On Unimodular Gauge of Quantum Gravity

BRST-invariant action of general relativity in the unimodular gauge proposed by Baulieu is studied without using perturbative expansions. The expression for the path integral in the unimodular gauge is reduced to a form in which a functional measure is defined by a norm invariant under Transverse Diffeomorphism. It is shown that general relativity in the unimodular gauge with this action and the quantum unimodular gravity are equivalent. It is also shown that Vacuum Expectation Values (VEVs) of Diff invariant operators in the unimodular gauge and other gauge such as the harmonic gauge take distinct values. A path integral for harmonic gauge is found to be gauge equivalent to a superposition of that for unimodular gauge obtained by performing constant Weyl transformation of the metric, after a non-dynamical cosmological term is introduced into the action of the unimodular gauge.