The Deformation Quantizations of the Hyperbolic Plane

Bieliavsky, Pierre;Detournay, S.;Spindel, Ph.
(2009) Communications in Mathematical Physics — Vol. 289, n° 2, p. 529-559 (2009)

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Abstract
We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane 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 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.
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Bieliavsky, P., Detournay, S., & Spindel, Ph. (2009). The Deformation Quantizations of the Hyperbolic Plane. Communications in Mathematical Physics, 289(2), 529-559. https://doi.org/10.1007/s00220-008-0697-9 (Original work published 2009)