On Weyl Quantization from geometric Quantization

TL;DR

This paper derives a formula for a deformed product of functions on symmetric symplectic spaces using geometric quantization, explicitly computing cases like the Euclidean plane, sphere, and hyperbolic plane, and discusses the interpretation of Weinstein's conjecture.

Contribution

The paper provides a derivation of Weinstein's conjectured product formula for symmetric symplectic spaces using geometric quantization techniques, with explicit computations for key examples.

Findings

Recovered the Moyal-Weyl product for the Euclidean plane

Computed deformed products for the sphere and hyperbolic plane

Highlighted interpretational nuances of Weinstein's original idea

Abstract

A. Weinstein has conjectured a nice looking formula for a deformed product of functions on a hermitian symmetric space of non-compact type. We derive such a formula for symmetric symplectic spaces using ideas from geometric quantization and prequantization of symplectic groupoids. We compute the result explicitly for the natural 2-dimensional symplectic manifolds: the euclidean plane, the sphere and the hyperbolic plane. For the euclidean plane we obtain the well known Moyal-Weyl product. The other cases show that Weinstein's original idea should be interpreted with care. We conclude with comments on the status of our result.