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Quantization of multiply-connected spaces requires tools which take thesespaces' global properties into account. Applying these tools exposes additionaldegrees of freedom. This was first realized in the Aharonov-Bohm effect, wherethis additional degree of freedom was a magnetic flux confined to a solenoid,which an electron cannot enter. Previous work using Feynman path integrals haseither only dealt with specific cases, or was limited to spaces with finitefundamental groups, and therefore, was in fact inapplicable to theAharonov-Bohm effect, as well as to interesting systems such as anyons. In thispaper we start from the fundamental groupoid. This less familiaralgebraic-topological object is oriented towards general paths. This makes it amore natural choice for the path integral approach than the more commonly knownfundamental group, which is restricted to loops. Using this object, we providea method that works for spaces with infinite fundamental groups. We adapt atool used in the previous, restricted result, in order to clearly delineatephysically significant degrees of freedom from gauge freedom. This allows us toexplicitly build phases which take account of symmetries of the space, directlyfrom topological considerations; we end by providing a pertinent examplerelating to anyons.

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