Gauge independent description of Aharonov-Bohm Effect

TL;DR

This paper demonstrates that the Aharonov-Bohm effect can be fully explained using only the gauge-invariant magnetic field 2, emphasizing a non-local but gauge-independent description that does not rely on the vector potential 1.

Contribution

The authors provide a general proof showing the AB effect can be described solely with the magnetic field 2, removing the need for gauge-dependent potentials and highlighting non-locality.

Findings

The AB phase can be derived from the change in the magnetic field's action.

The gauge-invariant magnetic field 2 suffices to explain the AB effect.

Non-locality replaces gauge dependence in the description of the AB effect.

Abstract

The Aharonov-Bohm (AB) effect is a pure quantum effect that implies a measurable phase shift in the wave function for a charged particle that encircles a magnetic flux located in a region \textit{inaccessible} to the particle. Classically, such a non-local effect appears to be impossible since the Lorentz force depends on only the magnetic field at the location of the particle. In quantum mechanics, the Hamiltonian, and thus the Schr\"odinger equation, has a local coupling between the current due to the particle, and the electromagnetic vector potential $\mathbf{A}$, which extends to the entire space beyond the region with finite magnetic field. This has sometimes been interpreted as meaning that in quantum mechanics $\mathbf{A}$ is in some sense more "fundamental" than $\mathbf {B}$ in spite of the former being gauge dependent, and thus unobservable. Here we shall, with a general proof followed by a few examples, demonstrate that the AB-effect can be fully accounted for by considering only the gauge invariant $\mathbf{B}$ field, as long as it is included as part of the quantum action of the entire isolated system. The price for the gauge invariant formulation is that we must give up locality -- the AB-phase for the particle will arise from the change in the action for the $\mathbf{B}$ field in the region inaccessible to the particle.