Smooth structures on the field of prequantum Hilbert spaces

TL;DR

This paper investigates how the smooth structure of prequantum Hilbert spaces varies with different complex structures on phase space, revealing the existence of inequivalent smooth Hilbert bundle structures.

Contribution

It demonstrates that the field of prequantum Hilbert spaces can admit multiple inequivalent smooth bundle structures depending on the complex structures.

Findings

Existence of multiple inequivalent smooth structures on the Hilbert bundle

Dependence of the bundle structure on the choice of complex structures

Insights into geometric quantization and bundle theory

Abstract

When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we obtain a field of Hilbert spaces $p:H^{pr Q}\rightarrow S$. We show that this field can have natural inequivalent smooth Hilbert bundle structures.