Mitchell, Alexander Valentine John
(2023)
The renormalization group and quantum gravity.
*University of Southampton, Doctoral Thesis*, 174pp.

## Abstract

A universally accepted theory of quantum gravity would play a crucial role in a complete understanding of Nature which has so far eluded physicists. This is illustrated by the variety and complexity of the many attempts to produce such a theory. A collection of analyses is presented which investigates a novel, natural approach which combines general relativity with quantum field theory in the framework of the Wilsonian renormalization group whilst respecting the conformal instability. We investigate the structure of this theory at first order in the coupling and verify that a continuum limit exists. We find that a continuum limit exists only if the theory is defined outside of the diffeomorphism invariant subspace. In the UV, interactions are associated to a coefficient function which is parametrised by an infinite number of fundamental couplings. In the physical limit diffeomorphism invariance is reinstated such that for a suitable choice of these couplings the coefficient functions trivialise. Dynamically generated effective diffeomorphism invariant couplings emerge, in particular Newton’s coupling. This investigation is continued to second order in perturbation theory. For pure quantum gravity, with vanishing cosmological constant, the result of the standard quantisation is recovered. Quantum gravity is renormalizable at second order for kinematic reasons but the structure is shown to hold in general. It may be the case that a continuum limit exists however with a, phenomenologically inconvenient, infinite number of fundamental couplings. However a possible non-perturbative resolution, based on the conformal instability and the parabolic properties of the flow equations, is investigated which would fix higher order effective couplings in terms of Newton’s constant and the cosmological constant. We then explore the properties of these flows with opposite natural direction through a related asymptotic safety problem, studying whether or not they have complete solutions. Finally we conclude with a discussion of the ramifications of this structure and possible applications to outstanding physical problems, in particular we investigate how well this structure is extended to arbitrary space-time dimensions and the physical consequences thereof.

**Doctoral Thesis Alex Mitchell - Version of Record**

**A. Mitchell Permission to deposit thesis**

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