Gauging non-Hermitian Hamiltonians

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 $\text{bs}x$ and $\text{bs}p$ appearing in the Hamiltonian, but quantities $\text{bs}X$ and $\text{bs}P$ 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 $\text{bs}X$ 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.