Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

TL;DR

This paper explores the geometric relationship between the Aharonov-Bohm effect and Dirac monopoles using bundle theory, revealing how topological and quantization conditions explain the effect's disappearance under certain flux values.

Contribution

It establishes a geometric link between the Aharonov-Bohm effect and Dirac monopoles via bundle isomorphisms and pull-backs, providing a new perspective on their topological connection.

Findings

The Aharonov-Bohm bundle is isomorphic to a pull-back of the Dirac monopole bundle.

The Aharonov-Bohm effect vanishes when magnetic flux is quantized as an integer multiple of the flux quantum.

Pull-back of the Dirac bundle's Chern class must vanish in the Aharonov-Bohm bundle.

Abstract

We discuss the Aharonov-Bohm ($A-B$) effect and the Dirac ($D$) monopole of magnetic charge $g={{1}\over{2}}$ in the context of bundle theory, exhibiting a purely geometric relation between them. If $\xi_{A-B}$ and $\xi_D$ are the respective $U(1)$-bundles, we show that $\xi_{A-B}$ is isomorphic to the pull-back of $\xi_D$ induced by the inclusion of the corresponding base spaces $\iota:(D_0^2)^*\to S^2$}. The fact that the $A-B$ effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux $\Phi_0={{2\pi}\over{\vert e\vert}}$ associated with the electric charge $\vert e\vert$, reflects here as a consequence of the pull-back by $\iota$ of the Dirac connection in $\xi_D$ to $\xi_{A-B}$, and the Dirac quantization condition. We also show the necessary vanishing in $\xi_{A-B}$ of the pull-back of the Chern class $c_1$ in $\xi_D$.