Gauging non-Hermitian Hamiltonians

TL;DR

This paper explores how to couple non-Hermitian Hamiltonians to electromagnetic fields by defining gauge transformations and minimal substitution, with applications to specific models like the Swanson Hamiltonian and imaginary cubic interactions.

Contribution

It introduces a method for gauging non-Hermitian systems using a metric operator, extending the concept of minimal substitution to these systems.

Findings

Gauge transformation in non-Hermitian systems leads to minimal substitution in momentum.

Calculation of electromagnetic transition matrix elements is demonstrated for specific models.

The approach applies to both local and non-local Hermitian equivalents of non-Hermitian Hamiltonians.

Abstract

We address the problem of coupling non-Hermitian systems, treated as fundamental rather than effective theories, to the electromagnetic field. In such theories the observables are not the $\bs{x}$ and $\bs{p}$ appearing in the Hamiltonian, but quantities $\bs{X}$ and $\bs{P}$ constructed by means of the metric operator. Following the analogous procedure of gauging a global symmetry in Hermitian quantum mechanics we find that the corresponding gauge transformation in $\bs{X}$ implies minimal substitution in the form $\bs{P}\to \bs{P} - e\bs{A}(\bs{X})$. We discuss how the relevant matrix elements governing electromagnetic transitions may be calculated in the special case of the Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian $h$ is local, and in the more generic example of the imaginary cubic interaction, where $H$ is local but $h$ is not.