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arXiv · 0806.4741

The deformation quantizations of the hyperbolic plane

Pierre Bieliavsky·Stéphane Detournay·Philippe Spindel
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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.

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Pierre Bieliavsky, Stéphane Detournay, Philippe Spindel. 2008-06-30. The deformation quantizations of the hyperbolic plane. https://doi.org/10.1007/s00220-008-0697-9

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