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arXiv:hep-ph/9707468

(Semi)classical motion in fields of Aharonov-Bohm and Aharonov-Casher

Ya. I. Azimov, R.M. Ryndin

Published Jul 25, 1997• 0 citations
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Abstract

Particle motion in the fields of Aharonov-Bohm and Aharonov-Casher is considered in framework of the classical theory to reveal conditions admitting duality of the two configurations. Important role of orientation of the magnetic dipole moment is demonstrated. Duality becomes totally destroyed by addition of electric dipole and/or higher multipole moments. Correspondence between quantum and classical considerations is also discussed.

DOI
10.48550/arxiv.hep-ph/9707468

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Particle motion in the fields of Aharonov-Bohm and Aharonov-Casher is considered in framework of the classical theory to reveal conditions admitting duality of the two configurations. Important role of orientation of the magnetic dipole moment is demonstrated. Duality becomes totally destroyed by addition of electric dipole and/or higher multipole moments. Correspondence between quantum and classical considerations is also discussed.
The Aharonov-Casher (AC) effect for quantum motion of a neutral magnetized particle in the electric field is believed to be a topological effect closely related to the known Aharonov-Bohm (AB) effect. We study how it depends on the spin of the particle involved. Duality of the AB and AC effects is demonstrated to exist only for two extreme spin (and magnetic moment) projections. Classical consideration confirms the conclusion. Motion of a classical magnetized particle with generally oriented magnetic moment in the AC field appears to be subjected to both forces and torques. Only for two special orientations of the magnetic moment (same as in the quantum case) the motion is effectively free and similar to the AB motion of a charged particle. Thus, the AC effect is not really topological. Presence of higher multipoles totally destroys the possible AB-AC duality.
We establish systematic consolidation of the Aharonov–Bohm (AB) and Aharonov–Casher effects including their scalar counterparts. Their formal correspondences in acquiring topological phases are revealed on the basis of the gauge symmetry in non-simply-connected spaces and the adiabatic condition for the state of magnetic dipoles. In addition, investigation of basic two-body interactions between an electric charge and a magnetic dipole clarifies their appropriate relative motions and discloses physical interrelations between the effects. Based on the two-body interaction, we also construct an exact microscopic description of the AB effect, where all the elements are treated on equal footing, i.e., magnetic dipoles are described quantum-mechanically and electromagnetic fields are quantized. This microscopic analysis not only confirms the conventional (semiclassical) results and the topological nature but also allows one to explore the fluctuation effects due to the precession of the magnetic dipoles with the adiabatic condition relaxed.
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.
It is shown that the vector potential of the Aharonov-Bohm effect is effectively the same as vector potentials of the quantum phases of electric and magnetic dipoles such as the Aharonov-Casher effect, when the Aharonov-Bohm solenoid is a straight cylinder with symmetric cross sections. With a circular cross section, the Aharonov-Bohm vector potential is effectively the same as the Aharonov-Casher vector potential, while with a rectangular cross section, it is effectively the same as the vector potential of the Casella phase. Therefore, we may say that the quantum phases of the dipoles belong to a subclass of the Aharonov-Bohm phase, and they must share the topological origin of the Aharonov-Bohm phase, even though their topological freedom is limited to the particle trajectories. However, the dipole phases have the extra relativistic effect that a magnetic (electric) dipole moving in an electric (magnetic) field feels an additional magnetic (electric) field, in its rest frame. This effect limits further the topological freedom of the dipole phases.