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Journal of Physics A Mathematical and General

Relativistic analogy of the Aharonov-Bohm effect in the presence of Coulomb field and magnetic charge

Van-Hoang Le, Ly Xuan Hai, L I Komarov, T S Romanova

Published Dec 7, 1992• 23 citations
Publisher

Abstract

The authors establish the exact solution of the Dirac equation for a charged particle moving in the Coulomb field plus Aharonov-Bohm field plus Dirac monopole field by using the equation in two-dimensional complex space. On the basis of the found solution, the relativistic analogy of the Aharonov-Bohm effect for the system of a diogen atom existing in an infinite cylindrical solenoid is studied. In this case the authors obtain some surprising results. One of them is that the Aharonov-Bohm effect is absent for certain states.

DOI
10.1088/0305-4470/25/23/038

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The authors establish the exact solution of the Dirac equation for a charged particle moving in the Coulomb field plus Aharonov-Bohm field plus Dirac monopole field by using the equation in two-dimensional complex space. On the basis of the found solution, the relativistic analogy of the Aharonov-Bohm effect for the system of a diogen atom existing in an infinite cylindrical solenoid is studied. In this case the authors obtain some surprising results. One of them is that the Aharonov-Bohm effect is absent for certain states.
The Aharonov-Bohm effect is considered by most authors as a quantum effect, but a generally accepted explanation does not seem to be available. The phenomenon is studied here under the assumption that hypothetical electric dipole distributions configured by moving charges in the solenoid act on the electrons as test particles. The relative motions of the interacting charged particles introduce relativistic time dilations. The massless dipoles are postulated as part of an impact model that has recently been proposed to account for the far-reaching electrostatic forces between charged particles described by Coulomb's law. The model provides a quantitative explanation of the Aharonov-Bohm effect.
The structure of additional electromagnetic fields to the Aharonov–Bohm field, for which the Schrödinger, Klein-Gordon, and Dirac equations can be solved exactly are described and the corresponding exact solutions are found. It is demonstrated that aside from the known cases (a constant and uniform magnetic field that is parallel to the Aharonov–Bohm solenoid, a static spherically symmetrical electric field, and the field of a magnetic monopole), there are broad classes of additional fields. Among these new additional fields we have physically interesting electric fields acting during a finite time or localized in a restricted region of space. There are additional time-dependent uniform and isotropic electric fields that allow exact solutions of the Schrödinger equation. In the relativistic case there are additional electric fields propagating along the Aharonov–Bohm solenoid with arbitrary electric pulse shape.
Exact analytic solutions are found to the Dirac equation in 2+12+12+1 dimensions for a combination of an Aharonov-Bohm potential and the Lorentz three-vector and scalar Coulomb potentials. By means of the solutions obtained the relativistic quantum Aharonov-Bohm effect is studied for the free (in the presence of a Lorentz three-vector Coulomb potential) and bound fermion states. We obtain the total scattering amplitude in a combination of the Aharonov-Bohm and Lorentz three-vector Coulomb potentials as a sum of two scattering amplitudes. This modifies the expression for the standard Aharonov-Bohm cross section due to the interference of these two amplitudes with each other. We discuss that the observable quantities can be the phases of electron wave functions or the energies of bound states.
We present new exact solutions (in 3+1 and 2+1 dimensions) of relativistic wave equations (Klein–Gordon and Dirac) in external electromagnetic fields of special form. These fields are combinations of Aharonov–Bohm solenoid field and some additional electric and magnetic fields. In particular, as such additional fields, we consider longitudinal electric and magnetic fields, some crossed fields, and some special nonuniform fields. The solutions obtained can be useful to study the Aharonov–Bohm effect in the corresponding electromagnetic fields.
We discuss the Aharonov-Bohm (A -B) effect and the Dirac (D) monopole of magnetic charge g = 1 2 in the context of bundle theory, which allows to exhibit a deep geometric relation between them.If ξA-B and ξD are the respective U (1)-bundles, we show that ξA-B is isomorphic to the pull-back of ξD induced by the inclusion of the corresponding base spaces.The fact that the A -B effect disappears when the magnetic flux in the solenoid equals an integer number of times the quantum of flux associated with the unit of electric charge, reflects here as a consequence of the pull-back of the Dirac connection in ξD to ξA-B, and the Dirac quantization condition.
We discuss the Aharonov-Bohm (A−BA-BA−B) effect and the Dirac (DDD) monopole of magnetic charge g=12g={{1}\over{2}}g=21​ in the context of bundle theory, exhibiting a purely geometric relation between them. If ξA−Bξ_{A-B}ξA−B​ and ξDξ_DξD​ are the respective U(1)U(1)U(1)-bundles, we show that ξA−Bξ_{A-B}ξA−B​ is isomorphic to the pull-back of ξDξ_DξD​ induced by the inclusion of the corresponding base spaces ι:(D02)∗→S2ι:(D_0^2)^*\to S^2ι:(D02​)∗→S2}. The fact that the A−BA-BA−B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux Φ0=2π∣e∣Φ_0={{2π}\over{\vert e\vert}}Φ0​=∣e∣2π​ associated with the electric charge ∣e∣\vert e\vert∣e∣, reflects here as a consequence of the pull-back by ιιι of the Dirac connection in ξDξ_DξD​ to ξA−Bξ_{A-B}ξA−B​, and the Dirac quantization condition. We also show the necessary vanishing in ξA−Bξ_{A-B}ξA−B​ of the pull-back of the Chern class c1c_1c1​ in ξDξ_DξD​.