Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

2017/12/30 by Socolovsky, Miguel
#FOS: Physical sciences #General Physics (physics.gen-ph)

paper · doi:10.48550/arxiv.1801.01411

Abstract

We discuss the Aharonov-Bohm (A-B) effect and the Dirac (D) monopole of magnetic charge g=1\over2 in the context of bundle theory, exhibiting a purely geometric relation between them. If ξ_A-B and ξ_D are the respective U(1)-bundles, we show that ξ_A-B is isomorphic to the pull-back of ξ_D induced by the inclusion of the corresponding base spaces ι:(D₀²)^*→ S². The fact that the A-B effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux Φ₀=2π\over\vert e\vert associated with the electric charge \vert e\vert, reflects here as a consequence of the pull-back by ι of the Dirac connection in ξ_D to ξ_A-B, and the Dirac quantization condition. We also show the necessary vanishing in ξ_A-B of the pull-back of the Chern class c₁ in ξ_D.

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