A gk k gk B1 (g
D
k ,λ
D
k )gk λk
(13.2b)
The explicit form of the functions ∂ (0)( ) ≡ ∂ (0)B k B (gD,λD1 1 k k ) and A∂ (0)(k) ≡ A∂ (0)(gD,λDk k ) can be
deduced from the beta-functions in chapter 8. For our purpose, it suffices to observe that these
functions are independent on the boundary couplings.
The remaining structure of the beta-functions is quite trivial, containing at most quadratic
orders in the couplings. Further notice, that in agreement with the hierarchy (13.1), the beta-
function of g∂ (0) does not depend on λ ∂ (0).
In the following we draw some conclusions about the global picture of the boundary sector,
very similar to those obtained for the background couplings in chapter 10. This is due to the
fact, that except for the explicit form of ∂ (0)A∂ (0), B B B1 or A , B1, both sectors satisfy the same class
of differential equations. We focus on d = 4, even though most of these properties remain valid
beyond.
13.2.1 The running UV attractor in the ∂ (0)-sector
Once we have chosen a solution k 7→ (gDk ,λDk ) in the dynamical sector, the differential equations
for ∂ (0)gk and λ
∂ (0)
k become non-autonomous, with explicit k-dependences introduced by
∂ (0)
B1 (k)
and A∂ (0)(k). Hence, the invariance of the beta-functions under translations in the RG-time
424 Chapter 13. The Gibbons-Hawking-York boundary term
t ≡ ln(k/k0) gets broken and the quantitative and possibly qualitative structure of the phase
diagram will change with k.
Nevertheless, considering deformation invariant properties of the RG flow, we will be able
to deduce the general picture of the boundary sector, again by means of k-dependent analogues
of fixed points. To search for these trivial points of the RG flow is actually a very common
technique in mathematics to understand the underlying differential system. Thus, let us consider
the set of solutions for which the beta-functions vanish:
( ) ( )
β ∂ (0) g∂ (0)( ),λ ∂ (0)( ); = 0 and β ∂ (0) ∂ (0)g • k • k k λ g• (k),λ
∂ (0)
• (k);k = 0 (13.3)
Here, we treated the dynamical sector as k-dependent parameters and thus solutions to eq. (13.3)
inherit a k-dependence. In quite general terms, if ∂ (0)B1 (k) 6= 0, the conditions of eq. (13.3) are
satisfied by the following two types of solutions
A. From the structure of the beta-functions it is apparent that the trivial solution, ∂ (0)g• = 0=
λ ∂ (0)• , always exists, independent of the dynamical sector and thus of k. This k-stable
solution represents a special point on the separating line g∂ (0) = 0, which divides the RG
flow in the boundary sector into two classes with either positive and negative g∂ (0) for all
k.
B. Furthermore there is a second, this time non-trivial solution which is explicitly k-dependent:2
∂ (0)( ) =−2/ ∂ (0)g k B (k) and λ ∂ (0)(k) =− 2A∂ (0)• 1 • 3 (k)/B
B
1(k) (13.4)
Since the underlying equations are of quadratic order in the couplings, this solution can be
associated to the non-Gaußian fixed point for asymptotically safe RG trajectories in the
limit → ∞. We denote this non-trivial solution ∂ (0)( ) ≡ ( ∂ (0),λ ∂ (0)k Attr k g• • ) as the running
UV-attractor of the boundary sector, for reasons which become clear in a moment.
The non-trivial solution Attr∂ (0) ≡ Attr∂ (0)(k) is very interesting, for it describes the evo-
lution of the later non-Gaußian fixed point and in addition decisively influences the RG flow
of the entire boundary sector. In fig. 13.1 we depict its k-dependent position for a typical
IIIa-trajectory of the D-sector corresponding to the class of solutions which are assumed to be
realized in Nature. The structure we observe for the UV part of this particular solution extends
to all asymptotically safe trajectory without any significant modification, since the D-couplings
approach the same coordinate, the fixed point values of NGD+-FP. In contrast to the bulk cou-
plings, notice that asymptotically safe trajectories evolve in the negative half-plane of Newton’s
coupling of the boundary sector, i.e. g∂ (0) < 0. This is reflected by the negative fixed point value
∂ (0) ∂ (0)
g∗ for NG− ⊕NGD+-FP and is also seen in fig. 13.1.
In the following, we comment on the general properties of Attr∂ (0)(k), valid as long as
∂ (0)
B1 (k) 6= 0.
A. Since Attr∂ (0) is not subject to the FRGE and does not correspond to an RG solution, the
running attractor can in principle disappear, change sign in its g∂ (0)-coordinate, or violate
any other condition which exact solutions of the FRGE have to satisfy.
B. Fig. 13.2 shows that the considered D-trajectory evolves in a very extended region of
theory space where the sign of the Attr∂ (0)-coordinates is preserved. The situation of
k→ 0 is difficult to answer for type IIIa trajectories, for the Einstein-Hilbert truncation in
the linear field parametrization is known to be plagued by a singularity of ηD for λD → 1/
2. We will postpone the discussion on the IR behavior of Attr∂ (0) to subsection 13.2.3.
However, notice that fig. 13.2 also indicates regions where Attr∂ (0) is situated in the upper
g∂ (0)-half-plane of the boundary sector. RG trajectories that emanate in the ultraviolet
2Notice that in comparison with the background result, λ ∂ (0)• (k) and λ B• (k) have different pre-factors of−(d−2)/
(d−1)|d=4 =−2/3 and −(d−2)/d|d=4 =−1/2, respectively. This is a result of their different canonical scaling.
13.2 Global properties of the boundary sector 425
g∂ (0) gD
λ ∂ (0) λD
(a) Running UV-attractor ∂ (0)Attr (k) (b) Type (IIIa)D trajectory
Figure 13.1: The position of the running UV attractor of the boundary sector, Attr∂ (0)(k) is
shown in the left panel for typical type IIIa trajectory k 7→ (gD,λD) as given in the right panel.
Though the g∂ (0) coordinate assumes very large values on intermediate scales 0 < k < ∞, it
always stays finite. For k→ ∞ (the direction of increasing RG-time t = ln(k/k0) is indicated by
∂ (0)
the transition from red to blue) Attr∂ (0)(k) approaches theNG D− ⊕NG+-FP coordinates. Hence,
the existence of the UV-limit for Attr∂ (0)(k) strengthens the evidence for the Asymptotic Safety
conjecture.
from NGD ∂ (0)+-FP, where g• is negative , and enter a region of positive
∂ (0)
g• at finite 0< k<
∞, experience a significant change of the topological landscape in the boundary-sector.
This is in particular the case for asymptotically safe type Ia trajectories being trapped in
the lower g∂ (0)-half-plane for all k, while in the IR they cross the boundary of a region for
which ∂ (0)g• turns positive.
C. While the ∂ (0)g• -coordinate assumes almost any negative value, the corresponding λ
∂ (0)
• -
coordinate changes only within a very small range around its fixed point value λ ∂ (0)∗ ≈
1.204. However, compared to its background counterpart AttrB, it never touches the
boundary of theory space and is always well-defined for a very large class of type IIa and
IIIa trajectories.
D. The name of a running UV-attractor is in fact justified. For each k, Attr∂ (0)(k) attracts
the RG flow in the boundary sector towards its current position, at least for all tra-
jectories in the same g∂ (0)-half plane. This can be seen by linearizing the boundary-
system (13.2) about ( ∂ (0) ∂ (0)g• ,λ• ) and deduce the k-dependent analogue of a stability ma-
trix, B( ∂ (0),λ ∂ (0)g• • ;k):
( )
−2 0
B(g∂ (0),λ ∂ (0)• • ;k) = (13.5)2A∂ (0)3 (k) −3
Since for each k both eigenvalues of(B, θ1;∂• a)nd θ2;∂•, are found to be positive we may
conclude that ∂ (0) ∂ (0)at all scales the point g• ,λ• is UV-attractive in both ∂ (0)-directions.
Independent on the imposed initial conditions for the ∂ (0)-couplings, RG trajectories
considered in the boundary sector of theory space, i.e. solutions 7→ ( ∂ (0),λ ∂ (0)k gk k ), are
always pulled towards Attr∂ (0) for k→ ∞.
Next, we consider the limits k → ∞ and k → 0 of Attr∂ (0)(k) and thus the topological structure
of the boundary level-(0) phase diagrams for the ultraviolet and infrared, respectively.
426 Chapter 13. The Gibbons-Hawking-York boundary term
gD gD
λD λD
(a) The coordinate g∂ (0) (b) The coordinate λ ∂ (0)
Figure 13.2: In the left (right) plot, we compute the coordinate value of Newton’s (cosmological)
coupling of Attr∂ (0)(gD,λD) for several points of the D-couplings. Negative (positive) values of
∂ (0) or λ ∂ (0)g• • are indicated by dark blue (light orange) regions. In the most interesting part of
theory space, associated to asymptotically safe RG trajectories of type-IIIa (as e.g. the white
superimposed curve with gD, λD > 0), the running UV-attractor is trapped in the region g∂ (0) < 0
and λ ∂ (0) > 0. For type Ia with a negative λD at finite k, the position Attr∂ (0) approaches the
boundary of theory space and in fact its ∂ (0)g• coordinate changes sign. On the left of each
plot, the sharp jump in the position of Attr∂ (0) is a truncation artifact associated to a pole in the
beta-functions of the D-couplings.
13.2.2 The UV behavior of the boundary sector
From the Asymptotic Safety point view, the most interesting part of the RG evolution is encap-
sulated in the UV regime where asymptotically safe RG trajectories have to enter the realm of a
suitable fixed point. In previous exploration of the Asymptotic Safety program studying space-
time manifolds without boundary, a very promising candidate in the shape of a non-Gaußian
fixed point was discovered [122, 123, 135, 147, 191]. This candidate corresponds to NGD+-FP
which from the discussion in chapter 9 is also the most stable one w.r.t. the ghost sector depen-
dence. The results obtained in chapter 9 further showed that NGD+-FP extends to the bi-metric
s∫ubs√pace of T∫ st√udied in this thesis which contains the lowest order of boundary invariants
∂M H̄ and ∂M H̄K̄.
In the last subsection, we discovered a running UV-attractor in the boundary-sector, Attr∂ (0),
that determines the RG evolution of the coefficients multiplying the two boundary invariants.
Its properties are essential to understand the global flow within the boundary part of theory
space and its UV limit for asymptotically safe RG trajectories coincides with the non-Gaußian
∂ (0)
fixed point NG− ⊕NGD+-FP. That is, for all RG solutions k 7→ (gD,λD ∂ (0) ∂ (0)k k ,gk ,λk ) which are
compatible with the Asymptotic Safety requirement, we have
lim Attr∂ (0)(gDk ,λ
D
k ) = Attr
∂ (0)(gD,λD)≡ (g∂ (0),λ ∂ (0)∗ ∗ ∗ ∗ ) (asymptotically safe)
k→∞
While an extensive discussion on the full set of fixed points an their UV-critical hypersurfaces
can be found in chapter 9, we here focus only on the results related to GD-FP and NGD+-FP in
d = 4. The latter is – as mentioned above – the most interesting fixed point from the Asymptotic
13.2 Global properties of the boundary sector 427
Safety point of view.3
d = 4 gD λD ∂ (0)∗ ∗ g∗ λ
∂ (0)
∗
∂ (0)
NG− ⊕GD-FP 0 0 −18.8 1.33
G∂ (0)⊕GD-FP 0 0 0 0 (13.6)
∂ (0)
NG− ⊕NGD+-FP 0.70 0.21 −2.14 1.2
G∂ (0)⊕NGD+-FP 0.70 0.21 0 0
The most important observation is the negative sign of ∂ (0)g∗ that we already mentioned above.
Conceptually, there is nothing wrong with g∂ (0) < 0, which is different for its gD counterpart.
!
Nevertheless, it has some severe consequence for the classical matching condition, where g(1) =
g∂ (1) is chosen in order ot have a well-posed variational principle that leads to the Einstein field
equations of General Relativity. While for the present truncation we only resolve g∂ (0) which in
general can differ from g∂ (1), in particular, a negative g∂ (0) would imply that split-symmetry to-
gether with Asymptotic Safety cannot be reconciled with with the classical matching condition.
We come back to this issue in the next section.
