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 ${\mathrm{T}}_{\omega }=\mathbf{R}\mathrm{/}{\mathrm{P}}_{\omega }$. 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, $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(\mathrm{X},\omega )$.