Configuration space for quantum gravity in a locally regularized path integral

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 parametrization 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 parametrization 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 parametrize 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 nonperturbative renormalization group setup.

Automated summary

This article discusses some aspects of the metric configuration space in quantum gravity in the background field formalism. A necessary and sufficient condition for the parametrization of Euclidean metric fluctuations is given, which preserves the signature of the metric in all configurations and provides a bijective map between full Euclidean metrics and metric fluctuations. For foliatable manifolds, a method to parametrize fluctuations to preserve foliatability is shown. The implementation of inequality constraints for preserving the signature on the configuration space in the Lorentzian quantum gravitational path integral is discussed, particularly in a nonperturbative renormalization group setup.