Field space parametrization in quantum gravity and the identification of a unitary conformal field theory at the heart of 2D Asymptotic Safety
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Abstract
Although only little is known about the precise quantum nature of the gravitational interaction, we can impose several essential requirements a consistent theory of quantum gravity must meet by all means: It must be renormalizable in order to remain well defined in the high energy limit, it must be unitary in order to admit a probabilistic interpretation, and it must be background independent as the spacetime geometry should be an outcome of the theory rather than a prescribed input. Being nonrenormalizable from the traditional, perturbative point of view, for a usual quantum version of general relativity already the first of these conditions seems to be ruled out. In the Asymptotic Safety program, however, a more general, nonperturbative notion of renormalizability is proposed, on the basis of which quantum gravity could be defined within the framework of conventional quantum field theory. The key ingredient to this approach is given by a nontrivial renormalization group fixed point governing the high energy behavior in such a way that the infinite cutoff limit is well defined. While there is mounting evidence for the existence of a suitable fixed point by now, investigations of background independence are still in their infancy, and the issue of unitarity is even more obscure.
In this thesis we extend the existing Asymptotic Safety studies by examining all three of the above conditions and their compatibility. We demonstrate that the renormalization group flow and its fixed points are sensitive to changes in the metric parametrization, where different qualified parametrizations, in turn, are seen to correspond to different field space connections. A novel connection is proposed, and the renormalization group flow resulting from the associated parametrization and a particular ansatz for the effective average action is shown to possess the decisive nontrivial fixed point required for nonperturbative renormalizability. For two special parametrizations we argue that background independence can be achieved in the infrared limit where all quantum fluctuations are completely integrated out.
In order to study the question of unitarity in an asymptotically safe theory we resort to a setting in two spacetime dimensions. We provide a detailed analysis of an intriguing connection between the Einstein-Hilbert action in d>2 dimensions and Polyakov's induced gravity action in two dimensions. By establishing the 2D limit of an Einstein-Hilbert-type effective average action at the nontrivial fixed point we reveal that the resulting fixed point theory is a conformal field theory, where the associated central charge, shown to be c=25, guarantees unitarity. Further properties of this theory and its implications for the Asymptotic Safety program are discussed.
In the last part of this work we present a strategy for conveniently reconstructing the bare theory pertaining to a given effective average action. For the Einstein-Hilbert case we prove the existence of a nontrivial fixed point in the bare sector and exploit the dependence of the bare action on the underlying functional measure to simplify the maps between bare and effective couplings. Applying this approach to 2D asymptotically safe gravity coupled to conformal matter we uncover a number of surprising consequences, for instance for the gravitational dressing of matter field operators and the KPZ scaling relations.