Geometry of the Aharonov-Bohm Effect

TL;DR

This paper explores the geometric and topological structures underlying the Aharonov-Bohm effect, demonstrating that the relevant connection resides in a trivial principal bundle and analyzing associated bundles involved in quantum effects.

Contribution

It establishes that the connection for the Aharonov-Bohm effect is in a trivial principal bundle and clarifies the roles of universal covering and associated bundles in the phenomenon.

Findings

The connection for the Aharonov-Bohm effect is in a trivial principal bundle.

Universal covering space is used for path integral computations.

Associated bundles host the wave function and its covariant derivative.

Abstract

We show that the connection responsible for any abelian or non abelian Aharonov-Bohm effect with $n$ parallel ``magnetic'' flux lines in $\R^3$, lies in a trivial $G$-principal bundle $P\to M$, i.e. $P$ is isomorphic to the product $M\times G$, where $G$ is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space $\tilde{M}\to M$, where path integrals are computed, and the associated bundle $P\times_G \C^m \to M$, where the wave function and its covariant derivative are sections.