Gauges and functional measures in quantum gravity I: Einstein theory

TL;DR

This paper computes one-loop divergences in Einstein gravity, exploring how different parametrizations and gauges affect results, and identifies a duality symmetry that singles out unimodular gravity as a special case.

Contribution

It systematically analyzes measure and gauge dependence in quantum Einstein gravity and reveals a duality symmetry that highlights unimodular gravity as a unique formulation.

Findings

Divergences are invariant under a Z2 duality transformation.

Certain parametrizations eliminate dependence on gauge and measure choices.

Unimodular gravity emerges as a self-dual theory under this symmetry.

Abstract

We perform a general computation of the off-shell one-loop divergences in Einstein gravity, in a two-parameter family of path integral measures, corresponding to different ways of parametrizing the graviton field, and a two-parameter family of gauges. Trying to reduce the gauge- and measure-dependence selects certain classes of measures and gauges respectively. There is a choice of two parameters (corresponding to the exponential parametrization and the partial gauge condition that the quantum field be traceless) that automatically eliminates the dependence on the remaining two parameters and on the cosmological constant. We observe that the divergences are invariant under a $\mathbf{Z}_2$ "duality" transformation that (in a particularly important special case) involves the replacement of the densitized metric by a densitized inverse metric as the fundamental quantum variable. This singles out a formulation of unimodular gravity as the unique "self-dual" theory in this class.