Path integrals in a multiply-connected configuration space (50 years after). (English) Zbl 1483.81094
Summary: The proposal made 50 years ago by L. S. Schulman [Phys. Rev. 176, No.5,:1558–1569 )1968);M. G. G. Laidlaw andC. Morette-DeWitt (Phys. Rev. D 3, No. 9), 1375–1378 (1971; doi:10.1103/PhysRevD.3.1375)] and ;J. S. Dowker [J. Phys. A 5, 936–943 (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.
MSC:
| 81S40 | Path integrals in quantum mechanics |
| 46T12 | Measure (Gaussian, cylindrical, etc.) and integrals (Feynman, path, Fresnel, etc.) on manifolds |
| 14H45 | Special algebraic curves and curves of low genus |
| 81V27 | Anyons |
| 14F35 | Homotopy theory and fundamental groups in algebraic geometry |
Keywords:
path integrals and topology; path integrals in multiply-connected space; topological phases; anyonsReferences:
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