On the relation between geometric and deformation quantization

TL;DR

This paper explores how geometric and deformation quantization can be combined into a complete quantization framework, extending the class of quantizable observables and representing the deformed algebra on a Hilbert space.

Contribution

It identifies conditions under which geometric and deformation quantization can be unified, especially for cotangent bundles and Kähler manifolds.

Findings

Extended geometric quantization to include all smooth functions.

Established conditions for representing deformed algebras on Hilbert spaces.

Analyzed special cases of cotangent bundles and Kähler manifolds.

Abstract

In this paper we investigate the possibility of constructing a complete quantization procedure consisting of geometric and deformation quantization. The latter assigns a noncommutative algebra to a symplectic manifold, by deforming the ordinary pointwise product of functions, whereas geometric quantization is a prescription for the construction of a Hilbert space and a few quantum operators, starting from a symplectic manifold. We determine under which conditions it is possible to define a representation of the deformed algebra on this Hilbert space, thereby to extend the small class of quantizable observables in geometric quantization to all smooth functions, as well as to give a natural representation of the algebra. In particular we look at the special cases of a cotangent bundle and a K\"ahler manifold.