Configuration space for quantum gravity in a locally regularized path integral

TL;DR

This paper explores the metric configuration space in quantum gravity, establishing conditions for parameterizations that preserve metric signature and foliatability, and discusses implementing these constraints in a non-perturbative renormalization group framework.

Contribution

It provides a comprehensive analysis of parameterizations that maintain metric signature and foliatability in quantum gravity path integrals, including inequality constraints for Lorentzian metrics.

Findings

Necessary and sufficient conditions for metric parameterization preserving signature.

Explicit parameterization for foliatable manifolds.

Implementation of inequality constraints in non-perturbative RG.

Abstract

We discuss some aspects of the metric configuration space in quantum gravity in the background field formalism. We give a necessary and sufficient condition for the parameterization of Euclidean metric fluctuations such that i) the signature of the metric is preserved in all configurations that enter the gravitational path integral, and ii) the parameterization provides a bijective map between full Euclidean metrics and metric fluctuations about a fixed background. For the case of foliatable manifolds, we show how to parameterize fluctuations in order to preserve foliatability of all configurations. Moreover, we show explicitly that preserving the signature on the configuration space for the Lorentzian quantum gravitational path integral is most conveniently achieved by inequality constraints. We discuss the implementation of these inequality constraints in a non-perturbative renormalization group setup.