Strict Deformation Quantization via Geometric Quantization in the Bieliavsky Plane

P Hurtado,J B Moreno,A Leones

doi:10.1155/2020/6794709

Abstract

Using standard techniques from geometric quantization, we rederive the integral product of functions on ℝ2 (non-Euclidian) which was introduced by Pierre Bieliavsky as a contribution to the area of strict quantization. More specifically, by pairing the nontransverse real polarization on the pair groupoid ℝ2×ℝ¯2, we obtain the well-defined integral transform. Together with a convolution of functions, which is a natural deformation of the usual convolution of functions on the pair groupoid, this readily defines the Bieliavsky product on a subset of L2ℝ2.

Highlights

Let M be a symplectic symmetric space, TM its tangent bundle, and let M × M be the symplectic pair groupoid
The regularity condition fails if M is compact but is satisfied if M is noncompact with no compact factors
The main idea for this derivation is already found in the aforementioned paper by Gracia-Bondia and Varilly ([5], for Euclidian case R2n). Appropriate generalizations of this technique to other noncompact hermitian symmetric spaces can in principle be helpful. This fact shall be thoroughly explored in subsequent papers and constitutes the main motivation for our working out this technique in detail for the case of R2b (Bieliavsky plane), in this present note

Summary

Introduction

Let M be a symplectic symmetric space, TM its tangent bundle, and let M × M be the symplectic pair groupoid. Appropriate generalizations of this technique to other noncompact hermitian symmetric spaces can in principle be helpful (for instance, if M = H2 is the hyperbolic plane) This fact shall be thoroughly explored in subsequent papers and constitutes the main motivation for our working out this technique in detail for the case of R2b (Bieliavsky plane), in this present note. E.g., [12,13,14]), and the generalized Segal-Bargmann transform for Lie groups of compact type can be developed using geometric quantization (cf [15, 16]) In this short note, again via geometric quantization, we shall obtain the 2-d integral transforms given by.

The Integral Transform Generated by the Geometric Quantization

Rederiving the Bieliavsky Product