On the Path-Integral Derivation of the Anomaly for the Hermitian Equivalent of the Complex $PT$-Symmetric Quartic Hamiltonian

TL;DR

This paper investigates the path-integral derivation of the anomaly in the Hermitian equivalent of a complex PT-symmetric quartic Hamiltonian, clarifying the necessary careful treatment to correctly reproduce the linear term as a parity anomaly.

Contribution

It provides a more precise path-integral derivation of the anomaly, correcting previous oversights and introducing a simpler alternative method using curvilinear coordinate transformations.

Findings

Naive path-integral derivation misses the linear term.

A more careful discretized approach reproduces the anomaly.

An alternative derivation uses coordinate change in functional integrals.

Abstract

It can be shown using operator techniques that the non-Hermitian $PT$-symmetric quantum mechanical Hamiltonian with a "wrong-sign" quartic potential $-gx^4$ is equivalent to a Hermitian Hamiltonian with a positive quartic potential together with a linear term. A naive derivation of the same result in the path-integral approach misses this linear term. In a recent paper by Bender et al. it was pointed out that this term was in the nature of a parity anomaly and a more careful, discretized treatment of the path integral appeared to reproduce it successfully. However, on re-examination of this derivation we find that a yet more careful treatment is necessary, keeping terms that were ignored in that paper. An alternative, much simpler derivation is given using the additional potential that has been shown to appear whenever a change of variables to curvilinear coordinates is made in a functional integral.