Nonlocal phases of local quantum mechanical wavefunctions in static and time-dependent Aharonov–Bohm experiments

Abstract

We show that the standard Dirac phase factor is not the only solution of the usual 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 (spatial or temporal integrals over potentials), also contains terms of classical fields that act nonlocally (in spacetime) on the local solutions of the time-dependent Schrödinger equation. As a result, the phases of wavefunctions in the Schrödinger picture are affected nonlocally by spatially and temporally remote magnetic and electric fields, in specific 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 full static magnetic or electric fields leads to cancellations of Aharonov–Bohm phases at the observation point; these cancellations are linked to behaviors at the semiclassical level (i.e. the old Werner and Brill experimental observations, or their ‘electric analogs’—or to more recent reports of Batelaan and Tonomura) but are shown to be far more general (true not only for narrow wavepackets but also for completely delocalized (spread-out) 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 (concerning an uncritical use 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 Schrödinger picture. This may open a new and direct way to address time-dependent double-slit experiments and the associated causal issues.