Now, let us linearize the GR flow about the latter two fixed points of eq. (13.6). We find the
following set of their associated critical exponents:
G∂ (0)⊕NGD+-FP: θ1,2 = 3.60±4.28ı , θ3 =−2 , θ4 = 1 (13.7a)
∂ (0)
NG− ⊕NGD+-FP: θ1,2 = 3.60±4.28ı , θ3 = 3 , θ4 = 2 (13.7b)
Whereas in the case of the G∂ (0)⊕GD-FP the g∂ (0)-direction is classified to be irrelevant, for
∂ (0)
the doubly non-Gaußian fixed point, NG− ⊕NGD+-FP, all four scaling fields are relevant, i.e.
the corresponding values of ℜ(θ) are all positive. Thus, the dimensionalities of the respective
UV critical manifolds are dimS (G∂ (0)UV ⊕ ∂ (0)GD-FP) = 3 and dimSUV(NG− ⊕GD-FP) = 4,
demanding a total of 3 or 4 additional initial conditions, respectively, in order to uniquely spec-
ify an RG solution. Notice that the critical scaling behavior of the boundary part is found to be
the same for the present bi-metric and the corresponding single-metric calculation [166].
13.2.3 The IR behavior of the boundary sector
Now, let us briefly comment on the k → 0 limit of the running UV attractor of the boundary
sector. Since we shown that it is indeed UV-attractive in the g∂ (0)- and λ ∂ (0)-direction, there is
at most a single RG solution which starts at Attr∂ (0)(0) for k = 0. Following the flow in the
direction of decreasing k, i.e. towards the infrared, Attr∂ (0)(k) has to be understood as a running
IR ‘repeller’. Still, it determines the non-trivial structure of the phase diagram, however now
RG solutions depart from the current position of Attr∂ (0).
In order to have a clear as possible picture of the global behavior of RG solutions in the
boundary sector, we have to consider the influence of Attr∂ (0) on asymptotically safe trajectories
also in the IR. The D-sector, which drives the k-dependence of Attr∂ (0)(k), can be classified
into three types of asymptotically safe RG trajectories which differ in the IR behavior of the
cosmological constant λD, see fig. 13.3. The position of Attr∂ (0)(k) for these three cases is listed
in the following.
3Here, and in what follows all numerical computations, if not stated otherwise, are based on the optimized shape
function.
428 Chapter 13. The Gibbons-Hawking-York boundary term
gD
Type Ia
Type IIIa
Type IIa λD
Figure 13.3: The schematic structure of the phase portrait on gD-λD including the three types of
asymptotically safe RG trajectories. As always, the arrows point in the direction of decreasing
k. Thus, classification is based on the IR behavior of λD which either becomes negative, zero,
or positive corresponding to type Ia, IIa (separatrix), or IIIa solutions, respectively.
k→0
Ia The defining property for this type of RG solutions is λDk −−→−∞. If we consider the D-
system for very large negative values of λDk we obtain the following asymptotic solutions
(for gDk > 0):0
2πgD 2
gD = ( k
k
0
k )− −
k−→→0 0 (13.8a)
gDk k
2 k2 +2πk2
0 (0 ) 0
gD k4k( 0− k
4 −6πk4 D
D 0 ( ) 0
λk0 ) −kλk = −
→→0 ±∞ (13.8b)
3k2 gD k2− k2k 0 −2πk20 0
k→0
Notice that this result is also valid for solutions of λDk −−→+∞ (the type-IIIa trajectories
discussed below). The IR behavior of the underlying trajectory is specified by the sign
of λD0 . For type (Ia)
D trajectories we insert solution (13.8) for negative λDk and obtain the0
corresponding IR position of Attr∂ (0)(k):
( )
3π 4
limAttr∂ (0)(k) = , (type Ia) (13.9)
k→0 2 3
Notice that opposite to the ultraviolet part of the trajectories Attr∂ (0)(k) now resides on
the upper g∂ (0)-half plane and thus has no impact on the IR evolution of asymptotically
safe type Ia trajectories.
IIa Due to the negative critical exponent in the gD direction, there is a unique asymptotically
safe RG solution (up to rescaling) that interconnects NGD+-FP with the Gaußian fixed
point GD-FP. It defines the dividing line between the regions of type Ia and IIIa trajec-
tories and is therefore also denoted separatrix. In the vicinity of GD-FP we can linearize
the RG flow and determine the separatrix solution of the D-couplings:
2 3gD = gDk k (k/k ) and λ
D
k = g
D
k (k/k
2)
0 8π 0
When this solution is inserted into the differential equations of the boundary-sector, we
can solve for the zeros of the corresponding beta-functions yielding the IR limit of
Attr∂ (0)(k):
( )
4
limAttr∂ (0)(k) = −6π, (separatrix) (13.10)
k→0 3
13.2 Global properties of the boundary sector 429
In this case, as is also apparent from fig. 13.2, ∂ (0)g• remains negative.
IIIa The problem in studying the IR behavior of type IIIa trajectories for the Einstein-Hilbert
truncations in the linear field parametrization is the occurrence of a singularity in the
beta-functions of the D-sector for λD close to 1/2. Extensions to non-local truncations
and exponential type field parametrization suggests that this pole is in fact an artifact
of truncating theory space and disappears for exact solutions [146, 175]. Thus, for the
present discussion we ignore this singularity and assume we can prolong the type IIIa
trajectories up to λD →+∞. Then, we can make use of the IR solution (13.8) and derive
the Attr∂ (0) position for k = 0:
( )
∂ ( ) 3π 4limAttr 0 (k) = , (type IIIa) (13.11)
k→0 2 3
It turns out that independent on the sign of λD in the limit λD → ±∞, the running UV-
attractor of the boundary sector approaches the same position in theory space.
Notice that whenever gD approaches zero, which is the case for the IR limits of the RG trajec-
tories and in particular for the Gaußian fixed point solutions, the value of λ ∂ (0)• assumes 4/3,
independent on λD.
13.2.4 The semiclassical regime
In between the infrared and ultraviolet limits we have just discussed, the dynamical RG solu-
tions pass a region in theory space which is best described by the semiclassical approximation.
Hence, for some intermediate scales 0 < k < ∞, we can easily solve the D-sector and study
the corresponding boundary equations within this semiclassical regime. As already mentioned,
‘semiclassical’ refers to an approximation in which the RG equations are reduced to their one
loop form. Thereby, the ‘improvement’ terms proportional to ηD and η ∂ (0) in the beta-functions
are neglected and evaluating all threshold functions are evaluated at zero cosmological constant
λD. Thus, approximating the anomalous dimensions η in this way, we obtain
ηD =−( ∂ (0)d−2)ω D ∂ (0) ∂ (0)d g , η =−(d−2)ωd g (13.12)
Here, ωDd and ω
∂ (0)
d denote numerical constants, independent on k, that classify the respective
semiclassical behavior. Using η ≡ ∂t lnGk the differential equations of the dimensionful New-
ton couplings admit the following exact solutions:
GI
GI 0k = − (13.13)1+ω I I d 2dG0 k
If the dimensionless Newton coupling gI = GI kd−20 0 is sufficiently small in the semiclassical
regime, we can expand eq. (13.13) for small GI d−20 k , which yields:
[ ]
Gk = G 1−ω d−20 d G0 k + · · · (13.14a)
[ ]
∂ (0) = ∂ (0)Gk G0 1−ω
∂ (0) ∂ (0)
d G0 k
d−2+ · · · (13.14b)
Along the same lines we next solve for the dimensionful running cosmological constants in the
same regime. Since λ̄ and λ̄ ∂ (0) show different canonical scaling, the respective semiclassical
approximations develop different k-dependencies:
λ̄k = λ̄0 + νd G
d
0 k + · · · (13.14c)
λ̄ ∂ (0) ∂ (0) ∂ (0) ∂ (0) d−1k = λ̄0 + νd G0 k + · · · (13.14d)
430 Chapter 13. The Gibbons-Hawking-York boundary term
Here , ∂ (0)G0 G0 , λ̄0, and λ̄
∂ (0)
0 are free constants of integration, and the dots represent higher
orders in G kd−20 and G d−2∂ (0)0 k , respectively.
In the following we consider the case of d = 4 and employ the ‘optimized’ threshold func-
tions. For a more detailed discussion and results for general spacetime dimensions d, we refer
to chapter 8 of this thesis. We will focus on the qualitative structure of the semiclassical results
and consider the signs of ω Id and ν
I
d coefficients, which in fact agree for any suitable shape
function. In detail, we obtain
5 3
ωD4 =− > 0 and νD4 =+ > 0 (13.15a)12π 8π
ω (0)
11 1
4 =+ > 0 and ν
(0)
4 =+ > 0 (13.15b)6π 8π
ω ∂ (0) − 1 24 = < 0 and ν
∂ (0)
4 =− < 0 (13.15c)6π 9π
Here, we included also the level-(0) bulk couplings, for it is very instructive to compare the
relative signs of the full sector. In fact, as discussed in chapter 10 the semiclassical results of
the level-(0) couplings agrees with the ones obtained by a single-metric calculation. Hence,
knowing ω (0)4 = ω
sm
4 and ν
(0) sm
4 = ν4 holds true, we can directly compare the present results of
the bi-metric truncation with the previous one derived in ref. [166]. Based on eq. (13.15) we
make the following observations:
A. Within the approximation (13.14), the bulk Newton constant GDk and the boundary coun-
terpart ∂ (0)Gk have both negative coefficient ω4. Hence, in contrast to the single-metric cal-
culation and the level-(0) results, GDk increases for increasing k. Nevertheless, away from
the semiclassical regime this behavior changes when leaving the vicinity of the Gaußian
fixed points and we re-obtain the familiar picture of gravitational anti-screening [122],
see chapter 10 and 12. On the other hand, ω ∂ (0) < 0 does not generally imply that ∂ (0)4 Gk
also increases towards the UV. In the boundary sector, there is a priori no experimen-
tal observation that would favor a positive ∂ (0)Gk and in fact the condition of Asymptotic
Safety requires ∂ (0)Gk < 0. Thus, albeit the different signs of ω
(0)/∂ (0)
4 , the semiclassical
behavior of asymptotically safe trajectories agrees on level-(0).
B. It can be checked that the negative signs of ωD4 and ω
∂ (0)
4 in the semiclassical regime, are
robust with respect to changes of the cutoff shape function.
C. In fact, while there is sign change of the anomalous dimension in the dynamical sector,
the results on the boundary, η ∂ (0), extends beyond the semiclassical approximation. In
d = 4 and for the ‘optimized’ threshold functions (8.4) we obtain
∂ (0) 1− 5
∂ ( ) g
D D
η 0 = 4
η +8λ
− (13.16)3π (1 2λD)
This result is exact in the sense that the improvement term ∝ ηD on the RHS of the FRGE
was retained and λD has not been set to zero in the arguments of the Φ’s. It is obvious
that the expression (13.16) is always negative in the regime of interest, g∂ (0) < 0, ηD < 0,
and λD ∈ [0,1/2).
D. As a result of the previous remarks, we can conclude that within this truncation we
find that the equality (0) = ∂ (0)Gk Gk is inconsistent with Asymptotic Safety as long as split-
symmetry is preserved. Hence, at least on the basis of the here derived beta-functions we
can say that different to the previous single-metric results there is no point in the asymp-
totically space branch of theory space for which the average action Γk has the desired
well posed variational principle by simply including a Gibbons-Hawking-York boundary
term. This also includes the ‘physical point’ k= 0. We are going to discuss this mismatch
13.3 Variational problems 431
in the next section, but mention only that there are general arguments indicating that the
technique introduced by Gibbons, Hawking, and York is in general insufficient and has
to be replaced by different surface corrections, see for instance [301, 302].
The well known decrease of the bulk Newton constant in the UV has been interpreted as an
indication for the anti-screening character of QEG at short distances [122]. In the bi-metric
setting, we have already seen in chapter 10 that the dynamical sector enters a regime of positive
ηD and thus has a screening property in the semiclassical regime. While this changes beyond
this region, the level-(0) boundary Newton’s coupling has the same sign for all k similar to its
bulk counterpart.
The analysis of the global properties for the RG flow in the boundary sector revealed that
opposite to the bulk part, Asymptotic Safety requires a negative g∂ (0). Since once we start
with an initial negative Newton coupling g∂ (0)(k0) < 0 there is no solution in theory space that
crosses the g∂ (0) = 0 and thus would lead to a positive g∂ (0) on some intermediate scale. From this
perspective the Gibbons-Hawking-York technique is incompatible with the two requirements of
Asymptotic Safety and Background Independence, at least on the results obtained within the
present truncation. In the next section, we briefly comment on the classical idea of introducing
boundary terms to obtain Einstein’s field equations form a well-posed variational principle.
13.3 Variational problems
Classical General Relativity is defined by Einstein’s field equation which can also be derived
from the requirement that the Einstein-Hilbert functional,
[ ] ∫1 √
S dEH gµν =− d x gR (13.17)
16πG M
becomes stationary, provided the spacetime manifold, M, has no boundary. In order to general-
ize the variational principle to very interesting case of spacetimes with a non-empty boundary
∂M the problem arises that SEH responds to a change δgµν = g+ vµν of the metric, with vµν
vanishing on ∂M, by producing a certain surface term, over and above the desired ‘bulk’ term
containing the Einstein tensor Gµν :
(∫ √ ∫ √ )1
d S = ddx g Gµνv EH v
d−1 αβ
µν + d x H H n
µ∂µ vαβ (13.18)16πG M ∂M
In eq. (13.18) we have introduced the normal vector field nµ related to the embedding of the
boundary in M and the tensor Hαβ which is related to the induced metric on the boundary.
