Three Comments on the Aharonov-Bohm Effect

Abstract

The discovery by Aharonov and Bohm in 1959 (and its partial anticipation ten years earlier by Ehrenberg and Siday) was surprising and important and has been a fertile source of further discoveries. Neverthless some aspects of it have since become the subject of unnecessary mystification, embodied in three commonly held views on which I will comment. 1. THE AB EFFECT CONTRADICTS COMMON SENSE. The 'common sense' is that fields are physical entities, whereas the potentials the AB effect relies upon (to describe nonlocal influences of fields) are mere mathematical conveniences. This opinion is ahistorical because it ignores the fact that fields were themselves originally conceived as mathematical conveniences-to help describe the distant (that is nonlocal) effects of charges and gravitating masses. These actions at a distance were certainly not common sense, and children who play with magnets or are told that the tides are caused by the moon find these phenomena miraculous today (so do I). One of AB's achievements was to contribute to the process whereby yesterday's mathematical constructions prove their worth by pointing to new phenomena, and so become regarded as physical entities today. 2. THE AB EFFECT IS NONCLASSICAL. This is doubly misleading. First, although distant fields play no part in Newtonian classical mechanics (which deals only with forces), they contribute in Hamiltonian classical mechanics via the action (which depends on canonical rather than kinetic momentum and so involves the vector potential). Second-and in a deeper sense-the effect is a general one of wave physics, not confined to quantum mechanics. Any wave on a current, that is in a medium moving with velocity u(r) (much smaller than the wave speed), satisfies an equation with the same structure as Schrodinger's, in which u(r) plays the role of a vector potential. It follows that the analogue of a vector potential which describes a field vanishing everywhere outside a line is a flow for which Vxu(r)=O outside the line, and this is the flow of an irrotational vortex. Therefore one way to display the AB wavefunction is to let ripples on the surface of water encounter a bathtub vortex. (The analogue of the dimensionless AB quantum flux parameter is the circulation of the vortex, divided by the product of the • Following the lectures by A. Tonomura (this volume).