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Prequantisation from the path integral viewpoint

P A Horvathy

TL;DR

This work links the Feynman path integral phase factor to Kostant-Souriau prequantisation, showing that quantum admissibility requires a prequantisable evolution space and that nontrivial topology (non-simply connected configuration spaces) yields inequivalent quantisations classified by characters of the fundamental group. It provides a coordinate-free, holonomy-based expression for the path-integral integrand via the prequantum 1-form $\omega$ and clarifies how transition functions depend only on space-time endpoints. The main contribution is a concrete classification theorem tying inequivalent prequantisations to $\pi_1$-characters, illustrated by the Aharonov-Bohm effect and identical-particle statistics, with explicit forms for the character and its decomposition across Betti components. This framework illuminates how topology and bundle geometry govern quantum phases in path integrals and offers a principled route to multiple quantisations on spaces with nontrivial topology.

Abstract

The quantum mechanically admissible definitions of the factor $\exp\big[(i/\hbar)S(γ)\big]$ in the Feynman integral are put in bijection with the prequantisations of Kostant and Souriau. The different allowed expressions of this factor -- the inequivalent prequantisations -- are classified. The theory is illustrated by the Aharonov-Bohm experiment and by identical particles.

Prequantisation from the path integral viewpoint

TL;DR

This work links the Feynman path integral phase factor to Kostant-Souriau prequantisation, showing that quantum admissibility requires a prequantisable evolution space and that nontrivial topology (non-simply connected configuration spaces) yields inequivalent quantisations classified by characters of the fundamental group. It provides a coordinate-free, holonomy-based expression for the path-integral integrand via the prequantum 1-form and clarifies how transition functions depend only on space-time endpoints. The main contribution is a concrete classification theorem tying inequivalent prequantisations to -characters, illustrated by the Aharonov-Bohm effect and identical-particle statistics, with explicit forms for the character and its decomposition across Betti components. This framework illuminates how topology and bundle geometry govern quantum phases in path integrals and offers a principled route to multiple quantisations on spaces with nontrivial topology.

Abstract

The quantum mechanically admissible definitions of the factor in the Feynman integral are put in bijection with the prequantisations of Kostant and Souriau. The different allowed expressions of this factor -- the inequivalent prequantisations -- are classified. The theory is illustrated by the Aharonov-Bohm experiment and by identical particles.
Paper Structure (5 sections, 6 theorems, 24 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Quantum mechanically Admissible (Q.M.A.) Systems
  3. Geometric expression for the integrand
  4. A classification scheme
  5. Classification of prequantizations

Key Result

Theorem 1

$(E,\sigma)$ is Q.M.A. iff it is prequantisable with transition functions depending only on the space-time, $X$. Then the Feynman factor Ffactor is defined for any curve $\gamma$ whose end points lie in a single $U_j$ as shown in FIG.Fffig. The consistency relation ijintegral is satisfied.

Figures (2)

  • Figure 1: In $d\geq3$ spatial dimensions, the inequivalent quantisations correspond to the characters of the fundamental group of the configuration space, $\pi_1(Q)$, \ref{['chiofg']}. The topologically distinct bundles correspond to the components of the character group and can be labelled by choosing (non canonically) a character $\chi_i$ in its respective component. The inequivalent connections on a chosen bundle are in bijection with the characters in the identity component which correspond physically to curl-less "Aharonov-Bohm - type" vector potentials.
  • Figure 2: The Feynman factor \ref{['Ffactor']} for a path $\gamma$ whose initial and end points $y,\,y'$ (but necesarilly the entire $\gamma$) belong to a chosen contractible subset $U_j$, is defined by lifting $\gamma$ horizontally to the prequantum bundle $Y$ as $\bar{\gamma}$ and then reading off from the change of the vertical coordinate in a local trivialisationHPAix79SimmsAix79.

Theorems & Definitions (7)

  • Definition 1
  • Theorem 1
  • Lemma 3.1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1