# Gauge transformation in macroscopic quantum electrodynamics near polarizable surfaces

## TL;DR

This paper explores gauge transformations in macroscopic quantum electrodynamics near polarizable surfaces, clarifying how different gauges affect the Hamiltonian and ensuring gauge invariance in boundary-influenced electromagnetic interactions.

## Contribution

It provides an explicit gauge transformation between generalized and true Coulomb gauges near dielectric boundaries, including the derivation of the corresponding Hamiltonian and commutation relations.

## Key findings

- Derived explicit gauge transformation for boundary conditions
- Identified additional terms in the Hamiltonian due to surface charge fluctuations
- Confirmed gauge invariance reproduces known electrostatic results

## Abstract

To describe charged particles interacting with the quantized electromagnetic field, we point out the differences of working in the so-called generalized and the true Coulomb gauges. We find an explicit gauge transformation between them for the case of the electromagnetic field operators quantized near a macroscopic boundary described by a piece-wise constant dielectric function. Starting from the generalized Coulomb gauge we transform operators into the true Coulomb gauge where the vector potential operator is truly transverse everywhere. We find the generating function of the gauge transformation to carry out the corresponding unitary transformation of the Hamiltonian and show that in the true Coulomb gauge the Hamiltonian of a particle near a polarizable surface contains extra terms due to the fluctuating surface charge density induced by the vacuum field. This extra term is represented by a second-quantised operator on equal footing with the vector field operators. We demonstrate that this term contains part of the electrostatic energy of the charged particle interacting with the surface and that the gauge invariance of the theory guarantees that the total interaction energy in all cases equals the well known result obtainable by the method of images when working in generalized Coulomb gauge. The mathematical tools we have developed allow us to work out explicitly the equal-time commutation relations and shed some light on typical misconceptions regarding issues of whether the presence of the boundaries should affect the field commutators or not, especially when the boundaries are modelled as perfect reflectors.