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Physical Review

Aharonov-Bohm Paradox

G. T. Trammell

Published Jun 8, 1964• 36 citations
Publisher

Abstract

It is remarked that even in the classical limit the behavior of a charged particle interacting with a stationary current distribution exhibits paradoxical behavior. This comes from the inequality of action and reaction forces. It is shown that the vector potential term appearing in the expression for the canonical momenta represents the electromagnetic field momentum.

DOI
10.1103/physrev.134.b1183

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It is remarked that even in the classical limit the behavior of a charged particle interacting with a stationary current distribution exhibits paradoxical behavior. This comes from the inequality of action and reaction forces. It is shown that the vector potential term appearing in the expression for the canonical momenta represents the electromagnetic field momentum.
Abstract The situation of a charged particle passing down the symmetry axis through a magnetic toroid presents a relativity paradox; different inertial frames suggest different forces on the charge and on the toroid due to the unperturbed systems. We review the charge-toroid interaction and suggest that the magnetic Aharonov–Bohm situation is misunderstood because of unfamiliarity with the acceleration fields following from the Darwin Lagrangian, which go unmentioned in recent textbooks of classical electromagnetism.
In the context of the Aharonov-Bohm effect, it is well-known that the local action of the vector potential or the non-local action of the magnetic field are properties attributable to quantum physics, which can explain the observed Aharonov-Bohm phase. Here we point out that, surprisingly, these properties are also attributable to classical physics and that this result follows from examining the electromagnetic angular momentum of a static electric charge outside an infinitely-long solenoid of finite radius carrying a uniform magnetic flux. This angular momentum may be expressed in three equivalent forms. The first form shows that there is an electromagnetic angular momentum even in regions for which the magnetic field vanishes but the potential does not, which stresses the physical nature of the vector potential in classical physics. The second form shows that there is an electromagnetic angular momentum produced by the non-local action of the magnetic field. The third form shows that the Aharonov-Bohm phase is proportional to the magnitude of the electromagnetic angular momentum, which sheds light on the physical meaning of this phase.
The motion of a system of particles under electromagnetic interaction is considered. Under the assumption that the force acting on an electric charge is given by the sum of the electromagnetic fields produced by any other charged particles in its neighborhood, we prove that the vector potential of the electromagnetic field has to be considered for the balance of kinetic momentum. The theory cannot be quantized in the usual form -- because it involves a mass matrix that depends on spatial variables -- and the Hamilton's function becomes singular at a distance equal to the geometric mean of the electrodynamic radiuses of electrons and protons.
It is often said that the Aharonov-Bohm (AB) effect (1959) is an effect that is due to the vector potential. This is considered remarkable since a quantity that is not uniquely determined has a measurable effect. The authors show that, in the symmetric Coulomb gauge, (e/c)A(r) is simply the momentum of the electromagnetic field. The effect may equally well be considered an effect of the electromagnetic momentum. Thus, it is possible to discuss the AB effect strictly in terms of observables. The idea that the effect is due to the vector potential is correct, but sheds little light on the physics of the 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.
We compare different ways to the Aharonov–Bohm (A–B) phase for an electrically charged point-like particle in an electromagnetic field and disclose three mathematically equivalent representations for the A–B phase, which explicitly demonstrate its Lorentz invariance and gauge invariance. Based on these results, we obtain a linear dependence of the wave vector of a moving charged particle on the vector composition of mechanical and electromagnetic momenta, which sheds new light on the physical origin of the A–B effect.