Under Dirichlet conditions variations of the metric do not affect its boundary value; hence,
by assumption, vµν vanishes on ∂M. This implies that the derivatives of vµν in d∣irections
tangential to ∂M will also vanish. On the other hand, its normal derivative nµ∂µ v ∣αβ is in∂M
general unrestricted and thus non-zero. For this reason the surface term in (13.18) remains for
general vµν and the condition of stationarity dvSEH = 0 is not equivalent to Einstein’s (vacuum)
field equation Gµν = 0.
Notice that these kind of difficulties are not uncommon in Lagrangian or Hamiltonian sys-
tems. In order to re-establish the standard variational problem, one includes an additional
boundary contribution to the action whose variation exactly cancels the unwanted surface term
originating from the bulk part.
In General Relativity, the most popular proposal4 for a surface correction with this property
is the Gibbons-Hawking-York term [79],
∫
1 − √S = dd 1GHY x H (−2K) (13.19)
16πG ∂M
4For recent proposals of different surface corrections see [301, 302]; for a general discussion see [76, 197, 198].
432 Chapter 13. The Gibbons-Hawking-York boundary term
where K denotes the trace of the extrinsic curvature of the embedded boundary.5 If one applies
the variation on the sum SEH + SGHY we find that the obstructing surface term on the RHS of
(13.18) is precisely compensated by the variation of the Gibbons-Hawking-York action of eq.
(13.19). Hence, the stationary points of the total action are exactly those metrics satisfying
Einstein’s (vacuum) field equation: d µνv (SEH+SGHY) = 0 ⇔ G vµν = 0 for all vµν .
This generalization of GR to spacetimes with boundary has lead to interesting applications
in cosmology. For instance, the Gibbons-Hawking-York term has played an important role in
the Euclidean functional integral approach to black hole thermodynamics [79, 303]. In the lead-
ing order of the semiclassical expansion the black hole’s free energy is given by the ‘on-shell’
value of the classical action functional. Since, for vacuum solutions, Rµν = 0 implies a van-
ishing contribution from the bulk term, the free energy, and hence all derived thermodynamical
quantities such as the entropy, for instance, stem entirely from the surface term. It is important
to ask how this picture presents itself in full fledged quantum gravity [73]. There exist already
detailed investigations within Loop Quantum Gravity [72, 75, 93], for instance [304].
At this point however, one has to be careful in translating these classical techniques into
the framework of Quantum Gravity. Fixing the relative size of coefficients of two independent
basis invariants, as done by the matching of boundary and bulk couplings, will in general be
spoiled by quantum effects. Only if certain symmetry principles or other fundamental algebraic
constraints require the precise matching of the classical conditions, there is a reasonable chance
that we encounter this tuning also on a quantum level. For the classically preferred ratio of
boundary and bulk coefficient there is actually a weak geometric property, namely it turns out
that the heat kernel expansion preserves the relative factor of 2. Whether or not this is sufficient
cannot be answered within the present truncation, for which details of the level-(1) boundary co-
efficients are needed. However, already in the classical description, there are severe constraints
on the reliability of the Gibbons-Hawking-York implementation, especially when spacetime is
non-compact, see ref. [301, 302].
13.3.1 Effective field equations and stationary points
From the FRG point of view, the motivation to study truncations which include the Gibbons-
Hawking-York term are not primary related to the classical variational problem. Rather, when
allowing for spacetimes with non-empty boundary, ∂M =6 0/ , which is very interesting to study
quantum effects in cosmological models, theory space gets extended by an infinite number of
basis invariants defined on ∂M. The FRGE on this enlarged space in general only closes if
we include all field monomials on ∂M which are compatible with the symmetry principle and
the field content, the latter being equipped with a set of suitable boundary conditions. Thus,
the relevance of the Gibbons-Hawking-York term in the EAA approach to QEG is better un-
derstood from a technical perspective, in that it makes its appearance in Γk when employing a
systematic derivative expansion up to second order in D or D̄. It is thus the natural candidate
to start exploring directions in theory spaces associated to ∂M 6= 0/ . For the exact solutions, we
have to include all further boundary terms will also comprises the proposed surface terms in
[301, 302].
In the Effective Average Action approach based on the FRGE classical field equations
get replaced by the running stationarity condition and the running tadpole equation, which in
absence of split-symmetry are not equivalent [166]. In following we present a short summary
of both concepts for vanishing spacetime boundaries, ∂M= 0/ .
5In the literature, SGHY is usually normalized by replacing K→K−K0 in (13.19), withK0 the trace of the extrinsic
curvature tensor appropriate for an embedding of the boundary manifold in flat space. While subtracting K0 does
not change dvSGHY, we shall refrain from it here for reasons to be discussed below.
13.3 Variational problems 433
Running stationarity condition
For the first equation, notice that it is Γ̃k ≡ Γk+∆kS rather than the Effective Average Action,
Γk, for which the k-dependent field-source relationship between sources and expectation fields
gµν = 〈γµν〉, hµν ≡ 〈hµν〉 is established. On an arbitrary theory space involving fluctuation
fields, ϕ , coupled to sources Jr, the corresponding stationary condition evaluated at background
fields Φ̄ reads
δ Γ̃k[ϕ ;Φ̄] ≡ δΓk[ϕ ;Φ̄] δ∆kS[ϕ ;Φ̄]+ = Jr (13.20)
δϕr δϕr δϕr
This equation can be interpreted as a first notion of a scale dependent ‘effective field equation’:
the running stationary point condition of Γ̃k. The complicate character of this equation resides
in its dependence on both the fluctuations, ϕ , and the background fields Φ̄.
In case of full-fledged Quantum Einstein Gravity, here evaluated for vanishing sources, eq.
(13.20) therefore assume the following form
δΓk[h; ḡ] √
+ ḡRkhµν(x) = 0 (13.21)
δhµν(x)
This coupled system yields the solutions hµν(x) ≡ hµν [ḡ](x) as functionals of the fixed but
arbitrary background metric ḡ.
Running tadpole equations
A very special, and interesting class of solutions is obtained by evaluating (13.21) for hµν = 0.
This gives rise to the definition of ‘self-consistent’ backgrounds, which satisfy the running
tadpole equation:
δ ∣∣
Γ s.c. ∣k[h; ḡ
δh (x) k
]∣ = 0 (13.22)
µν h=0
Notice that the Rkhµν term vanishes for hµν = 0. While the functional variation addresses the
fluctuation field h, the solutions are obtained for another field, ḡµν . This implies that in general
the tadpole equation cannot be written as the ḡ-derivative of any diffeomorphism invariant
functional F[ḡ]. Furthermore, the integrability of this system is not automatic, but under special
circumstances solutions can exist, see ref. [137] for further details.
s.c.
R Even though self-consistent backgroundmetrics, ḡk , define k-dependent solutions of the
tadpole equation for hµν = 0, this does not correspond to the idea of small fluctuations
in the perturbative sense. It rather corresponds to restricting the class of theories µ for
which the expectation value hµν = 〈ĥµν〉µ vanishes.
We have thus found another generalization of the classical field equations: a scale dependent
version of the tadpole condition. Though it derives from the running stationary point condition,
it has the significant difference of depending only on the background metric ḡµν and defines the
so-called self-consistent background solutions.
Summary
Both the running stationarity condition and the running tadpole equation generalize the field
equations of classical gravity theory in a way which goes far beyond replacing the Einstein-
Hilbert action SEH[g] with some other diffeomorphism invariant single-metric functional S[g]. In
the first case one deals with field equations involving two metrics ḡµν and ḡµν + hµν ≡ gµν ;
their structure is constrained only by the requirement that they must be representable as the h-
derivative of some invariant functional. In the second case the effective field equations contain
434 Chapter 13. The Gibbons-Hawking-York boundary term
only one metric, namely ḡ, but there is no longer the requirement to be the ḡ-derivative of an
invariant action. Thus the gravitational average action allows for a great variety of potential
modifications of the classical Einstein equations. A purely phenomenological analysis of such
bi-metric actions should therefore be a worthwhile complement to the flow equation studies.
Thus the gravitational average action allows for a great variety of potential modifications of
the classical Einstein equations. A purely phenomenological analysis of such bi-metric actions
should therefore be a worthwhile complement to the flow equation studies.
13.3.2 Variational principle in presence of a boundary
We now come back to the case of manifolds with boundaries, ∂M 6= 0/ and address the problem
of giving a consistent meaning of the variational principle in the presence of boundary terms.
Therefore, consider the truncation ansatz of eq. (5.15) with a bi-metric gravitational effective
action comprising the Einstein-Hilbert term of dynamical and background field as well as a
Gibbons-Hawking-York term of ḡ and further volume contributions.
For the time being, we extend Γk by incorporating a second, dynamical Gibbons-Hawking-
York term and allow for some matter field content, A, in form of an energy-momentum tensor
T µν [A; ḡ]. Thus, the EAA assumes the form (whereby for simplicity we consider only terms up
to quadratic order in h):
∫
1 √ { }
Γ [h; ḡ] = Γ [A; ḡ]+ · ddx ḡ Ḡµν + λ̄ (1) ḡµνk k k −8π
(1)
Gk T
µν [A; ḡ] hµν
∫ 16π
(1)
Gk M
[ ]
+ 1
√
ddx ḡ hρσ
µν
2 Ĥessh Γk[A,h; ḡ] hρσ µν
M( ) ∫
− 1 1
√ ( )
− 1 · dd−1x H̄ n̄µD̄ν − ḡµν n̄ρD̄ h (13.23)
16π (1) ∂ (1)
ρ µν
Gk Gk ∂M
If the ∂M-integral in (13.23) is absent, the running stationary condition, eq. (13.21), reads:
( ) [ ]
(8π (1))−1 µν + λ̄ (1) µν − µν8π (1)G Ḡ ḡ G T µνk k k [A; ḡ] =−Ĥess ρσh Γk[A,h; ḡ] h −Rkhµνρσ
(13.24)
The term on the RHS is reminiscent of Einstein’s field equation, however with k-dependent
coefficients associated to level-(1). The Hessian operator introduces further terms of at least
level-(2) and contributions of the fluctuation field h. Apparently, this equation depends on both
h and ḡ and its solution gives rise to fields of the form hk[A; ḡ].
In order to obtain the running tadpole equation we have to evaluate (13.24) at hµν = 0,
yielding a k-dependent analog of the Einstein field equations:
(1) (1)
Ḡµν [ḡs.c.]+ λ̄ (ḡs.c.)µν = 8πG T µν [A; ḡs.c.k k ] (13.25)
By construction, this equation involves only level-(1) couplings and defines the class of self-
consistent background metrics. Obviously, eq. (13.24) and (13.25) are in general not equivalent.
However, this familiar structure of the effective field equations can only be obtained if the
boundary term in eq. (13.23) vanishes.
Let us consider the running tadpole equation for simplicity. A functional variation of eq.
(13.23) and subsequently setting h = 0, results in the following condition for self-consistent
background solutions:
∫
1 · √
{ }
0= dd µν + λ̄ (1) µν −8π (1)x ḡ Ḡ ḡ G µν
16π (1) k k
T [A; ḡ] vµν
Gk( M ) ∫
− 1 1 1
√ ( )
− · dd−1x H̄ n̄µD̄ν − ḡµν n̄ρ D̄ v (13.26)
16π (1) ∂ (1)
ρ µν
Gk Gk ∂M
13.3 Variational problems 435
This condition has to hold for all tangent vectors vµν of hµν . In order to get rid of the ∂M-
integral, there are basically two possibilities. Either one imposes the matching condition ∂ (1)Gk =
(1)
Gk or one modifies the space of expectation fields, hµν ≡ 〈ĥµν〉, such that the integral van-
ishes.6 We comment on both ideas in the following:
A. Already in a first single-metric and bi-metric matter induced truncation comprising a
Gibbons-Hawking-York boundary term, we observed that the classical matching condi-
tion can only be satisfied at a single value of k [166]. The reason for this is founded in
the opposite sign in the respective anomalous dimensions.
