Path integrals in a multiply-connected configuration space (50 years after)

TL;DR

This paper critically examines the historical development and mathematical foundations of path integrals in multiply-connected spaces, emphasizing the necessity of the unitary representation of the fundamental group for quantum topological effects.

Contribution

It provides a critical analysis clarifying why the unitary representation of the first homotopy group is both necessary and sufficient in this context.

Findings

Clarifies the role of homotopy classes in path integrals

Analyzes the arguments supporting the unitary representation

Highlights the conditions for topological quantum effects

Abstract

The proposal made 50 years ago by Schulman (1968), Laidlaw & Morette-DeWitt (1971) and Dowker (1972) to decompose the propagator according to the homotopy classes of paths was a major breakthrough: it showed how Feynman functional integrals opened a direct window on quantum properties of topological origin in the configuration space. This paper casts a critical look at the arguments brought by this series of papers and its numerous followers in an attempt to clarify the reason why the emergence of the unitary linear representation of the first homotopy group is not only sufficient but also necessary.