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The Deformation Quantizations of the Hyperbolic Plane

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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.

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Authors and Affiliations

  1. Géométrie et Physique mathématique, Université Catholique de Louvain, 2, Chemin du Cyclotron, B-1348, Louvain-la-Neuve, Belgium

    P. Bieliavsky

  2. INFN, Sezione di Milano, Via Celoria, 16, 20133, Milano, Italy

    S. Detournay

  3. Mécanique et Gravitation, Université de Mons-Hainaut, 20 Place du Parc, 7000, Mons, Belgium

    Ph. Spindel

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  1. P. Bieliavsky
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  2. S. Detournay
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  3. Ph. Spindel
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Correspondence to Ph. Spindel.

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Communicated by A. Connes

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Bieliavsky, P., Detournay, S. & Spindel, P. The Deformation Quantizations of the Hyperbolic Plane. Commun. Math. Phys. 289, 529–559 (2009). https://doi.org/10.1007/s00220-008-0697-9

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