Quantum gravity, renormalizability and diffeomorphism invariance

TL;DR

This paper explores how the Wilsonian renormalization group can regularize quantum gravity, constructing a space of renormalizable interactions and addressing diffeomorphism invariance within this framework.

Contribution

It introduces a novel RG-based regularization of the Quantum Master Equation and constructs a Hilbert space of renormalizable quantum gravity interactions.

Findings

RG properties of the conformal factor enable a non-perturbative Hilbert space of interactions.

Diffeomorphism invariance is violated within the space but can be recovered at the boundary.

The framework sets the stage for second-order calculations in quantum gravity.

Abstract

We show that the Wilsonian renormalization group (RG) provides a natural regularisation of the Quantum Master Equation such that to first order the BRST algebra closes on local functionals spanned by the eigenoperators with constant couplings. We then apply this to quantum gravity. Around the Gaussian fixed point, RG properties of the conformal factor of the metric allow the construction of a Hilbert space $\mathfrak{L}$ of renormalizable interactions, non-perturbative in $\hbar$, and involving arbitrarily high powers of the gravitational fluctuations. We show that diffeomorphism invariance is violated for interactions that lie inside $\mathfrak{L}$, in the sense that only a trivial quantum BRST cohomology exists for interactions at first order in the couplings. However by taking a limit to the boundary of $\mathfrak{L}$, the couplings can be constrained to recover Newton's constant, and standard realisations of diffeomorphism invariance, whilst retaining renormalizability. The limits are sufficiently flexible to allow this also at higher orders. This leaves open a number of questions that should find their answer at second order. We develop much of the framework that will allow these calculations to be performed.