On the time-dependent Aharonov-Bohm effect and the 4-dimensional Stokes theorem

TL;DR

This paper reexamines the time-dependent Aharonov-Bohm effect using the 4-dimensional Stokes theorem, challenging previous claims that the effect is canceled by induced electric fields, and suggests the effect likely exists.

Contribution

It provides a detailed quantum mechanical analysis and clarifies that the cancellation argument is incomplete, supporting the possible existence of the time-dependent AB-effect.

Findings

The cancellation argument by Singleton et al. is only partially correct.

The time-dependent AB-effect is likely to exist contrary to previous claims.

A careful application of the 4D Stokes theorem shows the effect is not fully canceled.

Abstract

The time-dependent Aharonov-Bohm (AB) effect considers the situation in which the magnetic flux inside the solenoid changes time-dependently. Different from the standard AB-effect, the problem is unexpectedly subtle and not easy to solve without any doubt, which is the reason why it is still in a state of unsettlement even theoretically. The difficulty originates from the fact that its theoretical analysis requires line-integrals of the time-dependent vector potential along paths in the 4-dimensional Minkowski space. Owing to the 4-dimensional Stokes theorem, this closed line-integral of the vector potential can be related to the integral of the electric and magnetic fields over the 2-dimensional area, the boundary of which is given by the above-mentioned closed path. The central controversy concerns the success or failure of the claim by Singleton and collaborators based on the 4-dimensional Stokes theorem, which states that the time-dependent part of the AB-phase shift due to the magnetic vector potential is precisely cancelled by the effect of induced electric field generated by the time-variation of the magnetic flux. In the present paper, we carefully reanalyze their cancellation argument by going back to the basic quantum mechanical analysis of the charged particle motion under the presence of the time-dependent external electromagnetic potential combined with more careful treatment of the 4-dimensional Stokes theorem. Careful analysis of the 4-dimensional Stokes theorem shows that the cancellation argument by Singleton et al. is partially correct but their central claim that only the time-independent part of the magnetic field contributes to the AB-phase shift so that there is no time-dependent AB-effect is not justified, thereby supporting likely existence of the time-dependent AB-effect.