B. In order to fully resolve the issue of the matching condition, we have to resolve the
level-(1) boundary and bulk coefficients. While the present truncation makes a first step
towards this end by lifting the degeneracy in the bulk sector, more involved techniques
are necessary to obtain the same results on the boundary. Nevertheless, if the level-(0)
bulk and boundary results are mainly reliable there is the following constraint on a pos-
sible matching. We have seen that asymptotically safe trajectories are described by the
family of RG solutions with g∂ (0) < 0 and gD ≡ g(1) > 0. Furthermore, the requirement
of Background Independence translates to split-symmetry in the limit k → 0. Hence,
combining these fundamental constraints with the result of the present truncation, we
obtain the subset of theory space containing all candidates of a theory of Quantum Grav-
ity, namely Ttrunc (p) ∂ (p) (p) ∂ (p) (p) ∂ (p)AS& BI ≡ {(g , g , λ , λ ) |g > 0 and g < 0}. That the observation
g∂ (0) < 0 restricts all higher level boundary Newton couplings to be also negative, is a
consequence of split-symmetry at k= 0 and the topological property of theory space that
there is no transition from the negative, gI < 0, to the positive half-plane, gI > 0. From
this it follows that if the present level-(0) results qualitatively remain valid even in the
exact case, the matching condition for the coefficients of the boundary term in eq. (13.26)
would be inconsistent with the joined assumption of Asymptotic Safety and Background
Independence. This would then completely rule out the classical matching technique.
C. The second option can be understood as restricting the class of possible action functionals,
S[g], and thus theories µ , to only those which yield(expectation values)of the fluctuation
fields hµν ≡ 〈ĥµν〉µ which simultaneously satisfy n̄µD̄ν − ḡµν n̄ρD̄ρ hµν |∂M = 0 and
Dirichlet constraints. The former condition cancels the boundary term in eq. (13.26)
(independent of its)coefficient. It is reminiscent of the gauge condition,
ρσ
F[ḡ]µ hρσ =
D̄ρD̄σ −ϖ ḡρσ D̄2 hρσ (similar for vµν ) however with ϖ =−1 instead of the harmonic
choice ϖ = 1/2 utilized in this work. In fact rewriting the surface term into a bulk contri-
bution, we obtain:
∫ √ ∫
dd−
( ) ( )
1x H̄ n̄µD̄ν − ḡµν ρ √n̄ D̄ v = ddx ḡ D̄ρD̄σ + ḡρσ 2ρ µν D̄ vρσ
∂M ∫M √ ( )
= ddx ḡ D̄µ ρσF[ḡ]µ +
3 ḡρσ D̄22 vρσ
M
Thus, part of the boundary term can be absorbed in the gauge sector and what remains is
proportional to nρ D̄ρvµν . From this we can deduce that the difficulty of the ‘disturbing’
surface term concerns only one of several irreducible components of the metric fluctua-
tion. If one performs a transverse-traceless decomposition of hµν , ref. [123, 135, 147] or
section 7.2, it is only its trace part hTrḡ /d, with hTr ≡ ḡµνµν ∫ hµν√, which is affected by the
surface term in eq. (13.26) which assumes the form dd−1∂M x H̄ n
α ∂αφ . Obviously the
other (i.e. TT, TL, and LL) parts of the York decomposition do not contribute so that those
irreducible components, at fixed hTr, enjoy a standard variational principle. Note that hTr
6Though we fully appreciate the possibility of obtaining a well-posed variational problem by means of different
surface terms, we here focus only on the Gibbons-Hawking-York approach.
436 Chapter 13. The Gibbons-Hawking-York boundary term
amounts to a fluctuation of the conformal factor of gµν . Thus, we can recover Einstein’s
field equations for all tensor components for Dirichlet constraints, if we restrict the metric
fluctuation to have a vanishing derivative of the trace part on the boundary, or likewise if
the conformal factor and its normal derivative are simultaneously fixed on ∂M.
D. If we generalize the previous restriction of the conformal factor to all fluctuation fields, we
consider a smaller function space of expectation fields, which satisfy Dirichlet conditions
and also have a vanishing normal derivative D̄n on the boundary:
′ { }vµν ∈ TΦ̄F ≡ sµν tensor on M ; sµν = 0 and D̄nsµν = 0 on ∂M (13.27)
For an EAA defined over F′ the surface terms in the first functional variation disappear
always, and the effective field equation is unambiguously given by (13.25).
At first glance, the conventional point of view on this reduction is rather reserved, in
particular when the underlying functional integral is intended to represent a transition
amplitude between 3-geometries; M is the portion of spacetime between ‘initial’ and
‘final’ time slices, making up ∂M, then. If the dynamical field and its normal derivative
are fixed on the initial and final slice, the underlying equation is in general overdetermined
since we imposed twice too many boundary conditions for a second order field equation.
Hence, there may be no solutions that also satisfy the additional constraints and while
having preserved the form of the field equations, we have lost all of its information.
However, following the lines of [166], we argue that the FRG perspective on this restric-
tion is far more optimistic. A generic (second order in the derivatives) ansatz of the EAA,
Γk[h; ḡ], depends on two independent metrics, a result of the implementation of Back-
ground Independence and the explicit breaking of split-symmetry by cutoff and gauge
fixing action. The running tadpole equation than yields a special class of stationary points,
the self-consistent backgrounds evaluated for hµν = 0. In this context, dvΓk[h; ḡ] = 0 has
to have a meaning, not for all h, but only near hµν ≡ 0, where ḡ becomes self-consistent.
The essential observation is that the self-consistent solution hµν ≡ 0 on all of M, if it
exists, is not lost when we restrict field space to F′, the trivial reason being that the zero
solution has vanishing derivatives everywhere on M and, by continuity, a vanishing nor-
mal derivative on ∂M.
In this section we are far from exhaust the investigation of the variational problem in the context
of the Asymptotic Safety program to Quantum Gravity. Even though we have some first indica-
tion of a possibly global ‘mismatch’ of the bulk-boundary coefficients, it still remains an open
issue if a full resolution of the different levels for the Newton couplings will change this first
impression or if the classical matching technique has to be replaced. Most likely, we therefore
have to fully appreciating that Γk is not a classical but an effective action containing arbitrarily
high derivatives acting on the metric and correspondingly complicated surface terms. This will
require a major structural generalization of the FRGE, for the following reason.
For the present, and although in ref. [166], the boundary contributions vanish under[sec]ond
variation and Dirichlet conditions and thus its couplings do not enter the Hessian Ĥessh Γk on
the RHS of the flow equation. This implies that they cannot back-react on the RG evolution
which rather is fully determined by the dynamical bulk couplin[gs. ]In more complicated trun-
cations, and at the exact level this situation will change; Ĥessh Γk will consist of bulk-bulk,
bulk-boundary, and boundary-boundary blocks, and also the cutoff operator Rk has an analo-
gous block structure. At this point one must take a decision about how to coarse-grain fields
living on the boundary. A priori there is a considerable freedom in choosing a cutoff for them,
and clearly this choice will be crucially important for the bulk-boundary matching.
Returning to the more restricted scope of the present paper we shall consider it legitimate
to extract effective field equations from the bulk action at level-(1) and assume that no surface
13.4 Counting field modes 437
terms interfere with that. This is justified by either invoking the restriction from Fto F′ or, very
conservatively, by narrowing down the truncation to the ‘perfect’ one of (B) above; this will not
take anything away from the non-trivial results of the next section, in particular on black hole
thermodynamics.
13.4 Counting field modes
C This section, as well as part of the conclusion, follows closely the single-metric andmatter
induced bi-metric discussion published in [166].
In this subsection we present a number of examples which illustrate the ‘state counting’ property
of lnZ .k .= −Γk[0; ḡs.c.k ], see chapter 11 for more details. Here we focus on the relevance of
surface terms. For simplicity we fix d = 4 in this section.
13.4.1 Split-symmetry and ∂M= 0/
In a first example, let us assume ∂M = 0/ for a moment. With vanishing ghosts, self-consistent
backgrounds are solutions of the (conventional looking, but ‘running’) Einstein equation
( s.c.) =−λ̄ (1)G ḡ (ḡs.c.µν k k k )µν (13.28)
Notice that only level-(1) couplings, here identified with all higher levels and subsumed in the
D-sector, enter the running tadpole equation. For every given solution to (13.28), the bulk part
of eq. (13.23) leads to the running on-shell -action
λ̄ (0)
∫ ∣∣
Γgrav[0; ḡs.c.] =− k 4 √k k d x g∣ (13.29)8π (0)G ∣k M g=ḡs.c.k
This quantity is strictly negative, and lnZk positive therefore. (We assume λ̄
(0)
k and
(0)
Gk positive
here.)
As an example, consider the maximally symmetric solution to (13.28) for λ̄ (1)k > 0, namely
the 4-sphere S4∫ (L)√with radius = (3/λ̄
(1)
L )1/2k k . It has scalar curvature R = 12/
2 = 4λ̄ (1)Lk k and
the volume7 d4 (1)x g= s L4 24 k = 9s4/(λ̄k ) . Hence
9 (0) (0)s4 λ̄ 9s4 λ̄
ln k kZk = = (13.30)
8π (0) (1) 2 8π (0) (1)G 2k (λ̄k ) gk (λk )
In case of split-symmetry, λ ≡ λ (0) ≡ λ (1)k k k = · · · and gk ≡
(0) ≡ (1)gk gk = · · · , this further reduces to
lnZk =
9s4 1
8π λ . This is a very intriguing and important result. It shows that the weighted num-gk k
ber of modes integrated out between infinity and the IR cutoff k depends only on the properties
of the dimensionless combination of couplings, which for the split-symmetric case is given by
Gkλ̄k = gkλk.
Along a RG trajectory of type IIIa, for instance [191], this product decreases from its fixed
point value limk→∞ gkλk = g∗λ∗ = O(1) to the infrared value Gobsλ̄obs which is observed at low
scales; in real Nature it is of the order 10−120.
It is known [190, 209] that for all type IIIa trajectories admitting a long classical regime
there is a huge hierarchy Gobsλ̄obs ≪ g∗λ∗. Hence, as expected from the monotonicity results
of chapter 11
lnZk→0 ≫ lnZk→∞ (13.31)
( ) (
7Here and in the following we write s ≡ Sn(1) = 2π(n+1)/2) n Vol /Γ (n+1)/2 and bn ≡
n n/2
VolB (1) = π /Γ n/
2+1 for the volume of the unit n-sphere and n-ball, respectively.
438 Chapter 13. The Gibbons-Hawking-York boundary term
Along the hypothetical trajectory realized in Nature [209, 212] the Boltzmann weighted
number of modes integrated out when the cutoff approaches zero is about ln → ≈ 10120Zk 0 ,
while the NGFP value lnZk→∞ is basically zero.
13.4.2 Euclidean space with non-vanishing boundary
As a second example we consider a four dimensional Euclidean spacetime with a non-empty
boundary. We use the associated tadpole equation (13.26) to explore its contents employing
the results of the bi-metric-bulk – pure-background-boundary truncation. For simplicity we
specialize for a regime of the underlying RG trajectory in which the cosmological constant in
this equation, the level-(1) coupling λ̄ (1)k , is negligible, yielding
R̄µν = 0 with λ̄
(1)
k = 0 (13.32)
Clearly, the simplest Ricci flat solution is flat space. So let us assume M is a subset of R4, with
∂M 6= 0/ , and equipped with a flat metric. Inserting the configuration
ḡs.c.µν = δµν with λ̄
(1)
k = 0 , (13.33)
into Γk[h= 0; ḡµν ], the ‘mode counting’ device assumes the following form
λ̄ (0)− λ̄
∂ (0)
lnZk = Γ [0;δ
k k
k µν ] = Vol(M)+ Vol(∂M)
8π (0) 8π ∂ (0)Gk G∫ k√
− 1 d3x H̄ K̄ (13.34)
8π ∂ (0)Gk ∂M
A perhaps surprising property of this equation is that it involves a non-zero (bulk) cosmo-
logical constant term even though it applies to flat space. However, the condition for flat space
to be a self-consistent background is that the cosmological constant at level-(1) is negligible.
Its counterpart at level-(0), the one appearing in (13.34) may have any value. Only when split-
symmetry happens to be intact we have λ̄ (0) = λ̄ (1)k k so that the bulk term on the RHS of (13.34)
indeed vanishes. This results in a kind of ‘holographic’ property of the function Zk which
then is expressed by surface terms only. While the Gibbons-Hawking-York term is the most
important contribution, the condition λ̄ (0) (1)k = λ̄k = 0 still leaves room for a non-zero boundary
cosmological constant λ̄ ∂ (0)k , leading to a term proportional to Vol(∂M).
In the language of thermodynamics equation (13.34) defines a certain ‘free energy’. The
terms proportional to λ̄ (0)k Vol(M) and λ̄
∂ (0)
k Vol(∂M) amount to a homogeneously distributed
bulk and surface energy density reminiscent of the volume and surface energy of a liquid droplet.
In this picture the last term in (13.34), the Gibbons-Hawking-York contribution, is equally nat-
ural and describes how the droplet gains or looses energy by developing a curved or crumpled
surface.
