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Journal of Physics A Mathematical and General

A non-Abelian Aharonov-Bohm effect in the framework of pseudoclassical mechanics

Luiz C. L. Botelho, J C de Mello

Published Sep 11, 1985• 3 citations
Publisher

Abstract

The authors analyse a non-Abelian Aharonov-Bohm effect in the framework of pseudoclassical mechanics using a Gedanken experiment. A colour O(N) charged spinless particle is considered.
DOI
10.1088/0305-4470/18/13/038

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The Aharonov-Bohm effect is relatively little known, but it poses fundamental questions in quantum theory. It concerns physical effects on charged particles in field-free regions. The present paper provides an over-all critical review and commentary on the literature of the Aharonov-Bohm effect. The two fundamental questions raised by the effect concern the role of the force concept in quantum mechanics, and the localizability of physical effects principle. A recent letter from Professor Bohm concerning his evaluation of the significance of the effect is included in the Appendix.

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The Aharonov–Bohm effect in general gauge theories, for particles in gauge-curvature-free regions, is studied using the quantum mechanical propagator in the form of a Feynman sum over paths. Following Schulman [L. S. Schulman, Techniques and Applications of Path Integration (Wiley, New York, 1981)], such paths are divided into their homotopy equivalence classes, and the contributions from each class of paths of the Feynman sum are identified with propagators of a wave equation in the universal covering manifold of M, resulting in a simple form for the propagator on M. A group homomorphism from ℋ, the fundamental homotopy group of M, to the gauge group G is shown to characterize possible Aharonov–Bohm effects, which can be divided into two types, Abelian and non-Abelian, according to whether ℋ*, the image of this homomorphism, is Abelian or non-Abelian. For a non-Abelian Aharonov–Bohm effect, it is necessary that both ℋ and G be non-Abelian. Simple examples illustrate the theory.
The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrddoto\\ddot{o}ddotodinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrddoto\\ddot{o}ddotodinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively.
The Aharonov-Bohm effect is relatively little known, but it poses fundamental questions in quantum theory. It concerns physical effects on charged particles in field-free regions. The present paper provides an over-all critical review and commentary on the literature of the Aharonov-Bohm effect. The two fundamental questions raised by the effect concern the role of the force concept in quantum mechanics, and the localizability of physical effects principle. A recent letter from Professor Bohm concerning his evaluation of the significance of the effect is included in the Appendix.