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Path Integrals in a Multiply-Connected Configuration Space (50 Years After)

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We must neglect our models and study our capabilities.

Edgar Allan Poe ([1], p. 122)

Abstract

The proposal made 50 years ago by Schulman (Phys Rev 176(5):1558–1569, 1968), Laidlaw and Morette-DeWitt (Phys Rev D 3(9):1375–1378, 1971) and 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.

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Notes

  1. Not referring explicitly to topology does not mean, of course, that it is absent, all the more so when the mathematical concepts were not stabilized: even before Poincaré works at the turn of the xix–xxth centuries, topological arguments irrigated fluid dynamics and electromagnetism thought the works of Helmholtz and of the Anglo-Irish-Scottish school including Stokes. In his paper, Dirac follows repeatedly a topological reasoning.

  2. For an historical survey on the quantum phase operator see [12] and for a compilation of papers see [13].

  3. Whereas, in differential geometry, (oriented) paths or curves are usually defined up to a (monotonically increasing) bijective continuous reparametrisation that can always be taken to be, say, \(s\in [0,1]\).

  4. If \(k_n(s_0^+)=k_{\sigma (n)}(s_0^-)\ne k_n(s_0^-)\) for a permutation \(\sigma\) of the discrete labels n, then the function \({\tilde{k}}_n(s)\smash {\overset{\text { def}}{=}}k_n(s)\varTheta (s-s_0)+k_{\sigma (n)}(s)\varTheta (s_0-s)\) is continuous at \(s_0\) and one can express the discontinuous functions \(k_n\) with the continuous functions \({\tilde{k}}_n\): \(k_n(s)= {\tilde{k}}_n(s)\varTheta (s-s_0)+{\tilde{k}}_{\sigma ^{-1}(n)}(s)\varTheta (s_0-s)\).

  5. The same line of thought leads to the construction of the Riemann surfaces from the complex plane. The latter is unfolded into n connected sheets to avoid the line cuts of the function \(z\mapsto z^{1/n}\).

  6. In the context of the Ehrenberg–Siday–Aharonov–Bohm effect, see Berry [43]’s fair denunciation of the use of multivalued wavefunctions.

  7. Supposedly, this motivated Dowker ([6], Introduction) “to present a somewhat neater and more attractive derivation of [the result (1)].”

  8. To reuse their notations, they write p. 1376 that if a path is given by the concatenation \(q(a,a')=q(a,b)q(b,a')\) hence \(S[q(a,a')]>S[q(a,b)]\) or, translated into the notations of the present article, \(\mathscr {C}=\mathscr {C}_1\cdot \mathscr {C}_2\Rightarrow S[\mathscr {C}]>S[\mathscr {C}_1]\).

  9. When \(\pi _1(Q)\) is finite, (18) is sufficient for the E’s to be given by the roots of unity.

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Acknowledgements

It is a pleasure to acknowledge my deep gratitude to Alain Comtet of the Laboratoire de Physique Théorique et de Modèles Statistiques de l’Université Paris Saclay for his precious advices on this work; all the more that he somehow initiated it some decades ago while guiding my first steps in physics research. Many thanks also to Dominique Delande for his continuous support and hospitality at the Laboratoire Kastler-Brossel.

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  1. Institut Denis Poisson de Mathématiques et de Physique Théorique, Université de Tours — CNRS (UMR 7013), Parc de Grandmont, 37200, Tours, France

    Amaury Mouchet

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Mouchet, A. Path Integrals in a Multiply-Connected Configuration Space (50 Years After). Found Phys 51, 107 (2021). https://doi.org/10.1007/s10701-021-00497-y

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