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arXiv:hep-ph/9710433 · Physics of Atomic Nuclei

The Aharonov-Casher Effect for Particles of Arbitrary Spin

Ya. I. Azimov, R.M. Ryndin

Published Oct 22, 1997• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.hep-ph/9710433

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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 1984, 25 years after the prediction of the →Aharonov-Bohm (AB) effect, Aharonov and Casher [1] predicted a “dual ” effect. In both effects, a particle is excluded from a tubular region of space, but otherwise no force acts on it. Yet it acquires a measurable quantum phase that depends on what is inside the tube of space from which it is excluded. In the AB effect, the particle is charged and the tube contains a magnetic flux. In the Aharonov-Casher (AC) effect, the particle is neutral, but has a magnetic moment, and the tube contains a line of charge. Experiments in neutron [2], vortex [3], atom [4], and electron [5] interferometry bear out the prediction of Aharonov and Casher. Here we briefly explain the logic of the AC effect and how it is dual to the AB effect. We begin with a two-dimensional version of the AB effect. Figure 1 shows an electron moving in a plane, and also a “fluxon”, i.e. a small region of magnetic flux (pointing out of the plane) from which the electron is excluded. In Fig. 1 the fluxon is in a quantum superposition of two positions, and the electron diffracts around one of the positions but not the other. Initially, the fluxon and electron are in a
The scattering of relativistic spin-1/2 neutral particles with a magnetic dipole moment by a long straight charged line and a magnetic flux line at the same position is studied. The scattering cross sections for unpolarized and polarized particles are obtained by solving the Dirac-Pauli equation. The results are in general the same as those for pure Aharonov-Casher scattering (by the charged line alone) as expected. However, in special cases when the incident energy, the line charge density, and the magnetic flux satisfy some relations, the cross section for polarized particles is dramatically changed. Relations between the polarization of incident particles and that of scattered ones are presented, both in the full relativistic case and the nonrelativistic limit. The characteristic difference between the general and special cases lies in the backward direction: in the general cases the incident particles are simply bounced while in the special cases their polarization is turned over simultaneously. For pure Aharonov-Casher scattering there exist cases where the helicities of all scattered particles are reversed. This seems to be remarkable but appears unnoticed previously. Two mathematical approaches are employed to deal with the singularity of the electric and magnetic field and it turns out that the physical results are essentially the same.
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.
It is demonstrated that the Aharonov-Casher (AC) phase is a geometric phase that, in general, depends on the details of the closed path taken by a particle with a magnetic moment that is subject to an electric field. Consequently, it is not a topological phase. The proof of this statement is obtained by developing a counterexample that elucidates the dependence of the AC phase on the details of the path. Furthermore, we demonstrate that, in the particular example considered here, paths having an Abelian AC phase factor, also have an AC phase that is path-independent, whereas paths having a non-Abelian AC phase factor may have an AC phase that is path-dependent (i.e., not topological).