2008/06/30 by Pierre Bieliavsky, Stéphane Detournay, Ph. Spindel +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #hep-th #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-008-0697-9
26 pages, 5 figures
arxiv created 2008/06/30 · openalex publication_date 2009/01/07 · arxiv updated 2009/12/01 · openalex created_date 2020/11/23 · openalex updated_date 2026/07/29
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.