# Gauging non-Hermitian Hamiltonians - Quantum Physics

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Abstract: We address the problem of coupling non-Hermitian systems, treated asfundamental rather than effective theories, to the electromagnetic field. Insuch theories the observables are not the $\bs{x}$ and $\bs{p}$ appearing inthe Hamiltonian, but quantities $\bs{X}$ and $\bs{P}$ constructed by means ofthe metric operator. Following the analogous procedure of gauging a globalsymmetry in Hermitian quantum mechanics we find that the corresponding gaugetransformation 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 elementsgoverning electromagnetic transitions may be calculated in the special case ofthe Swanson Hamiltonian, where the equivalent Hermitian Hamiltonian $h$ islocal, and in the more generic example of the imaginary cubic interaction,where $H$ is local but $h$ is not.

Author: ** H. F. Jones**

Source: https://arxiv.org/