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arXiv:1107.1459

Homotopy and Path Integrals

Fumika Suzuki

Published Jul 7, 2011• 10 citations
arXivHTMLPDF

Abstract

This is an introductory review of the connection between homotopy theory and path integrals, mainly focus on works done by Schulman [23] that he compared path integral on SO(3) and its universal covering space SU(2), DeWitt and Laidlaw [15] that they proved the theorem to the case of path integrals on the multiply-connected topological spaces. Also, we discuss the application of the theorem in Aharonov-Bohm effect given by [20,24]. An informal introduction to homotopy theory is provided for readers who are not familiar with the theory.

DOI
10.48550/arxiv.1107.1459

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This is an introductory review of the connection between homotopy theory and path integrals, mainly focus on works done by Schulman [23] that he compared path integral on SO(3) and its universal covering space SU(2), DeWitt and Laidlaw [15] that they proved the theorem to the case of path integrals on the multiply-connected topological spaces. Also, we discuss the application of the theorem in Aharonov-Bohm effect given by [20,24]. An informal introduction to homotopy theory is provided for readers who are not familiar with the theory.
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Physics defined on multiply connected manifolds is an old topic in theoretical physics. In the context of the path integral formalism it was studded by the first time by Schulman in 1968 and rigorously formulated by Laidlaw and the Witt and Dowker in 1971. The central point is that in a multiply connected manifold the paths have different weights in the sum over histories and the problema - which does not exist in the "standard" quantum mechanicsis how to define a quantum theory taking this fact into account.