What is the statistical mechanics, and what are the pertinent degrees of freedom which
underlie this thermodynamics at the microscopic level?
In the context of the effective average action the answer is clear: It is the statistical mechan-
ics of the matter and geometry fluctuations about their respective backgrounds. The generalized
harmonic modes of those fluctuations are ‘counted’ by the partition function Zk when the IR
cutoff k is lowered from infinity to zero. The various running coupling constants contained in
Γk and lnZk parametrize how the number of fluctuation modes, contributing to the functional
integral and weighted with the ‘Boltzmann factor’ e−S, decreases when we ‘zoom’ deeper and
deeper into the microscopic structure of spacetime by increasing k.
Another remark is in order at this point. We stress that the occurrence of surface terms in the
partition function Zk on empty flat space is both unavoidable and natural from the physics point
13.4 Counting field modes 439
of view. It is unavoidable because the RG flow generates such terms when ∂M 6= 0/ . Therefore,
contrary to the classical action underlying the variational principle of General Relativity we
may not subtract any terms on an ad hoc basis ‘by hand’ from Γk. The surface terms are also
natural because the number of field modes counted by Zk will depend on the shape of ∂M in
general, and this dependence can lead to observable effects, the most famous example being the
Casimir effect.
In classical relativity, in order to obtain a finite action for asymptotically flat spacetimes, one
usually replaces K→K−K0 in the Gibbons-Hawking-York action SGH. Here K0 is the extrinsic
curvature of ∂M when embedded into a flat spacetime. From the above remarks it should be
clear that within the average action approach this procedure would not only be unmotivated but
wrong since we might loose essential physics.
Closed Euclidean 4-ball
To be more concrete about M let us take M to be a 4-ball, M = B4(L), with arbitrary radius
L. Embedding the boundary ∂M = S3(L) into R4 its extrinsic curvature equals K̄ = 3/L. As a
result,
λ̄ (0) 4 ∂ (0)b 3 2− 4 k L s3 λ̄k L − 3s3 LlnZk = + (13.35)
8π (0) 8π ∂ (0) 8π ∂ (0)Gk Gk Gk
Here it is particularly obvious that the asymptotic series for the heat kernel gives rise to a
systematic expansion in powers of 1/L.
‘Cylindrical’ foliation
As a second example consider a simple Euclidean caricature of the foliated cylinder type space-
times one considers in relation with the initial or boundary value problem of Lorentzian gravity.
Again we embed M into (R4,δµν). We fix a foliation of R4 in terms of flat 3-dimensional
hypersurfaces labeled by a parameter t referred to as ‘Euclidean time’. This gives rise to a cor-
responding foliation on M. We consider M foliated by hypersurfaces Σt , t ∈ [t1, t2] which are
bounded by closed 2-surfaces St . Thus ∂M consists of the hypersurfaces Σt1 , Σt2 , and the union
of all St = ∂Σt .
For simplicity we takeM to be the direct product of a 3-ball of radius ℓ, B3(ℓ), with the time
interval. Thus Σ 3t =B (ℓ) and St = S2(ℓ) for any t ∈ [t1, t2] so that we have Vol(M)= b ℓ33 (t2−t1)
and Vol(∂M) = 2b ℓ33 + s ℓ22 (t2 − t1). The extrinsic curvature of the flat t = const surfaces
in R4 vanishes and so Σt1 and Σt2 do not contribute to the surface integral over K̄. The only
contribution comes from the union of all 2-spheres St . Since the trac∫e of t√he extrinsic curvature
of S2(ℓ) embedded in flat R3 is given by K = 2/ℓ we therefore get d3x H̄ K̄ = (t2− t1) · (2/
ℓ) ·VolS2(ℓ) = 2s2(t2− t1)ℓ.
This brings us to the final result
[ ]
λ̄ (0) λ̄ ∂ (0) λ̄ ∂ (0)− ln k 3 k 2 ℓ k 3Zk = ℓ + ℓ − (t2− t1)+ ℓ (13.36)
6 (0) 2 ∂ (0) ∂ (0)Gk Gk G 3G
∂ (0)
k
The coefficient of (t2− t1) on the RHS of (13.36) may be thought of as a certain energy asso-
ciated to the empty flat 3-dimensional space (not ‘spacetime’) interior to a 2-sphere of radius
ℓ.
13.4.3 Thermodynamics of the Schwarzschild black hole
We continue to restrict ourselves to the self-consistent backgrounds of the type (13.32), i.e.
Ricci-flat metrics, Rµν(ḡ) = 0. Perhaps the most prominent representative of this class is the
440 Chapter 13. The Gibbons-Hawking-York boundary term
Euclidean Schwarzschild solution ds2 = f (r)dt2+ f (r)−1dr2+ r2dΩ2 with
RS
f (r) = 1− (13.37)
r
Here r ∈ (RS,∞), t ∈ [0,β ], and time is now compactified to a circle of circumference β ≡ 4πRS.
Note that the Schwarzschild radius RS has the status of a free constant of integration with
the dimension of a length. It is usually re-expressed in terms of a mass, M, so as to recover
Newtonian gravity asymptotically. Then RS = 2GM where G is the Newton constant of the
classical theory. Since in quantum gravity it is not a priori obvious which constant (0), ∂ (0)Gk Gk ,
(1)
Gk , · · · should be used to convert RS to a mass we shall refrain from doing this and continue to
label the family of Schwarzschild metrics gSchwµν by the length parameter RS.
The running on-shell action
We consider the Euclidean Schwarzschild manifold M foliated by 3D hypersurfaces Σt of con-
stant t. They carry the metric ds2Σ = f (r)
−1dr2+ r2dΩ2. With the time compactified and the
t
period chosen as8 β = 4πRS, the only boundary of M is the union of the asymptotic 2-spheres
∂Σt = S
2 on which r = const ≡ r̂, r̂→ ∞.
Let us insert this background into the truncation ansatz (13.23) with A = 0 where, by as-
sumption, λ̄ (1)k = 0. All that needs to be evaluated is − lnZk = Γk[h = 0; ḡ = gSchw]. Since
all terms involving R̄, will vanish we are left only with the bulk and boundary cosmological
constant terms, respectively, together with the extrinsic curvature term:
∫ ∫
− 2
√
3 2
√
d x H̄ (K̄− K̄ 30)− d x H̄ K̄0 (13.38)
16π ∂ (0)Gk ∂M 16π
∂ (0)
Gk ∂M
Here we rewrote the extrinsic curvature ofM in ∂M by adding and subtracting K̄0, the curvature
of ∂M embedded in flat space, K̄ ≡ (K̄− K̄0)+ K̄0. As a result, the first integral of (13.38) is the
usual subtracted Gibbons-Hawking-York term, while the second, to leading order in RS/r → 0,
becomes independent of the spacetime curvature caused by the black hole. For very large r̂ it
corresponds to the surface term of flat spacetime considered in Section 13.4.2. When added to
the bulk and boundary cosmological constant terms it yields the free energy of flat spacetime.
Thus the on-shell average action boils down to
∫ √
− lnZk =−
1
d3x H̄ (K̄− K̄
∂ (0) 0)+ · · · (13.39)8πGk ∂M
where the dots stand∫for th√e contributions of flat space. The evaluation of the integral in (13.39)
is standard; it yields d3x H̄ (K̄− K̄0) =−2πRS β when r̂→ ∞. For the partition function this
leads us to
− βRS AlnZk = + · · ·= + · · · (13.40)
4 ∂ (0)Gk 4
∂ (0)
Gk
where A≡ 4πR2S denotes the area of the event horizon (in the Lorentzian interpretation).
Eq. (13.40) is a very instructive formula. Structurally it coincides with the familiar semi-
classical Gibbons-Hawking-York result [79]. However, the (unique and truly constant) classical
Newton constant appearing there got replaced by a specific member of the various infinite fam-
ilies of running Newton-type couplings which parametrize a general bi-metric average action.
Thus we see that the scale dependence of the partition function Zk and the derived thermody-
namical quantities are governed by the boundary Newton constant at level zero.
8If β 6= 4πRS there is another boundary at the horizon, however. We shall not consider this generalization here.
13.4 Counting field modes 441
Thermodynamics at finite scale
Note that the Bekenstein-Hawking temperature is scale independent,
1 1
T = = , (13.41)
β 4πRS
while the free energy Fk ≡−β−1 lnZk inherits its k-dependence from ∂ (0)Gk :
RS 1 1
Fk = = (13.42)
4 ∂ (0) ∂ (0)G Tk 16πGk
If we apply the standard relations U =−T 2 ∂∂ (F/T ) and S =−∂F/∂T at every fixed value ofT
k we obtain for the internal energy and entropy, respectively:
RS 1 1
Uk = = = 2F (13.43)
2 ∂ (0) 8π ∂ (0)
k
G Tk Gk
R2
S = π S
A 1 1
k = = (13.44)∂ (0)
Gk 4
∂ (0) ∂ (0)
G T 2k 16πGk
Likewise C = ∂U/∂T yields the specific heat capacity
− R
2
C = 2π S − 1 1k = (13.45)∂ (0) ∂ (0)
Gk 8πG T
2
k
Obviously the RG running of all thermodynamical functions of interest is governed by a single
running coupling, namely 1/ ∂ (0)Gk .
Running ADM mass
Up to now we never ascribed any mass to the Schwarzschild spacetime. As we mentioned
already, it is more natural to characterize it by a length such as RS. Its conversion to a mass
is a matter of convention, strictly speaking, which in quantum gravity becomes particularly
ambiguous. Nevertheless, our result (13.40) suggests that a natural way of relating a mass to a
black hole with a given parameter RS is by means of
∂ (0)
Gk :
≡ RSMk (13.46)
2 ∂ (0)Gk
We emphasize again that RS has no k-dependence; it labels different solutions of the truncated
tadpole equation, R̄µν = 0, which happens to be independent of any running coupling. So the
k-dependence of the running massMk is entirely due to
∂ (0)
Gk . It can be seen as a scale dependent
generalization of the classical ADM mass.9 The definition (13.46) is motivated by observing
that all relations of semiclassical black hole thermodynamics retain their form when quantum
gravity effects are included via the average action provided we replace the classical mass by the
running mass Mk. The mass (13.46) controls the partition function of a single black hole,
− 1ln ∂ (0) 2Zk = 4πGk Mk = β Mk , (13.47)2
and the ensuing thermodynamical relations
1
Fk = Mk Uk =Mk , = 4π
∂ (0)
S G M2k
2 k k
, (13.48)
look like their semiclassical counterparts with the replacement M →Mk. In its other roles, the
classical mass M might possibly get replaced by running masses with a different k-dependence.
9 ∮In more general backgrounds it becomesMk =−(8π ∂ (0)G )−1 (K−K0)where the integral is over an asymptotick
sphere.
442 Chapter 13. The Gibbons-Hawking-York boundary term
The range of validity of self-consistent Schwarzschild solutions
In order to understand the range of validity for the present discussion, let us go back to the
running tadpole equation that defines the k-dependent ‘self-consistent’ background solutions:
Gµν(ḡs.c.k ) =−λ̄ (1) ḡs.c.µνk k ⇐⇒ R(ḡs.c.k ) = 4 λ̄
(1)
k (13.49)
This defines ḡs.c.k to be an Einstein-metric for all k associated to some constant, but k-dependent
curvature R(ḡs.c.k ) = 4 λ̄
(1)
k . For the Schwarzschild solution, we impose certain homogeneity and
isotropy constraints on the spatial part of the metric and consider the case of λ̄ (1)k ≈ 0. This
leads to ḡs.c. ≡ ḡSchwk . However, the assumption λ̄
(1)
k ≈ 0 is only justified for some range of scales
k ∈ (kSchwIR ,kSchwUV ), in particular there exists no RG trajectory for which λ̄ (1) ≡ 2λ (1)k k k ≈ 0 for all k.
Figure 13.4 shows the k-dependence of the cosmological bulk coupling, λ̄ (1)k , for a typical type
IIIa trajectory. While towards decreasing values of k, the approximation λ̄ (1)k ≈ 0 is well justified,
λ̄D
k/mPlanck
Figure 13.4: The k-dependence of the dimensionful dynamical cosmological constant on the
bulk, λ̄ (1) 2 (1)k ≡ k λk , evaluated for a typical IIIa trajectory (inset) in units of Planck mass. From
this result we see that the range in which the self-consistent background can be described by a
Euclidean Schwarzschild metric with vanishing λ̄ (1)k , extends to arbitrary small values of k, i.e.
kSchwIR → 0. Towards the ultraviolet, leaving the semiclassical regime, λ̄ (1)k increases significantly
and cannot be neglected anymore. Thus, from this scale kSchwUV onwards, the k-dependence of ḡ
s.c.
k
becomes important.
implying in particular kSchwIR = 0, we see that when the trajectory leaves the semiclassical regime
towards the UV fixed point the dimensionful cosmological constant increases and becomes non-
negligible, hence kSchwUV < ∞. This implies that solutions of the running tadpole equations (13.49)
turn k-dependent above kSchwUV and Ricci-flatness has to be replaced by the constant curvature
condition, i.e.
