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European Journal of Physics

Application of topological ideas to the Aharonov-Bohm effect

A. Heil, Alfred Kersch, B. Reifenhäuser, Helmut Vogel

Published Jul 1, 1988• 2 citations
Publisher

Abstract

The Aharonov-Bohm effect is studied to elucidate the existence of global effects in field theory, originating from the topology of the field configuration. The problem is formulated in the language of Maxwell's equations, cohomology and principal fibre bundle theory. The authors offer a motivation for the use of these concepts. In particular, the idealised and realistic descriptions of the Aharonov-Bohm effect are distinguished.

DOI
10.1088/0143-0807/9/3/007

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The Aharonov-Bohm effect is studied to elucidate the existence of global effects in field theory, originating from the topology of the field configuration. The problem is formulated in the language of Maxwell's equations, cohomology and principal fibre bundle theory. The authors offer a motivation for the use of these concepts. In particular, the idealised and realistic descriptions of the Aharonov-Bohm effect are distinguished.
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The Aharonov-Bohm (A-B) effect has been a major focus of the foundations of physics. And yet, much confusion persists. In particular, the effect purportedly leads to a dilemma: on one horn, we have a non-local action of a gauge-invariant quantity on charged particles; on the other, we get a local action on these particles, but of a non-gauge invariant quantity. This is the folklore, but the folklore is filled with misconceptions. Here, by deploying a recently defended formulation of gauge theory that dispenses with principal bundles, gauge potentials, and explicit gauge symmetries, I argue, with previous authors, that the A-B effect can be understood gauge-independently. But here my argument will go further: I will show that the A-B effect, when expressed in terms of the covariant derivative of a vector bundle, is entirely analogous to the holonomy of spacetime vectors, and can be understood completely locally. The only surprising idea illustrated by the A-B effect is that, in some circumstances, there is more to the covariant derivative than can be accounted for by the curvature and underlying topology of a vector bundle.
Abstract The Aharonov–Bohm (AB) effect has been highly influential in fundamental and applied physics. Its topological nature commonly implies that an electron encircling a magnetic flux source in a field-free region must close the loop in order to generate an observable effect. In this paper, we study a variant of the AB effect that apparently challenges this concept. The significance of weak values and nonlocal equations of motion is discussed as part of the analysis, shedding light on and connecting all these fundamental concepts.
Since its discovery in 1959 the Aharonov-Bohm effect has continuously been the cause for controversial discussions of various topics in modern physics, e.g. the reality of gauge potentials, topological effects and nonlocalities. In the present paper we juxtapose the two rival interpretations of the Aharonov-Bohm effect. We show that the conception of nonlocality encountered in the Aharonov-Bohm effect is closely related to the nonseparability which is common in quantum mechanics albeit distinct from it due to its topological nature. We propose a third alternative interpretation based on the loop space of holonomies which serves to solve some of the problems and we trace back the topological nonlocality and thereby the Aharonov-Bohm effect to their quantum mechanical origin. All three discussed interpretations are, of course, empirically equivalent. In fact, they present us with an instructive case study for the thesis of theory underdetermination by empirical data.