The deformation quantizations of the hyperbolic plane

TL;DR

This paper provides an explicit, unified method for classifying all invariant deformation quantizations of the hyperbolic plane using non-commutative harmonic analysis, revealing a Lorentzian metric linked to the geometry.

Contribution

It introduces a comprehensive approach to construct all invariant deformation quantizations of the hyperbolic plane via hyperbolic differential operators.

Findings

Complete classification of invariant deformation quantizations.

Explicit solutions to Weinstein's WKB quantization in 2D.

Discovery of a Lorentzian metric associated with the hyperbolic plane.

Abstract

We describe the space of (all) invariant deformation quantizations on the hyperbolic plane as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of the hyperbolic plane, and provides, in the 2-dimensional context, an exact solution to Weinstein's WKB quantization program within geometric terms. The construction reveals the existence of a metric of Lorentz signature canonically attached (or `dual') to the geometry of the hyperbolic plane through the quantization process.