Is the Aharonov-Bohm phase shift for a non-closed path a measurable quantity ?

TL;DR

This paper investigates whether the Aharonov-Bohm phase shift for non-closed paths is measurable, concluding that it is gauge-invariant but not directly observable experimentally, based on a quantum mechanical analysis and path-integral formalism.

Contribution

It provides a gauge-invariant quantum mechanical analysis showing the AB-phase shift for non-closed paths is not directly measurable, challenging previous assumptions.

Findings

Interaction energies of Boyer and Saldanha are gauge invariant and cancel each other.

The AB-phase for non-closed paths is gauge-invariant but not an observable.

The origin of the AB-phase is linked to parts of the effective Hamiltonian beyond interaction energy.

Abstract

There recently appear some interesting attempts to explain the AB-effect through the interaction between the charged particle and the solenoid current mediated by the exchange of a virtual photon. A vital assumption of this approach is that AB-phase shift is proportional to the change of the interaction energy between the charged particle and solenoid along the path of the moving charge. Accordingly, they insist that the AB-phase change along a path does not depend on the gauge choice so that the AB-phase shift for a non-closed path is in principle measurable. We however notice the existence of two fairly different discussions on the interaction energy between the solenoid and a charge particle, the one is due to Boyer and the other is due to Saldanha and others. In the present paper, based on a self-contained quantum mechanical treatment of the combined system of a solenoid, a charged particle, and the quantized electromagnetic fields, we show that both interaction energies of Boyer and of Saldanha are in fact gauge invariant at least for non-singular gauge transformations but they are destined to cancel each other. Our analysis rather shows that the origin of the AB-phase can be traced back to other part of our effective Hamiltonian. Furthermore, based on the path-integral formalism with our effective Lagrangian, we explicitly demonstrate that the AB-phase shift for a non-closed path is not a gauge-variant quantity, which means that it would not correspond to direct experimental observables.