Curvature of fields of quantum Hilbert spaces

TL;DR

This paper investigates the geometric properties of quantum Hilbert space fields derived from phase space quantization of symmetric spaces, revealing flatness only in the case of the 3-sphere.

Contribution

It demonstrates that the quantum Hilbert space field is flat for the 3-sphere but not projectively flat for other rank-1 symmetric spaces.

Findings

The field is flat for the 3-dimensional sphere.

The field is not projectively flat for other rank-1 symmetric spaces.

Geometric quantization with half-form correction influences curvature properties.

Abstract

We show that using the family of adapted K\"ahler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank-1, the obtained field $H^{corr}$ of quantum Hilbert spaces produced by geometric quantization including the half-form correction is flat if $M$ is the 3-dimensional sphere and not even projectively flat otherwise.