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arXiv:2112.11171 · European Journal of Applied Physics

Comments on the Aharonov-Bohm Effect

Kazuyasu Shigemoto, Kunihiko Uehara B

Published Oct 20, 2022• 0 citations
arXivHTMLPDF

Abstract

In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is expressed by the gauge invariant physical current, gives the desired contribution to the Aharonov-Bohm effect. While the scalar mode of the vector potential, which changes under the gauge transformation so that it is the unphysical mode, give no contribution to the Aharonov-Bohm effect. Then Aharonov-Bohm effect really occurs by the physical longitudinal mode in the original Aharonov-Bohm’s setting. In the setting of Tonomura et al., where the magnet is shielded with the superconducting material, not only the magnetic field but also the longitudinal mode of the vector potential become massive by the Meissner effect. Then not only the magnetic field but also the physical longitudinal mode does not come out to the region where the electron travels. In such setting, only the scalar mode of the vector potential exists in the region where the electron travels, but there is no contribution to the Aharonov-Bohm effect from that mode.

DOI
10.24018/ejphysics.2022.4.5.209

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In the original setting of the Aharonov-Bohm, the gauge invariant physical longitudinal mode of the vector potential, which is expressed by the gauge invariant physical current, gives the desired contribution to the Aharonov-Bohm effect. While the scalar mode of the vector potential, which changes under the gauge transformation so that it is the unphysical mode, give no contribution to the Aharonov-Bohm effect. Then Aharonov-Bohm effect really occurs by the physical longitudinal mode in the original Aharonov-Bohm’s setting. In the setting of Tonomura et al., where the magnet is shielded with the superconducting material, not only the magnetic field but also the longitudinal mode of the vector potential become massive by the Meissner effect. Then not only the magnetic field but also the physical longitudinal mode does not come out to the region where the electron travels. In such setting, only the scalar mode of the vector potential exists in the region where the electron travels, but there is no contribution to the Aharonov-Bohm effect from that mode.
The vector potential in electrodynamics is investigated through the decomposition of its form to the following two parts; the one is the so-called transverse part represented by a divergenceless vector and the other the longitudinal part represented by an irrotational vector. The decomposition can be done by the Helmholtz theorem in the vector analysis because the conditions which should be required when we use the Helmholtz theorem are satisfied for the almost vector potentials of physically interesting problems. As an example of such interesting problems, we choose here the Aharonov-Bohm effect. As for the Aharonov-Bohm effect, the vector potential given in the original paper of Aharonov and Bohm has the singularities along the z-axis. We show that even for such a singular potential the Helmholtz theorem is held provided that the concept of the distribution is introduced in it. Generally, the transverse part of the vector potential obtained through such a decomposition is uniquely determined by the magnetic field and does not alter by a gauge transformation, on the other hand, the longitudinal part depends on the choice of special gauge. We show that the Aharonov-Bohm effect is due to the contribution of the transverse part of the vector potential and therefore should not be influenced by any gauge transformations.
The most popular interpretation of the Aharonov–Bohm (AB) effect is that the electromagnetic potential locally affects the complex phase of a charged particle’s wave function in the magnetic field free region. However, since the vector potential is a gauge-variant quantity, multiple researchers suspect that it is just a convenient tool for calculating the force field. This motivates them to explain the AB effect without using the vector potential, which inevitably leads to some sort of non-locality. This frustrating situation is shortly summarized by the statement by Aharonov et al. that the AB effect may be due to a local gauge potential or due to non-local gauge-invariant fields. In the present paper, we shall give several convincing arguments which support the viewpoint that the vector potential is not just a convenient mathematical tool with little physical entity. Despite its gauge arbitrariness, the vector potential certainly contains a gauge-invariant piece, which solely explains the observed AB phase shift. Importantly, this component has a property such that it is basically unique and cannot be eliminated by any regular gauge transformations. To complete the discussion, we also discuss the role of remaining gauge arbitrariness still contained in the entire vector potential.
This technical note gives a self-contained exposition of the Aharonov–Bohm effect, aimed at advanced undergraduate students. Starting from the role of electromagnetic potentials in classical electrodynamics, it introduces gauge freedom, the quantum Hamiltonian with minimal coupling, and the gauge covariance of the Schrödinger equation. It then derives the Aharonov–Bohm phase for both the time-dependent scalar configuration and the magnetic configuration, showing how a charged particle acquires an observable phase shift while travelling only through regions where the electromagnetic field vanishes. The final part reviews the experimental evidence, from the early electron-interference experiments of Chambers and of Möllenstedt and Bayh to the 1986 toroidal-magnet experiment of Tonomura and collaborators. In that experiment, superconducting shielding confined the magnetic flux completely and closed the leakage-field loophole, while flux quantization produced the characteristic half-fringe shift. The note is expository and pedagogical: it does not present original research results and has not been peer reviewed. Originally developed from study notes prepared for the PHYS02 scientific talk at Lady Margaret Hall, Oxford, in July 2026; an earlier version of the notes has been publicly available on GitHub since August 2026. Corrections and comments are more than welcome at giancarminesparso@gmail.com.
It is argued that much of the recent controversy over the Aharonov-Bohm effect has been fuelled by the widespread consideration of solenoids of infinite length: in this limit, the field and the vector potential have somewhat anomalous properties and it appears superficially that the Aharonov-Bohm phase shift can be changed by a gauge transformation. This incorrect impression is here removed by considering a solenoid of finite length and by carefully distinguishing the longitudinal and transverse parts of the vector potential. It is shown that the phase shift depends only on the transverse part of the vector potential and it cannot be changed by a gauge transformation. The nature of the difficulty that occurs in the limit of a solenoid of infinite length is examined in detail. The finite-length theory given here also provides guidelines for the design of experiments on the Aharonov-Bohm effect.
This article describes the Aharonov-Bohm (AB) effect of electron waves travelling in free space and its application to the observation of gauge fields (vector potentials). The AB effect is inconceivable in classical physics since an observable effect is produced on electrons passing through field-free spaces. Electrons can be affected only by Lorentz forces due to electromagnetic fields. The situation is different in quantum mechanics, where electrons show wave properties: the concept of force is no longer relevant, so electric field E and magnetic field B, defined as forces acting on a unit charge, take on secondary meanings. “Phase shifts” come into play, and the primary physical entities become neither E nor B but electrostatic potential V and vector potential A. These potentials interact with electron waves and shift their phases.