Asymptotically safe quantum gravity : bimetric actions, boundary terms, & a C-function
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Abstract
Today our understanding of Nature is based on two pillars, the classical theory of General Relativity 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 incorporate 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 important 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 generalized 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 introduced 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 a C-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 Asymptotic 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 primordial 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.