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arXiv:2111.00476

A Classical (Local) Account of The Aharonov-Bohm Effect

Vlatko Vedral

Published Oct 31, 2021• 2 citations
arXivHTMLPDF

Abstract

It is frequently stated that the electromagnetic vector potential acquires a fundamental role in quantum physics, whereas classically it only represents a convenient, but by no means necessary, way of representing the electromagnetic field. Here we argue that this is a historical accident due to the fact that the electromagnetic field was discovered before photons, while the electron itself was discovered first as a particle, before it became clear that it must also be treated as a wave and therefore as an excitation of the underlying electron field. We illustrate the fact that the vector potential ought to play a fundamental role classically using the Aharonov-Bohm effect. This effect is considered as the strongest argument for the role the vector potential plays in quantum physics, however, here we offer a fully classical account of it. This is a consequence of the fact that any account, be it classical or quantum, must involve the vector potential in order to preserve the local nature of the Aharonov-Bohm (as well as all the other) phases.

DOI
10.48550/arxiv.2111.00476

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It is frequently stated that the electromagnetic vector potential acquires a fundamental role in quantum physics, whereas classically it only represents a convenient, but by no means necessary, way of representing the electromagnetic field. Here we argue that this is a historical accident due to the fact that the electromagnetic field was discovered before photons, while the electron itself was discovered first as a particle, before it became clear that it must also be treated as a wave and therefore as an excitation of the underlying electron field. We illustrate the fact that the vector potential ought to play a fundamental role classically using the Aharonov-Bohm effect. This effect is considered as the strongest argument for the role the vector potential plays in quantum physics, however, here we offer a fully classical account of it. This is a consequence of the fact that any account, be it classical or quantum, must involve the vector potential in order to preserve the local nature of the Aharonov-Bohm (as well as all the other) phases.
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The AB (Aharonov-Bohm) effect is a pivotal quantum mechanical phenomenon that illustrates the fundamental role of the electromagnetic vector potential A in determining the phase of a charged particle's wave function, even in regions where the magnetic field B is zero.This effect demonstrates that quantum particles are influenced not only by the fields directly present but also by the potentials associated with those fields.In the AB effect, an electron beam is split into two paths, with one path encircling a solenoid and the other bypassing it.Despite the absence of a magnetic field in the regions traversed by the beams, the vector potential A associated with the magnetic flux  through the solenoid induces a phase shift in the electron's wave function.This phase shift, quantified by qc     , manifests as a change in the interference pattern observed in the detection screen.The phenomenon underscores the principle of gauge invariance in QED (quantum electrodynamics), where physical observables remain invariant under local gauge transformations of the vector and scalar potentials.This reinforces the notion that the vector potential A has a profound impact on quantum systems, beyond its classical role.This article outlines the AB effect, including its theoretical framework, experimental observations, and implications.The focus on the role of the vector potential in quantum mechanics provides a comprehensive understanding of this important phenomenon.
We present a review and discussion of the physical meaning of the vector potential in electromagnetism by means of a classical experiment in which a long solenoid, connected to a sinusoidal voltage supply, produces an electric field and a vector potential in a circuit exterior to the solenoid itself. Some basic considerations are recalled concerning the meaning of action-at-a-distance and local description, in terms of 'real fields' of the induced electromotive force on the circuit outside the solenoid. From this viewpoint the roles played by the magnetic field, the electric field and the vector potential to account for the physical effects are reviewed. Finally, the connections between these considerations and the magnetic Aharonov–Bohm effect are recalled in the light of recent theoretical work aimed at explaining the Aharonov–Bohm phase shift in terms of the magnetic field without resorting to the vector potential.
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.
The Aharonov-Bohm effect is a quantum mechanical effect, that is, has no classical counterpart. The effect was predicted in 1959 in a seminal paper of Y. Aharonov and D. Bohm [AB59] in which they demonstrated that a beam of electrons is affected by the existence of the electric/magnetic field even though electrons travel through field-free regions. Aharonov and Bohm carried out two hypothetical experiments to support their claim that potentials are more fundamental than fields and they are respon-sible of the effect. Since then, the debate has arisen around whether potentials are mathematical tools or fundamental entities in physics. Different arguments have been set to explain the results predicted by Aharonov and Bohm and experimentally confirmed. Amongst these arguments, the first argument adopted by Aharonov and Bohm was that potentials are physically significant. Many have claimed that fields do have non-local features, i.e. action at a distance. Others have claimed that topological effects may interpret the effect in which potentials are modeled as connections in higher-dimensional fiber bun-dle geometries. The most recent argument has been proposed by Vaidman [Vai12] who claimed that the the composite system is represented by one state, an entangled state, and due to the electromagnetic interactions part of this state is changed, hence, the total state. In the present essay, I will discuss the latter argument as well as reviewing some other arguments. i
There is a consensus today that the the main lesson of the Aharonov-Bohm effect is that a picture of electromagnetism based on the local action of the field strengths is not possible in quantum mechanics. Contrary to this statement, it is argued here that when the source of the electromagnetic potential is treated in the framework of quantum theory, the Aharonov-Bohm effect can be explained without the notion of potentials. It is explained by local action of the field of the electron on the source of the potential. The core of the Aharonov-Bohm effect is the same as the core of quantum entanglement: the quantum wave function describes all systems together.