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We formally prove the existence of a quantization procedure that makes thepath integral of a general diffeomorphism-invariant theory of gravity, withfixed total spacetime volume, equivalent to that of its unimodular version.This is achieved by means of a partial gauge fixing of diffeomorphisms togetherwith a careful definition of the unimodular measure. The statement holds alsoin the presence of matter. As an explicit example, we consider scalar-tensortheories and compute the corresponding logarithmic divergences in bothsettings. In spite of significant differences in the coupling of the scalarfield to gravity, the results are equivalent for all couplings, includingnon-minimal ones.

Can quantum fluctuations differentiate between standard and unimodular gravity?