Gauge invariance of the Aharonov-Bohm effect in a quantum electrodynamics framework

TL;DR

This paper demonstrates the gauge invariance of the Aharonov-Bohm effect within a quantum electrodynamics framework, showing that the phase difference is gauge-invariant for closed paths but may depend on the gauge for nonclosed paths.

Contribution

It provides an exact quantum treatment of the electromagnetic field in the AB effect and proves gauge invariance for closed paths, clarifying theoretical foundations.

Findings

AB phase difference is gauge-invariant for closed paths

Electromagnetic ground energy depends on quantum particle paths

Gauge dependence may occur for nonclosed paths

Abstract

The gauge invariance of the Aharonov-Bohm (AB) effect with a quantum treatment for the electromagnetic field is demonstrated. We provide an exact solution for the electromagnetic ground energy due to the interaction of the quantum electromagnetic field with the classical charges and currents that act as sources of the potentials in a classical description, in the Lorenz gauge. Then, we use first-order perturbation theory to compute an extra change on the electromagnetic ground energy due to the presence of a quantum charged particle with known wave function in the system. This energy in general depends on the quantum particle path in an interferometer, what results in an AB phase difference between the paths. The gauge invariance of this AB phase difference is then shown for the magnetic, electric, and the recently proposed electrodynamic versions of the AB effect. However, the AB phase difference could depend on the gauge for nonclosed paths, what reinforces the view that it only can be measured in closed paths.