Geometric Quantization by Paths Part II: The General Case

TL;DR

This paper extends geometric quantization to general parasymplectic diffeological spaces by constructing a Prequantum Groupoid that accounts for periods and symmetries, offering a new perspective on quantum systems.

Contribution

It introduces a construction of a Prequantum Groupoid for arbitrary connected parasymplectic spaces, addressing obstructions via a total group of periods and linking the quantum system to classical symmetries.

Findings

Constructed a Prequantum Groupoid with torus of periods.

Identified the obstruction as non-additivity of loop integration.

Proposed the groupoid as the quantum system itself.

Abstract

In Part I, we established the construction of the Prequantum Groupoid for simply connected spaces. This second part extends the theory to arbitrary connected parasymplectic diffeological spaces $(\mathrm{X}, \omega)$. We identify the obstruction to the existence of the Prequantum Groupoid as the non-additivity of the integration of the prequantum form $\mathbf{K}\omega$ on the space of loops. By defining a Total Group of Periods $\mathrm{P}_\omega$ directly on the space of paths, which absorbs the periods arising from the algebraic relations of the fundamental group, we construct a Prequantum Groupoid $\mathbf{T}_\omega$ with connected isotropy isomorphic to the torus of periods $\mathrm{T}_\omega = \mathbf{R}/\mathrm{P}_\omega$. Furthermore, we propose that this groupoid $\mathbf{T}_\omega$ constitutes the Quantum System itself. The classical space $\mathrm{X}$ is embedded as the Skeleton of units, surrounded by a Quantum Fog of non-identity morphisms. We prove that the group of automorphisms of the Quantum System is isomorphic to the group of symmetries of the Dynamical System, $\mathrm{Diff}(\mathrm{X}, \omega)$.