Comparing Hermitian and Non-Hermitian Quantum Electrodynamics

TL;DR

This paper compares Hermitian and non-Hermitian quantum electrodynamics, revealing an equivalence between the two approaches and discussing their physical interpretations in modeling quantum systems.

Contribution

It introduces a non-standard inner product for local electric and magnetic fields, establishing a biorthogonal description and comparing it with the Hermitian approach.

Findings

An equivalence between Hermitian and non-Hermitian descriptions is found.

Different physical interpretations depend on the modeling context.

A non-standard inner product leads to a natural biorthogonal field description.

Abstract

In recent years, non-Hermitian quantum physics has gained a lot in popularity in the quantum optics and condensed matter communities in order to model quantum systems with varying symmetries. In this paper, we identify a non-standard inner product that implies bosonic commutator relations for local electric and magnetic field observables and leads to a natural local biorthogonal description of the quantised electromagnetic field. When comparing this description with an alternative local Hermitian description, in which the states of local photonic particles - i.e. of so-called bosons localised in position (blips) - are orthogonal under the conventional Hermitian inner product, we find that there is an equivalence between the two approaches. Careful consideration needs to be given to the physical interpretation of the different descriptions. Whether a Hermitian or a non-Hermitian approach is more suitable depends on the circumstances that we want to model.