ḡs.c. = ḡSchw ∀k ∈ (0,kSchwk UV ≈ 0.7mPlanck) else ( s.c.) = 4 λ̄ (1) λ̄ (1)R ḡk k k 6= 0
(13.50)
In the following we make use of the observation that as long as the background curvature effects
are small, i.e. R(ḡs.c.k ) ≈ 0, the self-consistent solution is k-independent. Since the range of
validity includes the classical and semiclassical regime, we will be most interested in the results
at the edge, just below kSchwUV , where the semiclassical approximation is still valid.
Explicit k-dependence of the boundary Newton constant
Let us see now what we obtain for Mk from the running of
∂ (0)
Gk . Its k-dependence is governed
by the anomalous dimension η ∂ (0) which depends on the bulk couplings g(1) ≡ g(2) ≡ ·· · and
13.4 Counting field modes 443
λ (1) ≡ λ (2) ≡ ·· · . In the range of the Schwarzschild solution k ∈ (0,kSchwUV ≈ 0.7mPlanck) we can
use the semiclassical approximation that yields
η ∂ (0) =−(d−2)ω ∂ (0)g∂ (0)d (13.51)
with the crucial coefficient obtained in eq. (8.63)
∂ (0) − d(d−3) 1ωd = Φ1 (0)< 0 (13.52)3(4π)d/2−1 (d−2) d/2−1
( ) ( )
The following semiclassical solution for the RG equation ∂ 1/ ∂ (0) = −η ∂ (0) 1/ ∂ (0)t Gk Gk can
thus be obtained, which reads in 4 dimensions:
1 1
= +ω ∂ (0) 24 k (13.53)∂ (0) ∂ (0)
Gk G0
Notice, that this is the same result as obtained with the full-fledged single-metric- and the bi-
metric matter induced-truncation in ref. [166].
Explicit k-dependence of the ADM mass
Using (13.53) in (13.46) we associate the following running mass to the black hole with Schwarzschild
radius RS:
[ ] RS
Mk = 1+ω
∂ (0) ∂ (0)
4 G0 k
2 M0 where M0 ≡ (13.54)
2 ∂ (0)G0
Though, from the present truncation it seems incompatible with the Asymptotic Safety require-
ment, let us nevertheless match the surface and bulk Newton constants at k= 0 and identify this
quantity with the standard Newton constant, ∂ (0) = (0)G0 G0 ≡ G ≡ m−2Planck. Then we recover the
ordinary relationship M0 = RS/(2G) in the extreme infrared (at k = 0), but at higher scales the
mass associated to the very same geometry is smaller than M0:
[ ( ) ]2
k
Mk = 1−|ω ∂ (0)4 | M0 (13.55)mPlanck
In writing down (13.55) we made it manifest that the coefficient ω ∂ (0)4 turned out negative. As
a consequence, Mk decreases for increasing k, reaches zero at a scale near k = mPlanck, and
becomes negative for even larger k-values.
An attempt at interpreting this behavior could be as follows. It is known that in QEG the
bulk Newton constant Gk decreases for increasing k, and this was interpreted as an indication
of gravitational anti-screening due to the energy and momentum of the virtual particles sur-
rounding every massive body; because of the attractivity of gravity, they are pulled towards this
body, adding positively to its bare mass, whence the virtual cloud leads to an effective mass that
increases with increasing distance [122].
Now we have seen that the boundary Newton constant ∂ (0)Gk increases for increasing k at
least in the upper half plane. Interestingly enough, what at first sight might seem to contradict
the picture of gravitational anti-screening, in view of M = 1 ∂ (0)k 2RS/Gk , at least heuristically,
actually confirms it: According to this definition of ‘mass’, the running mass of any material
body decreases with increasing k, or decreasing distance. Somewhere near k = mPlanck it even
seems to vanish, indicating probably that in this regime a more elaborate treatment is necessary.
444 Chapter 13. The Gibbons-Hawking-York boundary term
Running thermodynamic quantities
The running free and internal energy, the entropy and the spec[ific heat capacity are g]overned
by the same function of k as Mk in (13.55): Fk,Uk, , ∝ 1−|ω ∂ (0)Sk Ck 4 |(k/m )2Planck . Note,
however, that the specific heat has a negative IR value, = −2π 2/ ∂ (0)C0 RS G0 , and tends to zero
in the Planck regime, see Fig. 13.5. At this point, when the underlying RG trajectory leaves
the semiclassical regime and approaches the vicinity of the non-Gaußian fixed point, we expect
further effects due to the k-dependence of the self-consistent background solution.
k/mPlanck
Fk,Uk,Sk,Mk
Ck
k/mPlanck
(a) Fk,Uk,Sk,Mk (b) Specific heat capacity Ck
Figure 13.5: The k-dependence of the thermodynamical quantities.
A natural cutoff identification.
Even though the shortcut to extracting physical information from the running couplings in Γk
by identifying k with some physical scale (‘RG improvement’) is notoriously ambiguous in
general, it is clear that the black hole spacetime specified by a given RS has a distinguished
intrinsic mass scale associated to it that does not rely on any artificial conversion factor, namely
1/RS, or the temperature T = (4πRS)
−1.
If we tentatively adopt the cutoff identification k ≈ 1/RS, and go from macroscopic astro-
physical black holes to microscopic ones with a Planckian Schwarzschild radius, we find thatMk
decreases monotonically, heading forMk = 0 near RS = ℓPlanck. Remarkably, by eqs. (13.47), the
thermodynamical quantities, the entropy in particular, all vanish in this limit: Fk → 0, Uk → 0,
Sk → 0 for RS ց ℓPlanck.
It is particularly intriguing that the specific heat capacity Ck exhibits the same behavior
however starting from negative values. This suggests that near ℓPlanck the notorious instability of
classical gravity possibly gets tamed in a dynamical way: The system no longer can lower its
energy by accreting further mass and the gravitational collapse might come to a halt.
This picture based on the boundary Newton constant is surprisingly similar to what we
found in [305–309] by a rather different reasoning, namely the ‘RG improvement’ of the bulk
Newton constant in the classical formula f (r) = 1− 2GM/r. Keeping M fixed we replaced
G→ Gk and identified 1/k with the radial proper distance.
Future work will have to clarify the precise relationship between the two treatments, in par-
ticular whether they are different pictures of the same phenomenon or should be superimposed
rather. In addition, the disentanglement of the boundary level couplings turns out very impor-
tant, in particular to understand if the level-(0) Newton coupling is indeed trapped in the lower
half plane for negative Newton couplings. In this case the previous results have to be evaluated
for negative ∂ (0)Gk which then implies that we need further surface invariants that compensate
this sign and lead to a positive entropy and ADM mass.
13.5 Conclusion 445
13.5 Conclusion
In this chapter we studied the results of the functional RG flows of Quantum Einstein Gravity
on spacetime manifolds with boundary. This extends our previous work [166] to a bi-metric
truncation where the running of the boundary coefficients is induced by a dynamical Newton
and cosmological coupling, now being disentangled from their level-(0) counterparts. Most of
the single-metric and matter induced bi-metric results of the boundary sector could be recovered.
All these truncations have in common that they contained various surface terms such as the
Gibbons-Hawking-York term for instance. While the motivation to incorporate these boundary
invariants into the EAA ansatz exceeds far beyond the classical matching condition that was
established to recover Einstein’s field equation from a variational principle.
We studied the global RG properties of the boundary sector, which due to their structural
similarity exhibits certain global features reminiscent of the bulk level-(0) part of theory space.
In particular, we discovered a running UV-attractor that however is mainly situated in the lower
half plane of negative g∂ (0) < 0 and coincides with the non-Gaußian fixed point for asymptoti-
cally safe trajectories. Though the present truncation is only a first step towards understanding
the boundary sector, if the results remain valid while including higher level invariants, it would
suggest that Asymptotic Safety together with Background Independence exclude the possibility
of a bulk-boundary matching as proposed by Gibbons, Hawking, and York. From the FRG
perspective, this is actually not a real drawback, but supposed to happen, if not for the Gibbons-
Hawking-York functional than for other boundary invariants. Leaving the structural generaliza-
tion of the FRGE which is needed for arbitrary (untruncated) action functionals to future work,
we justified the variational procedure for the second derivative actions considered here. For a
proper interpretation of the surface terms and the variational principle it was crucial to take the
bi-metric character of gravitational average action into account. In an expansion with respect to
(p)
hµν a generic functional Γk contains ‘towers’ of bulk Newton constants Gk , boundary Newton
constants ∂ (0)Gk , and many more similar couplings whereby the ‘level’ p = 0,1,2, · · · is indica-
tive of the hµν -power in the corresponding field monomial. Since the background-quantum
split-symmetry is broken by the IR cutoff, the different levels evolve independently under the
RG flow.
A key observation in this context is the following. The partial differential equation which
determines self-consistent backgrounds, that is, backgrounds which once prepared by external
means are not modified by the intrinsic quantum fluctuations, involve only the couplings of level-
(1). The (thermodynamical, etc.) properties of the backgrounds they imply are determined by
the level-(0) couplings in addition. In the example considered, in fact only those of level-(0)
happened to be relevant. It is therefore possible to have a bulk-boundary matching of Newton’s
(1) (1,∂ )
constant at level-(1), Gk = Gk , hence a standard variational principle, but nevertheless a
(0) (0,∂ )
mismatch at level zero: Gk =6 Gk . This mismatch can encode important information relevant
to the effective field theory description of physics at finite scales k; black hole thermodynamics
turned out to be a prime example, however special care has to be exercised concerning the sign
of G∂ (0).
In (semi-)classical black hole physics there exists only a single mass parameter, M, and
this parameter plays various conceptually rather different roles. It controls, for instance, the
semiclassical partition function of a single black hole but it also describes the strength of the
Newtonian force between two black holes at large distances, say. In quantum gravity, in the
context of the average action, there does not exist a single running mass which serves all these
purposes at a time. Likewise, there is not a single, but actually quite many different running
Newton constants. Each of them takes over one specific role played by the standard Newton con-
stant, or the classical concept of mass, respectively, and depending on this role its k-dependence
446 Chapter 13. The Gibbons-Hawking-York boundary term
is different in general. We saw that the partition function and the related thermodynamics of an
isolated black hole is governed by 1 (0,∂ )Mk ∝ /Gk , while the effective Einstein equation involve
the level-(1) bulk couplings only.
Based upon the concept of self-consistent backgrounds, solutions of the running tadpole
equation, we employed the proposed Ck-function of chapter 11, to count the number of field
modes integrated out by the average action. The associated partition function related to a given
RG trajectory and a background metric, ḡ, describes the statistical mechanics of the metric
fluctuations relative to this background. While this mode count is of interest also on spacetimes
without boundary, here it led us to an intriguing scale dependent generalization of black hole
thermodynamics which represents the physical basis and motivation for the specific definition
of Mk.
A more phenomenological analysis of the corresponding quantum effects in black hole
spacetimes is an interesting playground to test and understand the implications of the RG results.
This seems to be a very promising path to follow in future work.
14.1 Summary
Asymptotic Safety
Background Independence vs. Asymptotic
Safety
Towards a C-function in quantum gravity
Positivity vs. Asymptotic Safety
Asymptotic Safety on manifolds with bound-
ary
14.2 Outlook
14. CONCLUSION
In this chapter we give a brief summary of the results discussed in this part of the thesis and
then conclude with possible future projects in these directions.
14.1 Summary
On the path towards a deep understanding of Nature physicists are confronted with two theoreti-
cal concepts, the Standard Model of particle physics and General Relativity, each outstanding in
its predictivity and precision. In quest of a unified theory that intertwine the quantum nature of
matter with the geometrical construction of spacetime, the search still continues. Many decades
of research have brought deep insights in both concepts, have revealed their deficiencies and
their mathematical foundations, and many physicists nowadays believe that a solid theory of
Quantum Gravity may cure most, or even all, of it. Promoting General Relativity to a quantum
level is a very sophisticated task, since most of the methods available in the construction of
previous quantum field theories turned out to be inapplicable.
One of the reasons may be that, considered as a gauge theory, gravity is special because it
relates the geometry of spacetime (its base space) with the structure of its gauge group and thus
renders geometry as the mediating gauge field. Hence, Background Independence, the require-
ment that the formulation should be independent on any – possibly prescribed – background
geometry, is central to all constructions of quantum theories of gravity.
