  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.
