Geometric Quantization by Paths Part II: The General Case

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, $\mathrm{Diff}(\mathrm{X}, \omega)$.