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Journal of Physics B Atomic Molecular and Optical Physics

Power–Zienau–Woolley QED: centre of mass-energy and the Aharonov–Casher effect

S. A. R. Horsley, M. Babiker

Published Jul 24, 2006• 15 citations
Publisher

Abstract

The Aharonov–Casher (AC) effect is due to the interaction between an electric field and a moving particle that is endowed with a magnetic moment. This interaction is normally, but controversially, introduced by hand and by appeal to electromagnetic symmetry, as first done by Aharonov and Casher. We show here that the AC interaction arises rigorously within quantum electrodynamics (QED) in the Power–Zienau–Woolley formulation, provided that the centre of mass-energy is used as the multipolar reference centre, rather than its strictly non-relativistic limit—the centre of mass.

DOI
10.1088/0953-4075/39/15/s05

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The Aharonov–Casher (AC) effect is due to the interaction between an electric field and a moving particle that is endowed with a magnetic moment. This interaction is normally, but controversially, introduced by hand and by appeal to electromagnetic symmetry, as first done by Aharonov and Casher. We show here that the AC interaction arises rigorously within quantum electrodynamics (QED) in the Power–Zienau–Woolley formulation, provided that the centre of mass-energy is used as the multipolar reference centre, rather than its strictly non-relativistic limit—the centre of mass.
The classical electromagnetic interaction of a point charge and a magnet is discussed by first calculating the interaction of a point charge with a simple model magnetic moment and then suggesting a multiparticle limit. The Darwin–Lagrangian is used to analyse the electromagnetic behaviour of the model magnetic moment (composed of two oppositely charged particles of different masses in an initially circular Coulomb orbit) interacting with a passing point charge. Considerations of force, energy, momentum and centre of energy are treated through second order in 1/ c . The changing magnetic moment is found to put a force back on a passing charge; this force is of order 1/ c 2 and depends upon the magnitude of the magnetic moment. The limit of a many-particle magnet arranged as a toroid is discussed. It is suggested that in the multiparticle limit, the electric fields of the passing charge are screened out of the body of the magnet while the magnetic fields of the passing charge penetrate into the body of the magnet. This is consistent with our understanding of the penetration of electromagnetic velocity fields into ohmic conductors. The proposed multiparticle limit is consistent with the conservation laws for energy and momentum, as well as constant motion of the centre of energy, and Newton's third law for the net Lorentz forces on the magnet and on the point charge. The work corresponds to a classical electromagnetic analysis of the interaction which is basic to understanding the controversy over the Aharonov–Bohm and Aharonov–Casher phase shifts and represents a refutation of the suggestions of Aharonov, Pearle and Vaidman.
The relation of the Aharonov-Casher (AC) effect and the force on a particle having a magnetic moment is explored. The general form of the AC Hamiltonian is derived using the Foldy-Wouthuysen transformation to the Dirac equation. Geometries in which an analytic expression for the phase can be obtained are examined, as well as the relation of the AC phase to the Berry phase. The AC phase is determined for an arbitrary homogeneous electric field; it is quadratic (linear) in the field strength for small (large) electric field strengths.
The relation of the Aharonov-Casher (AC) effect and the force on a particle having a magnetic moment is explored. The general form of the AC Hamiltonian is derived using the Foldy-Wouthuysen transformation to the Dirac equation. Geometries in which an analytic expression for the phase can be obtained are examined, as well as the relation of the AC phase to the Berry phase. The AC phase is determined for an arbitrary homogeneous electric field; it is quadratic (linear) in the field strength for small (large) electric field strengths.
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.
In this paper we give a covariant expression for Aharonov–Casher phase. This expression is a combination of the canonical electric field, Aharonov–Casher phase plus a magnetic field phase shift. We use this covariant expression for the Aharonov–Casher phase to investigate the case of a neutral particle with a non-zero magnetic moment moving in the time dependent electric and magnetic fields of a plane electromagnetic wave background. We focus on the case where the magnetic moment of the particle is oriented so that both the electric and magnetic fields lead to non-zero phases, and we look at the interplay between these electric and magnetic phases.