The Deformation Quantizations of the Hyperbolic Plane

Abstract

We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane $${\mathbb D}$$ 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 $${\mathbb D}$$ , 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.