Nonlocal Phases of Local Quantum Mechanical Wavefunctions in Static and Time-Dependent Aharonov-Bohm Experiments

TL;DR

This paper reveals that quantum wavefunction phases are influenced nonlocally by electromagnetic fields, extending the Aharonov-Bohm effect and resolving paradoxes related to causality in time-dependent quantum phenomena.

Contribution

It introduces a general gauge function including nonlocal field terms, correcting previous misconceptions and providing a comprehensive framework for understanding phase effects in static and dynamic electromagnetic environments.

Findings

Nonlocal field contributions affect wavefunction phases beyond traditional Aharonov-Bohm effects.

Cancellations of phases occur in static fields, explaining experimental observations.

New phases in time-dependent cases restore relativistic causality and clarify quantum interference phenomena.

Abstract

We show that the standard Dirac phase factor is not the only solution of the gauge transformation equations. The full form of a general gauge function (that connects systems that move in different sets of scalar and vector potentials), apart from Dirac phases also contains terms of classical fields that act nonlocally (in spacetime) on the local solutions of the time-dependent Schr\"odinger equation: the phases of wavefunctions in the Schr\"odinger picture are affected nonlocally by spatially and temporally remote magnetic and electric fields, in ways that are fully explored. These contributions go beyond the usual Aharonov-Bohm effects (magnetic or electric). (i) Application to cases of particles passing through static magnetic or electric fields leads to cancellations of Aharonov-Bohm phases at the observation point; these are linked to behaviors at the semiclassical level (to the old Werner & Brill experimental observations, or their "electric analogs" - or to recent reports of Batelaan & Tonomura) but are shown to be far more general (true not only for narrow wavepackets but also for completely delocalized quantum states). By using these cancellations, certain previously unnoticed sign-errors in the literature are corrected. (ii) Application to time-dependent situations provides a remedy for erroneous results in the literature (on improper uses of Dirac phase factors) and leads to phases that contain an Aharonov-Bohm part and a field-nonlocal part: their competition is shown to recover Relativistic Causality in earlier "paradoxes" (such as the van Kampen thought-experiment), while a more general consideration indicates that the temporal nonlocalities found here demonstrate in part a causal propagation of phases of quantum mechanical wavefunctions in the Schr\"odinger picture. This may open a direct way to address time-dependent double-slit experiments and the associated causal issues