Geometric and deformation quantization

TL;DR

This paper introduces a geometric approach connecting geometric and deformation quantization, clarifying parameter dependencies and compatibility conditions, especially for cotangent bundles and Kähler manifolds, enhancing understanding of quantum observable quantization.

Contribution

It provides a simple geometric construction linking geometric and deformation quantization, analyzing parameter choices and compatibility conditions across different phase space types.

Findings

Compatibility conditions restrict parameters for real polarizations.

The approach reproduces geometric quantization for cotangent bundles.

A canonical choice of parameters exists for Kähler manifolds.

Abstract

We present a simple geometric construction linking geometric to deformation quantization. Both theories depend on some apparently arbitrary parameters, most importantly a polarization and a symplectic connection, and for real polarizations we find a compatibility condition restricting the set of admissible connections. In the special case when phase space is a cotangent bundle this compatibility condition has many solutions, and the resulting quantum theory not only reproduces the well-known geometric quantization scheme, but also allows to quantize all interesting observables. For K\"ahler manifolds there is no compatibility condition, but a canonical choice for the parameters. The explicit form of the observables however remains undetermined.