Violation of the phase space general covariance as a diffeomorphism anomaly in quantum mechanics

TL;DR

This paper investigates a topological quantum mechanics model based on phase space path integrals, revealing an anomaly in diffeomorphism invariance caused by measure issues, even after super-extension fixes divergences.

Contribution

It identifies and calculates a diffeomorphism anomaly in a phase space quantum mechanics model, highlighting the subtlety of invariance under nonlinear transformations.

Findings

Naive bosonic integral is divergent and non-invariant.

Super-extension removes divergences and restores naive invariance.

Anomaly persists under nonlinear diffeomorphisms, with calculable lowest-order contribution.

Abstract

We consider a topological quantum mechanics described by a phase space path integral and study the 1-dimensional analog for the path integral representation of the Kontsevich formula. We see that the naive bosonic integral possesses divergences, that it is even naively non-invariant and thus is ill-defined. We then consider a super-extension of the theory which eliminates the divergences and makes the theory naively invariant. This super-extension is equivalent to the correct choice of measure and was discussed in the literature. We then investigate the behavior of this extended theory under diffeomorphisms of the extended phase space and despite of its naive invariance find out that the theory possesses anomaly under nonlinear diffeomorphisms. We localize the origin of the anomaly and calculate the lowest nontrivial anomalous contribution.