Equivalence of a Complex $\cP\cT$-Symmetric Quartic Hamiltonian and a Hermitian Quartic Hamiltonian with an Anomaly

TL;DR

This paper demonstrates the equivalence between a non-Hermitian $ ext{PT}$-symmetric quartic Hamiltonian and a Hermitian counterpart using differential and functional integration methods, revealing an anomaly that leads to bound states.

Contribution

It provides a simple demonstration of the equivalence and uncovers an anomaly in the Hermitian form of a $ ext{PT}$-symmetric Hamiltonian, with implications for bound states.

Findings

The linear term in the Hermitian Hamiltonian is anomalous and has no classical analog.

The equivalence holds even when a harmonic term is added.

The anomaly causes the emergence of bound states.

Abstract

In a recent paper Jones and Mateo used operator techniques to show that the non-Hermitian $\cP\cT$-symmetric wrong-sign quartic Hamiltonian $H=\half p^2-gx^4$ has the same spectrum as the conventional Hermitian Hamiltonian $\tilde H=\half p^2+4g x^4-\sqrt{2g} x$. Here, this equivalence is demonstrated very simply by means of differential-equation techniques and, more importantly, by means of functional-integration techniques. It is shown that the linear term in the Hermitian Hamiltonian is anomalous; that is, this linear term has no classical analog. The anomaly arises because of the broken parity symmetry of the original non-Hermitian $\cP\cT$-symmetric Hamiltonian. This anomaly in the Hermitian form of a $\cP\cT$-symmetric quartic Hamiltonian is unchanged if a harmonic term is introduced into $H$. When there is a harmonic term, an immediate physical consequence of the anomaly is the appearance of bound states; if there were no anomaly term, there would be no bound states. Possible extensions of this work to $-\phi^4$ quantum field theory in higher-dimensional space-time are discussed.