How unimodular gravity theories differ from general relativity at quantum level

TL;DR

This paper compares the quantum path integral formulations of two unimodular gravity theories with general relativity, highlighting differences in the treatment of the cosmological constant and gauge conditions.

Contribution

It provides a detailed analysis of the path integral quantization of two unimodular gravity models and clarifies their relation to each other and to general relativity at the quantum level.

Findings

The fully diffeomorphism-invariant unimodular theory's path integral resembles GR but with an unconstrained cosmological constant.

The standard unimodular theory's path integral differs due to the average unimodular condition and zero-average constraints.

A canonical relation between the two unimodular theories is established.

Abstract

We investigate path integral quantization of two versions of unimodular gravity. First a fully diffeomorphism-invariant theory is analyzed, which does not include a unimodular condition on the metric, while still being equivalent to other unimodular gravity theories at the classical level. The path integral has the same form as in general relativity (GR), except that the cosmological constant is an unspecified value of a variable, and it thus is unrelated to any coupling constant. When the state of the universe is a superposition of vacuum states, the path integral is extended to include an integral over the cosmological constant. Second, we analyze the standard unimodular theory of gravity, where the metric determinant is fixed by a constraint. Its path integral differs from the one of GR in two ways: the metric of spacetime satisfies the unimodular condition only in average over space, and both the Hamiltonian constraint and the associated gauge condition have zero average over space. Finally, the canonical relation between the given unimodular theories of gravity is established.