Path integral measures and diffeomorphism invariance

Abstract: Much like the action, diffeomorphism invariance can be used to fix the form of the path integral measure in quantum gravity. Moreover, since there is a redundancy between what constitutes "the action" and what constitutes "the measure" one can always pick a minimal form of the latter. However, the authors of the papers arXiv:2412.14108, arXiv:2412.10194 have advocated a form of the path integral measure for quantum gravity, proposed long ago by Fradkin and Vilkovisky, that is not invariant. This is easily seen since it depends explicitly on the $g^{00}$ component of the inverse metric without being contracted to form a scalar. An equally non-invariant measure was proposed in arXiv:2009.00728. As noted by their proponents, when these measures are used, certain divergences that typically appear are absent. However, the divergences that remain with the proposed measures are, unsurprisingly, neither diffeomorphism-invariant nor is the regulated effective action. We demonstrate this explicitly by computing the free scalar field contribution to the divergent part of the gravitational effective action using different measures and a proper-time cutoff. We support our findings with a thorough discussion of the path integral measure. In particular, we see how the contributions from the measure, obtained in a canonical setting, could be reinterpreted in a relational way compatible with diffeomorphism invariance.