The deformation quantizations of the hyperbolic plane

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.

26 pages, 5 figures

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Cited by in corpus (16)

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• Symbol correspondences for spin systems
• Higher Spin Gravities and Presymplectic AKSZ Models
• Symplectic Microgeometry III: Monoids
• Explicit Formulae for Noncommutative Deformations of CP^N and CH^N
• Disentanglement of Topological and Dynamical Fields in 3d Higher-Spin Theory within Shifted Homotopy Approach
• Toeplitz Quantization and Asymptotic Expansions: Geometric Construction
• Convergence of Star Product: From Examples to a General Framework
• Homotopes of Symmetric Spaces II. Structure Variety and Classification
• Twisted Quadrics and Algebraic Submanifolds in ${\mathbb{R}}^n$
• Noncommutative Supergeometry and Quantum Supergroups
• Deformation quantization of the simplest Poisson Orbifold
• Homotopes of Symmetric Spaces I. Construction by Algebras with Two Involutions
• Multipliers of Hilbert algebras and deformation quantization
• Noncommutative Deformations of Locally Symmetric Kähler manifolds
• Group Actions in Deformation Quantisation