The covariant, time-dependent Aharonov-Bohm Effect

TL;DR

This paper presents covariant formulations of the time-dependent Aharonov-Bohm effect, enabling calculations in dynamic scenarios and revealing phase shift cancellations in specific cases like a varying flux solenoid.

Contribution

It introduces two covariant expressions for the Aharonov-Bohm effect applicable to time-dependent situations, expanding theoretical understanding.

Findings

Phase shift cancels out in a time-varying flux solenoid due to electric field effects.

The formulations allow calculation of the effect in dynamic electromagnetic fields.

The results align with existing experimental observations.

Abstract

We discuss two possible covariant generalizations of the Aharonov-Bohm effect - one expression in terms of the space-time line integral of the four-vector potential and the other expression in terms of the space-time "area" integral of the electric and magnetic fields written in terms of the Faraday 2-form. These expressions allow one to calculate the Aharonov-Bohm effect for time-dependent situations. In particular, we use these expressions to study the case of an infinite solenoid with a time varying flux and find that the phase shift is zero due to a cancellation of the Aharonov-Bohm phase shift with a phase shift coming from the Lorentz force associated with the electric field, ${\bf E} = - \partial_t {\bf A}$, outside the solenoid. This result may already have been confirmed experimentally.