Covariant integral quantization of the unit disk

TL;DR

This paper develops a covariant integral quantization method for functions on the unit disk, modeling phase space for a particle in Anti de Sitter space-time, incorporating SU(1,1) symmetry and coherent states.

Contribution

It introduces a new SU(1,1) covariant quantization approach on the unit disk, including the use of weight functions and Perelomov coherent states, for quantizing phase space functions.

Findings

Derived semi-classical portraits of key operators

Identified unitary irreducible representations of SU(1,1) for quantization

Established a flexible quantization framework depending on a weight function

Abstract

We implement a SU(1,1) covariant integral quantization of functions or distributions on the unit disk. The latter can be viewed as the phase space for the motion of a test "massive" particle on 1+1 Anti de Sitter space-time, and the relevant unitary irreducible representations of SU(1,1) corresponding to the quantum version of such motions are found in the discrete series and its lower limits. Our quantization method depends on a weight function on the phase space, and it includes Perelomov coherent states quantization. Semi-classical portraits or lower symbols of main physically relevant operators are determined.