Can quantum fluctuations differentiate between standard and unimodular gravity?

TL;DR

This paper demonstrates a formal equivalence between the path integrals of general diffeomorphism-invariant gravity theories and their unimodular counterparts, even with matter, through a specific gauge fixing and measure definition.

Contribution

It provides a rigorous proof of the equivalence of quantization procedures for standard and unimodular gravity, including matter fields, using a partial gauge fixing approach.

Findings

Path integrals of standard and unimodular gravity are equivalent with fixed spacetime volume.

The equivalence holds for scalar-tensor theories with various couplings.

Logarithmic divergences are identical in both gravity formulations.

Abstract

We formally prove the existence of a quantization procedure that makes the path integral of a general diffeomorphism-invariant theory of gravity, with fixed total spacetime volume, equivalent to that of its unimodular version. This is achieved by means of a partial gauge fixing of diffeomorphisms together with a careful definition of the unimodular measure. The statement holds also in the presence of matter. As an explicit example, we consider scalar-tensor theories and compute the corresponding logarithmic divergences in both settings. In spite of significant differences in the coupling of the scalar field to gravity, the results are equivalent for all couplings, including non-minimal ones.