Quantisation of a Family of Phase Spaces

TL;DR

This paper explores the geometric structure of quantum Hilbert spaces over parameter-dependent phase spaces, analyzing connections, curvature, and phases, with applications to symplectic and fermionic vector spaces.

Contribution

It introduces a natural unitary connection on the bundle of quantum Hilbert spaces for varying symplectic structures, extending geometric quantization theory.

Findings

Derived the connection and curvature for Hilbert space bundles over parameter spaces.

Analyzed phase factors and geometric phases in the context of varying symplectic structures.

Applied theoretical results to specific examples like the two-sphere family of symplectic structures.

Abstract

We explain that when quantising phase spaces with varying symplectic structures, the bundle of quantum Hilbert spaces over the parameter space has a natural unitary connection. We then focus on symplectic vector spaces and their fermionic counterparts. After reviewing how the quantum Hilbert space depends on physical parameters such as the Hamiltonian and unphysical parameters such as choices of polarisations, we study the connection, curvature and phases of the Hilbert space bundle when the phase space structure itself varies. We apply the results to the two-sphere family of symplectic structures on a hyper-K\"ahler vector space and to their fermionic analogue, and conclude with possible generalisations.