On the other hand, Quantum Gravity seem to manifest as a nontrivial theory at a fundamen-
tal level, i.e. on very small scales (in the ultraviolet, so to say), thus that perturbative approaches
may only get a glimpse of the full beauty of gravity well below the Planck energy. In addition,
it turned out that from the perspective of conventional perturbative approaches quantum gravity
seems to be non-renormalizable.
A variety of different approaches trying to circumvent these difficulties brought to light
hidden generalized concepts that only become relevant when gravity enters the game. A con-
servative approach to Quantum Gravity and thus to a complete, unified theory is Asymptotic
Safety. Based on the ideas of Weinberg [145] there is a (non-)perturbative notion of renormal-
izability as a (non-)trivial fixed point of the Renormalization Group (RG) flow. Employing the
448 Chapter 14. Conclusion
Effective Average Action (EAA) formulation, Wetterich [112] succeeded in deriving an exact
functional differential equation, the FRGE, which is UV- and IR-regulated. This non-linear
equation defines RG trajectories on theory space, a manifold containing all possible invariants
compatible with the required symmetries and field content, from which only those that emanate
from a suitable UV fixed point are physical relevant. The position of this fixed point defines the
fundamental theory, which is thus a prediction of the formalism instead of a mere input.
However, the non-linearity of the FRGE forces to employ, still non-perturbative, approxi-
mations, so called truncations, that consider only a subspace of all possible invariants. Imposing
different symmetries, varying the field contents, or changing the underlying topology has pro-
vided strong evidence for the existence of a suitable non-trivial UV fixed point, thus supporting
the Asymptotic Safety conjecture of Quantum Gravity. Besides increasing the complexity of
truncations by including further invariants, studying the general properties of the FRGE for
gravity and test its consistency with the requirement a theory of Quantum Gravity should fulfill,
is at least equally important.
In this thesis, we studied action functionals of bi-metric Einstein-Hilbert type augmented
with a boundary Gibbons-Hawking-York surface term, resolving the RG evolution of six basis
invariants. The derivation of the corresponding beta-functions was carried out in great detail
in part II. Therefore, we introduced a new technique that simplifies bi-metric calculations by
exploiting the properties of the employed truncation on the basis of the gauge sector.
In part III of this thesis we studied the derived beta-functions with respect to various central
questions of the Asymptotic Safety program to QG. We here present only a very brief summary
of the results and refer to the conclusions of the respective chapters for more details.
14.1.1 Asymptotic Safety
In chapter 9 the dimensionless beta-functions were analyzed for (non-trivial) fixed points of the
RG flow in four spacetime dimensions. We revealed a hierarchy among the beta-functions dom-
inated by the dynamical couplings which allows to study their influence on bulk- and boundary-
sector separately. For the dynamical part of theory space we found three fixed points, one
Gaußian and two non-Gaußian solutions. In particular we recovered a fixed point reminiscent
of the previous results in the Asymptotic Safety program to Quantum Gravity, with a positive
Newton and cosmological constant. It was found that for each dynamcial solution both, the
background and the bulk level-(0) sector, give rise to a trivial and non-trivial solution, which
yields a total of 12 fixed points.
We then discussed their critical behavior in detail showing that in the present case each
(non-)trivial component in the fixed point comes along with (two) one relevant and (zero) one
irrelevant direction. Thus dimensionality of their UV-critical hypersurfaces assumes values be-
tween 3 and 6, whereby the lower limit represents a triple-GFP and the upper limit corresponds
to a triple-NGFP.
14.1.2 Background Independence vs. Asymptotic Safety
Background Independence is a fundamental requirement every quantum theory of gravity has
to take care of. The principle that geometry is the dynamical object itself is found to be im-
plemented by invoking no background at all (e.g. LQG, CDT,. . . ), or by introducing a back-
ground geometry at intermediate states of the quantization while assuring that observations do
not depend on this particular choice. In the EAA approach the background field method real-
izes the latter option at the expense of a now bi-field (in QEG bi-metric) formulation of the
FRGE. Background Independence translates into a reduction of the EAA to a single-field de-
pendence at the physical scale where it agrees with the standard effective action Γ and is known
as split-symmetry. In chapter 10 we investigated in which way the conditions of Background In-
14.1 Summary 449
dependence and non-perturbative renormalizability can be fulfilled simultaneously. The results
uncover a topological feature of the Renormalization Group flow, the running UV-attractor, that
guarantees Asymptotic Safety on the one hand and assures Background Independence on the
other hand, thereby increasing the predictivity of the setting. Thus, there is a first evidence for
the coexistence of Background Independence and Asymptotic Safety. Furthermore, we studied
the conceptual and quantitative differences of bi-metric calculations and their single-metric ap-
proximation, revealing only small portions of theory space where the latter seems to be reliable.
Miraculously, the regime of the non-Gaußian fixed point is one part of theory space where the
single-metric approximation qualitatively agrees with its bi-metric counterparts, even though
especially the critical exponents differ in both cases.
14.1.3 Towards a C-function in quantum gravity
In chapter 11 we studied aC-function like feature of the Effective Average Action under Renor-
malization Group evolution. The conjectured function on theory space acts as a mode counting
device and at the same time should fulfill the monotonicity condition along suitable Renormal-
ization Group trajectories. To test the violation of this monotonicity condition for a particular
truncation would then be a very powerful tool to draw conclusions about the reliability and the
error of the corresponding results. The properties of the proposed candidate of a C-function
could be established for the bi-metric Einstein-Hilbert truncation for all RG solutions compati-
ble with Background Independence. In sharp contrast, a large class of split-symmetry violating-
and the the full set of single-metric RG trajectories fail to reproduce a monotonicity. The clash
between the proposed C-function and the non-monotonicity for those trajectories was shown to
be not a failure of theC-function but an indication for the unreliability of those solutions. While
our proposal consists of two components, the first which is manifestly monotone for exact solu-
tions and the second which ‘measures’ the split-symmetry violation along the RG trajectory, it
turns out to be the first term which generates the non-monotone behavior of the C-function in
case of the single-metric or split-symmetry violating trajectories. Thus in confirmation with the
observations of chapter 10, these solutions are only approximately valid in parts of theory space,
while the split-symmetry restoring bi-metric trajectories are in accordance with the properties
of the exact solutions.
14.1.4 Positivity vs. Asymptotic Safety
Bi-metric calculations, even though rare, were already studied several years ago. However, the
perspective we adopted in chapters 10 and 11 brought new insights into the finer structures un-
derlying the bi-metric nature of the gravitational FRGE. It turned out that the Einstein-Hilbert
single-metric approximation that was used occasionally for Renormalization Group-based pre-
dictions in cosmology for instance, come close to their bi-metric analogs only in the classical
and the UV regime. Especially on intermediate scales, where the single-metric approximation
was found unreliable, the full bi-metric picture revealed completely new, so far uncovered fea-
tures, one of them being the sign change in the anomalous dimension of Newton’s coupling,
η . In chapter 12 we studied the effect a positive anomalous dimension has on the propaga-
tion of gravitational modes and the particle interpretation of the fields. It is argued that when
approaching the nontrivial fixed point towards the UV, the anomalous dimension turns nega-
tive, indicating a break down of a Källén-Lehmann – and thus particle – representation. At the
same time the equation of motion predict a strongly curved spacetime where again the particle
concept looses its meaning for independent reasons. The appearance of dark matter could be
due to a cloaking of the actual sources by these ‘unparticle’ formation, as was pointed out in
section 12.3. A transition from a negative anomalous dimension, crucial for the existence of
a nontrivial fixed point, to a positive η had never been observed in the single-metric trunca-
450 Chapter 14. Conclusion
tions. Being negative in the UV and positive in the (semi-)classical regime, the η obtained by
bi-metric truncations was opening the possibility to have a coexistence of Asymptotic Safety
at the fundamental level with the standard Wightman axioms of flat spacetime QFT in the flat
space on-shell limit.
14.1.5 Asymptotic Safety on manifolds with boundary
In chapter 13 we then studied the effect of adding a Gibbons-Hawking-York like boundary term
to the groundbreaking Einstein-Hilbert truncation, [122] in the FRG setting. We explored its
Renormalization Group evolution, in particular the interplay of boundary and bulk terms. Em-
bedding the Asymptotic Safety program on manifolds having a non-vanishing boundary finally
made for instance black hole spacetimes accessible for a detailed study within this approach
without Renormalization Group improvement which suffers from the need for ambiguous cut-
off identification. Even under for this completely different setting the necessary conditions for
non-perturbative renormalizability in the shape of a nontrivial fixed point could be established,
extending the results we obtained in a previous single-metric and matter induced bi-metric study
[166]. Applying the Renormalization Group results to the Euclidean Schwarzschild geometry
provided an indication for a stabilization of the thermodynamical properties due to quantum
effects.
14.2 Outlook
A satisfactory theory of quantum gravity is one longterm objective in theoretical physics. In
order to achieve this ambitious goal, while lacking experimental data, it will be very impor-
tant to find parallels among different approaches to quantum gravity. Each of these theories
acknowledge the fact that gravity is somewhat special and at least on a mathematical level more
involved, revealing new features and open issues.
Following the path of the Asymptotic Safety program within the Effective Average Action
approach the difficulty resides in the functional non-linear differential equation, the FRGE,
which requires a completely new catalog of mathematical techniques, yet to be discovered.
Thus physicists invent (non-perturbative) approximation schemes, known as truncations, which
allow to stepwise uncover possible candidates for a theory of Quantum Gravity which are renor-
malizable in a non-perturbative sense. From this perspective there are basically two ways future
work will extend our understanding of the underlying Renormalization Group flow on theory
space, the manifold containing all possible action functionals. The first way is formulated on
the level of truncations, where the previous results are tested for its stability and extended by
including more and more basis invariants. The second way addresses the general properties of
the FRGE on the level of exact solutions, revealing possibly hidden features of the RG flow
that can be utilized to judge the reliability of truncations. These additional information may be
found very instructive to construct new truncations in a systematic way.
In this thesis we studied partially vaguely explored and completely untouched regions in
theory space. Both extensions, the bi-metric ansatz which is actually mandatory and the inclu-
sion of boundary invariants, are very interesting aspects of Quantum Gravity and it had been
only for the technically difficulty that previous investigations almost exclusively omitted their
influence. Encouraged by the results obtained in this thesis, further bi-metric truncations are
needed to test the reliability of the single-metric results and addresses fundamental questions as
Background Independence within the Asymptotic Safety program to Quantum Gravity. The in-
troduced method to reduce the complexity of bi-metric studies may be helpful in this direction.
Furthermore, we kept the assumptions and notation as general as possible, especially in view
of the boundary sector. Only at the very end, in order to obtain definite results, we specified the
14.2 Outlook 451
gauge fixing condition, the boundary constraints the fields satisfy, and the field parametrization
related to the field space connection. This has the great advantage that future works can continue
the investigation of bi-metric and/ or boundary type truncations from this point onwards.
In particular, we have pave the ground to study truncations that contain a dynamical Gibbons-
Hawking-York term that would require to substitute the ordinary conformal projection tech-
nique by an x-dependent analog and then employ the TT-decomposition to evaluate the func-
tional trace by means of the heat kernel expansion. We have mentioned the necessary steps in
detail. Further studies in this direction can be used to study non-perturbative results to black
hole physics or cosmology either with or without boundary contributions. Thereby, dissolving
the tied bound of different invariants on the classical level already revealed hidden features on
the quantum level and will in future help understanding to which extent those results may effect
observations on our scales.
Another, very promising aspect studied in this thesis relates to the proposed C-function
property of the Effective Average Action. Establishing a suitableC-function can be an excellent
guiding principle from the theoretical perspective that would give rise to many diverse applica-
tions also outside Quantum Gravity. One possible way can be to employ the Ward Identities
for split-symmetry and use its properties to prove or adjust our proposal, ultimately leading to
a non-perturbative constraint on the RG evolution.
Thus, there are many ways to continue the journey towards a better understanding of Nature.

