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Theoretical Physics

Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria, 04510, México D. F., México, Miguel Socolovsky

Published Aug 15, 2018• 3 citations
Publisher

Abstract

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.

DOI
10.22606/tp.2018.33002

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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​.
Within the context of infinite-dimensional representations of the rotation group, the Dirac monopole problem is studied in details. We argue that an arbitrary magnetic charge is allowed, and the Dirac quantization condition can be replaced by a generalized quantization rule yielding a new quantum number, the so-called topological spin, which is related to the weight of the Dirac string. We find that the emerging difficulties in explanation of Aharonov-Bohm effect for an arbitrary Dirac string may be removed by introducing nonassociative path-dependent wave functions. The latter provides the absence of the AB effect for an arbitrary magnetic charge. Nevertheless, a singular string can be observed in the quantum-mechanical scattering process.
Abstract Chapter 9 discusses the rigorous incorporation of electromagnetism in quantum mechanics via minimal substitution and the inclusion of spin, which leads to the Pauli equation. The gauge invariance of this formulation is inspected more closely, which leads us to understanding quantum mechanics as a gauge theory, and on to fascinating geometric effects such as the Aharonov-Bohm effect and the discovery of geometric phases such as the Berry phase and the Aharonov-Anandan phase. To give a better context to these effects, a compact introduction to the mathematical background of differential geometry is provided, introducing concepts such as differentiable manifolds, tangent and cotangent spaces, and maps between them, as well as from their elements to the real numbers, arriving at tensors and differential forms. With these tools at hand, we first define fibre bundles before formulating the previously discussed geometric phases as well as the so-called Dirac monopole in this language
In quantum mechanics, the magnetic field has a physical effect beyond its local presence. For axially symmetric vector potentials, A=A(s,z)ϕ̂, it is the magnetic flux that plays a central role in both interference experiments (as in the Aharonov-Bohm effect) and in bound state spectra for a particle constrained to move in a circle. The flux dependence is evident in various configurations, and we have chosen two applications to highlight the non-local nature of the quantum mechanical magnetic field dependence: A finite solenoid and a Dirac string with a non-zero radius. The finite solenoid is interesting because it represents the actual experimental apparatus for measuring the Aharonov-Bohm effect. The Dirac string provides an electric charge quantization mechanism that depends on where the quantization argument is applied, yielding different fundamental charge units for different locations.