Gauging 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.

The mimimal substitution rule given in v1 was incorrect. The corrected version appears in Eq. (21) of v2, and the necessary changes are made to the rest of the paper. In particular, gauge transformations are not now restricted, as they were in v1

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