Aharonov-Bohm Effect, Dirac Monopole, and Bundle Theory

Abstract

We discuss the Aharonov-Bohm ($A-B$) effect and the Dirac ($D$) monopole of magnetic charge $g=\frac{1}{2}$ 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 $\iota :(D_0^2{)}^{\ast }\to S^2$}. The fact that the $A-B$ effect disappears when the magnetic flux in the solenoid equals an integer times the quantum of flux ${\Phi }_0=\frac{2\pi }{|e|}$ associated with the electric charge $|e|$, 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 ${\xi }_D$.

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