Fundamental groupoids in quantum mechanics: a new approach to quantization in multiply-connected spaces

TL;DR

This paper introduces a novel approach to quantization in multiply-connected spaces using fundamental groupoids, enabling analysis of systems with infinite fundamental groups and capturing topological effects like the Aharonov-Bohm effect and anyons.

Contribution

It develops a method based on fundamental groupoids for quantization in spaces with infinite fundamental groups, extending previous approaches limited to finite groups.

Findings

Method effectively handles infinite fundamental groups.

Explicitly constructs phases incorporating space symmetries.

Provides an example related to anyons.

Abstract

Quantization of multiply-connected spaces requires tools which take these spaces' global properties into account. Applying these tools exposes additional degrees of freedom. This was first realized in the Aharonov-Bohm effect, where this additional degree of freedom was a magnetic flux confined to a solenoid, which an electron cannot enter. Previous work using Feynman path integrals has either only dealt with specific cases, or was limited to spaces with finite fundamental groups, and therefore, was in fact inapplicable to the Aharonov-Bohm effect, as well as to interesting systems such as anyons. In this paper we start from the fundamental groupoid. This less familiar algebraic-topological object is oriented towards general paths. This makes it a more natural choice for the path integral approach than the more commonly known fundamental group, which is restricted to loops. Using this object, we provide a method that works for spaces with infinite fundamental groups. We adapt a tool used in the previous, restricted result, in order to clearly delineate physically significant degrees of freedom from gauge freedom. This allows us to explicitly build phases which take account of symmetries of the space, directly from topological considerations; we end by providing a pertinent example relating to anyons.