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Europhysics Letters (EPL)

Interpretation of the Aharonov-Bohm and the Aharonov-Casher Effects in Terms of Classical Electromagnetic Fields

Gianfranco Spavieri, G. Cavalleri

Published Feb 21, 1992• 14 citations
Publisher

Abstract

When an electric charge passes past a stationary current loop the time-varying fields induce changes of the internal energy of the loop. The magnetic flux of the loop also changes and, by Faraday's law, creates a radiation electric field which acts back on the particle. This interaction produces a classical electromagnetic lag which generates the phase shifts of the wave function foreseen by these effects. A possible experimental test is proposed.

DOI
10.1209/0295-5075/18/4/004

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A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces.

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Modern Physics Letters B•Sep 20, 1998•1 citation
When an electric charge passes past a stationary current loop the time-varying fields induce changes of the internal energy of the loop. The magnetic flux of the loop also changes and, by Faraday's law, creates a radiation electric field which acts back on the particle. This interaction produces a classical electromagnetic lag which generates the phase shifts of the wave function foreseen by these effects. A possible experimental test is proposed.
A new classical electromagnetic analysis is presented suggesting that the Aharonov-Bohm phase shift is overwhelmingly likely to arise from a classical lag effect based upon classical electromagnetic forces. The analysis makes use of several aspects of classical electromagnetic theory which are unfamiliar to most physicists, including the Darwin Lagrangian, acceleration-based electric fields, internal electromagnetic momentum in a magnet, and a magnet model involving at least three mutually-interacting particles. Only when the acceleration-based electric forces acting on the passing charge are included do we find consistency with all the relativistic conservation laws: energy, linear momentum, angular momentum, and constant center-of-mass velocity. The electric forces on the passing charge lead to a lag effect which accounts quantitatively for the Aharonov-Bohm phase shift. Thus the classical analysis strongly suggests that the Aharonov-Bohm phase shift (observed when electrons pass a long solenoid which corresponds to a line of magnetic dipoles) is the analogue of the Matteucci-Pozzi phase shift (observed when electrons pass a line of electric dipoles). The classical electromagnetic analysis suggests experiments to distinguish the proposed classical-based lag effect from the presently accepted view that the Aharonov-Bohm phase shift is a quantum topological effect arising from magnetic fluxes in the absence of classical electromagnetic forces.
In 1984 Aharonov and Casher proposed that a beam of neutral particles with magnetic dipole moments passing around opposite sides of a line charge will undergo a relative quantum phase shift. This is currently under experimental investigation. Aharonov and Casher claimed that the magnetic dipole particles should undergo the phase shift despite experiencing no classical force. They suggested that this new effect has a ``duality'' with the solenoid Aharonov-Bohm effect where charged particles passing around a magnetic solenoid experience a phase shift despite, it is claimed, experiencing no classical force. Here it is pointed out that a magnetic dipole particle passing a line charge does indeed experience a classical electromagnetic force in the usual electric-current model for a magnetic dipole. This force will produce a relative lag between dipoles passing on opposite sides of the line charge, and the classical lag then leads to a quantum phase shift in exact agreement with that calculated by Aharonov and Casher. Thus actually the proposed Aharonov-Casher effect has a transparent explanation as a classical lag effect. It is emphasized that the solenoid Aharonov-Bohm effect is currently the only phase-shift effect which cannot be explained in an obvious fashion as arising from a classical lag and it is again proposed that also this shift may actually involve a classical electromagnetic lag effect. A natural experimental consequence suggested by the lag point of view is the breakdown of the interference pattern when the lag becomes comparable to the wave-packet coherence length.
The magnetic Aharonov-Bohm effect shows that charged particles may be affected by the vector potential in regions without any electric or magnetic fields [1]. The Aharonov-Bohm effect was experimentally confirmed [2-3] and has been found in many situations [4-6]. A common explanation is based on quantum mechanics, which states that the wavefunctions associated with the charges will accumulate a phase shift due to the vector potential. However, consensus about its nature and interpretation has not been achieved [7-14]. We here propose a simple but reasonable interpretation based on the theory for electromagnetic radiation and couplings [15]. The energy associated with a pulse radiator is divided into a Coulomb-velocity energy and a radiative energy, together with a macroscopic Schott energy accounting for the energy exchange between them. All these energies are expressed with terms including the potentials, so are the mutual coupling energies. There exists a force acting on the moving charges even though the fields completely vanish. This force makes the charges pass through the magnetic solenoid in different velocity with different path length, causing a phase shift the same as that obtained with quantum mechanics. The theory is originally aimed for providing an interpretation for electromagnetic radiation and mutual coupling. It is derived directly from the Maxwell theory with no modification but only substitution and reorganization.
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.
Classical electromagnetic lag effects can give rise to quantum interference pattern shifts such as that observed experimentally in the Aharonov-Bohm effect involving electrons passing a solenoid. This paper presents an extensive comparison between interference pattern shifts based upon classical electromagnetic fields, and based upon classical electromagnetic potentials as suggested by Aharonov and Bohm. Stress is placed upon the difference between two types of interference pattern shifts: those involving deflection of the entire interference pattern and those involving a deflection of only the double-slit pattern while leaving the single-slit envelope undisplaced. The first type of shift is produced by a classical deflecting force. The second type of shift can be produced by classical electromagnetic lag effects, and is also the type of shift associated with the Aharonov-Bohm effect. The two types are confused in the literature. A new experiment is proposed which shows the relationship between a classical lag effect, due to electrostatic fields on electrons passing along different paths, and the associated quantum interference pattern shifts. The experiment is analyzed in detail using the WKB approximation in the Schr\"odinger equation and also semiclassical ideas. The classical limit for the situation illustrates the Bohr correspondence principle, showing the relative lag between the electron wave packets becoming a measurable classical lag with a disappearance of the interference pattern as the lag becomes large compared to the wave-packet dimensions. For small shifts, the phase change predicted for the new experiment is identical with the scalar potential effect proposed by Aharonov and Bohm for a slightly different, time-varying experimental arrangement. The theoretical and experimental differences for large phase shifts are noted. The possibility is raised that a new classical electromagnetic lag effect may occur for electrons passing a small solenoid. Using a particular model for energy conservation, the predicted lag effect can be calculated and is associated with a quantum interference pattern shift of the same magnitude as predicted by Aharonov and Bohm based upon the electromagnetic vector potential. Thus the possibility exists that the experiments of Chambers and of M\"ollenstedt and Bayh may not confirm the ideas of Aharonov and Bohm on the vector potential in quantum theory. Several experiments are suggested which allow confirmation that the Aharonov-Bohm effect indeed involves local effects of the classical electromagnetic potential, rather than local electromagnetic fields leading to a new classical lag effect and hence to the observed quantum interference pattern shift.
When an electron beam passes outside a confined magnetic flux, the motional magnetic field of the electrons overlaps the confined flux. The potential energy of the magnetic dipolar interaction of the motional magnetic field with the confined magnetic flux causes the velocity of the electrons to change slightly, which shifts the phase of the electron de Broglie wave. When portions of an electron beam pass on different sides of a confined flux and are then mixed, they produce an interference pattern. There is a similar magnetic dipolar interaction, with resulting velocity and phase changes, when a beam of magnetic particles interacts with the motional magnetic flux produced by their crossing a strong electric field. These are classical electromagnetic dipolar (not Lorentz) interactions that causes changes in Berry's geometrical phase, but are not ESAB or AC effects. Data that were suggested to demonstrate the ESAB and AC effects have been quantitatively interpreted in terms of motional magnetic fields and velocity changes.