Appendix
A
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
Books
Articles
Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

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ABBREVIATIONS
Acronyms
AC axiom of choice
ADM Arnowitt-Deser-Misner
AS Asymptotic Safety
BH Black Hole
BI Background Independence
BRST Becchi, Rouet, Stora and Tyutin
CDT Causal Dynamical Triangulations
CMB Cosmic Microwave Background
DOF degrees of freedom
EAA Effective Average Action
EQFT Euclidean Quantum Field theory
ERGE Exact Renormalization Group Equation
FLRW Friedmann–Lemaître–Robertson–Walker
FP fixed point
FRG Functional Renormalization Group
FRGE Functional Renormalization Group Equation
GFP Gaußian fixed point
GR General Relativity
IR infrared
LHC Large Hadron Collider
LHS left-hand-side
LQG Loop Quantum Gravity
NGFP non-Gaußian fixed point
QCD Quantum chromodynamics
QED Quantum Electrodynamics
QEG Quantum Einstein Gravity
QFT Quantum Field theory
474
QG Quantum Gravity
RG Renormalization Group
RHS right-hand-side
SM Standard Model of particle physics
SR Special Relativity
TT transverse-traceless
UV ultraviolet
WISS Ward Identities for split-symmetry
ZF Zermelo–Fraenkel
ZFC Zermelo–Fraenkel set theory with the axiom of choice
Index
BRST transformation . . . . . . . . . . . . . . . . . . 89
Symbols
C
σ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Cameron-Martin theorem . . . . . . . . . . . . . . . 83
Cardinality
A Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Category
Ambrose-Singer theorem . . . . . . . . . . . . . . . 49 Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Associated bundle. . . . . . . . . . . . . . . . . . . . . .45 Closed set
Asymptotic Safety . . . . . . . . . . . . . . . . . . . . 151 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Atiyah-Singer index theorem . . . . . . . . . . . . 70 Conformal projection technique . . . . . . . . 223
Atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Connection
associated . . . . . . . . . . . . . . . . . . . . . . . . . 49
Levi-Civita . . . . . . . . . . . . . . . . . . . . . . . . 50
B
principal . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Background field covering . . . . . . . . . . . . . . 64 Continuous function
Background field method . . . . . . . . . . . . . . . 61 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Background Independence . . . . . . . . . . . . . 129 Cotangent space . . . . . . . . . . . . . . . . . . . . . . . 41
Basis invariants . . . . . . . . . . . . . . . . . . . . . . . . 73 Critical exponent . . . . . . . . . . . . . . . . . . . . . 296
Bochner theorem. . . . . . . . . . . . . . . . . . . . . . .87 Curvature
Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 13, 125 two-form . . . . . . . . . . . . . . . . . . . . . . . . . 49
hypersurface . . . . . . . . . . . . . . . . . . . . . 125 Cutoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
normal vector . . . . . . . . . . . . . . . . . . . . 125 action . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Quantum Gravity . . . . . . . . . . . . . . . . . 129 operator . . . . . . . . . . . . . . . . . . . . . . . . . 144
Boundary condition . . . . . . . . . . . . . . . . . . . . 59 shape function. . . . . . . . . . . . . . . . . . . .145
Dirichlet . . . . . . . . . . . . . . . . . . . . . . . . . . 62 smooth . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Neumann . . . . . . . . . . . . . . . . . . . . . . . . . 62 Cutoff shape function . . . . . . . . . . . . . . . . . 145
476 INDEX
Functional Renormalization Group Equation
145, 150
D derivation . . . . . . . . . . . . . . . . . . . . . . . . 145
Derivative
covariant exterior . . . . . . . . . . . . . . . . . . 48 G
exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Derivative Gauß-Codazzi equation . . . . . . . . . . . . . . . 130
Lie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Gaußian measure . . . . . . . . . . . . . . 82, 83, 114
Differential form . . . . . . . . . . . . . . . . . . . . . . . 42 Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Dimension Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
vector space . . . . . . . . . . . . . . . . . . . . . . . . 7 Gauge fixing condition . . . . . . . . . . . . . . . . . 90
Gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . 56
Gauge transformation
pure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
E Gauge transformations . . . . . . . . . . . . . . . . . 56
Effective action . . . . . . . . . . . . . . . . . . . . 97, 98 General Relativity . . . . . . . . . . . . . . . . . . . . 122
Effective Average Action . . . . . . . . . . . . . . 149 Generating functional . . . . . . . . . . . . . . . . . . 86
Einstein’s field equation . . . . . . . . . . . . . . . 125 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Gibbons-Hawking-York functional . . . . . 125
Euclidean quantum field theory . . . . . . . . 118
Gribov copies . . . . . . . . . . . . . . . . . . . . . . . . . 90
Euclidean Quantum Gravity . . . . . . . . . . . .131
Group theory
Euclidean space
Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
with boundary . . . . . . . . . . . . . . . . . . . . . . 9
Conjugacy class . . . . . . . . . . . . . . . . . . . 15
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Coset spaces . . . . . . . . . . . . . . . . . . . . . . 16
Exponential map . . . . . . . . . . . . . . . . . . . . . . . 51
Direct product . . . . . . . . . . . . . . . . . . . . . 17
Equivalent representation . . . . . . . . . . . 20
Equivariant map . . . . . . . . . . . . . . . . . . . 20
F General linear group . . . . . . . . . . . . . . . 19
Group action . . . . . . . . . . . . . . . . . . . . . . 17
Faddeev-Popov method . . . . . . . . . . . . . . . . . 94 Isotropy group. . . . . . . . . . . . . . . . . . . . .19
Fiber bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . 26
connection . . . . . . . . . . . . . . . . . . . . . . . . 46 Lie group . . . . . . . . . . . . . . . . . . . . . . . . . 25
covariant derivative . . . . . . . . . . . . . . . . 46 Linear representation . . . . . . . . . . . . . . . 19
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 39 Little group . . . . . . . . . . . . . . . . . . . . . . . 19
equivalence . . . . . . . . . . . . . . . . . . . . . . . 40 Normal subgroup . . . . . . . . . . . . . . . . . . 16
Exponential map. . . . . . . . . . . . . . . . . . .52 Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Field space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Product groups . . . . . . . . . . . . . . . . . . . . 16
Riemannian metrics . . . . . . . . . . . . . . . 127 Schur’s lemma. . . . . . . . . . . . . . . . . . . . . 21
Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Semidirect product . . . . . . . . . . . . . . . . . 16
Fixed point Stabilizer subgroup . . . . . . . . . . . . . . . . 19
Gaußian . . . . . . . . . . . . . . . . . . . . . . . . . 152 Subgroups . . . . . . . . . . . . . . . . . . . . . . . . 15
non-Gaußian . . . . . . . . . . . . . . . . . . . . . 152 Symmetry phase . . . . . . . . . . . . . . . . . . . 24
non-trivial . . . . . . . . . . . . . . . . . . . . . . . 152 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
trivial . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Free theory . . . . . . . . . . . . . . . . . . . . . . . 84, 101
Functional H
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 71
Functional derivative . . . . . . . . . . . . . . . . . . . 72 Heat kernel . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Functional Integro-Differential Equation 150 Hessian
INDEX 477
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 73
Hessian operator
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 73 O
Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Holonomy group. . . . . . . . . . . . . . . . . . . . . . .49 Observable
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 80
Operator
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 66
I determinant . . . . . . . . . . . . . . . . . . . . . . . 71
normal . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
Inner product . . . . . . . . . . . . . . . . . . . . . . . . . . 66 projection . . . . . . . . . . . . . . . . . . . . . . . . . 68
Isomorphism self-adjoint . . . . . . . . . . . . . . . . . . . . . . . . 68
Bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 symmetric . . . . . . . . . . . . . . . . . . . . . . . . 68
Homeomorphism . . . . . . . . . . . . . . . . . . . 5 trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70
unitary . . . . . . . . . . . . . . . . . . . . . . . . . . . .68
Osterwalder-Schrader axioms. . . . . . .87, 118
L
Lebesgue measure . . . . . . . . . . . . . . . . . . . . 110 P
Legendre-Fenchel transform . . . . . . . . . . . . 98
Parameter expansion . . . . . . . . . . . . . . . . . . 115
Level-expansion . . . . . . . . . . . . . . . . . . . . . . 203
Partition of unity . . . . . . . . . . . . . . . . . . . . . . . 88
Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Perturbation theory . . . . . . . . . . . . . . . . . . . 112
Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Pontryagin duality theorem . . . . . . . . . . . . . 87
Loop expansion . . . . . . . . . . . . . . . . . . . . . . 115 Predictivity . . . . . . . . . . . . . . . . . . . . . . . . . . 295
Principal bundle . . . . . . . . . . . . . . . . . . . . . . . 44
Product measure . . . . . . . . . . . . . . . . . . . . . . . 82
M
Matter fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Q
Maurer-Cartan form . . . . . . . . . . . . . . . . . . . . 52
Measurable space . . . . . . . . . . . . . . . . . . . . . . 77 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 104
Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 Quantum field theory. . . . . . . . . . . . . . . . . .103
construction . . . . . . . . . . . . . . . . . 105, 107
σ -finite . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Quantum Gravity. . . . . . . . . . . . . . . . .124, 129
complete . . . . . . . . . . . . . . . . . . . . . . . . . . 78
CDT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
equivalence . . . . . . . . . . . . . . . . . . . . . . . 83
experimental search . . . . . . . . . . . . . . . 134
Fourier transform . . . . . . . . . . . . . . . . . . 86
Loop Quantum Gravity . . . . . . . . . . . .132
invariant . . . . . . . . . . . . . . . . . . . . . . . . . . 78
String theory . . . . . . . . . . . . . . . . . . . . . 132
pushforward. . . . . . . . . . . . . . . . . . . . . . .83 theories of . . . . . . . . . . . . . . . . . . . . . . . 131
Measure theory Asymptotic Safety . . . . . . . . . . . . . . . . 151
Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Measurable function . . . . . . . . . . . . . . . 80
Simple function . . . . . . . . . . . . . . . . . . . 80 R
Support of a measure . . . . . . . . . . . . . . . 79
Mellin transform. . . . . . . . . . . . . . . . . . . . . .244 Radon-Nikodym theorem . . . . . . . . . . . . . . . 85
Method of steepest descent . . . . . . . . . . . . 115 Reflection positivity . . . . . . . . . . . . . . . . . . .118
Metric Regularization . . . . . . . . . . . . . . . . . . . . . . . . 116
metric space . . . . . . . . . . . . . . . . . . . . . . . . 7 Renormalizability . . . . . . . . . . . . . . . . . . . . .110
Minlos’ theorem . . . . . . . . . . . . . . . . . . . . . . . 86 Asymptotic Safety . . . . . . . . . . . . . . . . 152
478 INDEX
perturbative . . . . . . . . . . . . . . . . . . . . . . 116 Topological space
Renormalization . . . . . . . . . . . . . . . . . . . . . . 111 Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Renormalization Group . . . . . . . . . . . . . . . 140 Hausdorff space . . . . . . . . . . . . . . . . . . . 10
scale . . . . . . . . . . . . . . . . . . . . . . . . 140, 154 Open sets . . . . . . . . . . . . . . . . . . . . . . . . . . 3
time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 Second-countable space . . . . . . . . . . . . 10
Representation Compactness . . . . . . . . . . . . . . . . . . . . . . . 4
Character . . . . . . . . . . . . . . . . . . . . . . . . . 20 Connectedness . . . . . . . . . . . . . . . . . . . . . 3
Irreducible . . . . . . . . . . . . . . . . . . . . . . . . 21 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Reducible . . . . . . . . . . . . . . . . . . . . . . . . . 21 Locally Euclidean (with boundary) . . . 9
Restricted . . . . . . . . . . . . . . . . . . . . . . . . . 21 Subspace topology . . . . . . . . . . . . . . . . . . 8
Riemannian manifold . . . . . . . . . . . . . . . . . . .51 Triviality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Riemannian metric Triviality problem . . . . . . . . . . . . . . . . . . . . 152
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 51 Truncations . . . . . . . . . . . . . . . . . . . . . . . . . . 153
TT-decomposition . . . . . . . . . . . . . . . . . . . . 213
S
U
Saddle point approximation . . . . . . . . . . . . 115
Schwinger functional . . . . . . . . . . . . . . . 95, 97 UV-critical-hypersurface . . . . . . . . . . . . . . 295
Section
Global . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Local . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 V
Seeley-DeWitt coefficient . . . . . . . . . . . . . .246
Set Vector field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 2 fundamental . . . . . . . . . . . . . . . . . . . . . . . 47
Smooth manifold
atlas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Derivation . . . . . . . . . . . . . . . . . . . . . . . . 40 W
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 12 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . 131
Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . 1, 13 Wightman axioms . . . . . . . . . . . . . . . . . . . . 118
maximally symmetric . . . . . . . . . . . . . 232
singularities . . . . . . . . . . . . . . . . . . . . . . 127
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 13
Spectral theorem . . . . . . . . . . . . . . . . . . . 66, 68
Stability Matrix . . . . . . . . . . . . . . . . . . . . . . .296
Symmetry
extrinsic . . . . . . . . . . . . . . . . . . . . . . . . . . 65
intrinsic. . . . . . . . . . . . . . . . . . . . . . . . . . .65
T
Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . 40
Tensor bundles . . . . . . . . . . . . . . . . . . . . . . . . 40
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 42
Tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Theory space . . . . . . . . . . . . . . . . . . . . 139, 140
truncations . . . . . . . . . . . . . . . . . . . . . . . 153
Topological manifold with boundary
Definition . . . . . . . . . . . . . . . . . . . . . . . . . 11