2-D Covariant Affine Integral Quantization(s)

TL;DR

This paper explores covariant affine integral quantization on a punctured plane, analyzing different quantizer operators and illustrating their effects through examples including coherent states and affine inversion, leading to a real, marginal 2-D affine Wigner function.

Contribution

It introduces a framework for covariant affine integral quantization on the punctured plane and compares different quantizer operators derived from weight functions.

Findings

Different weight functions lead to distinct quantization schemes.

The affine inversion example reproduces canonical quantization.

The resulting 2-D affine Wigner function is real and marginal in position and momentum.

Abstract

Covariant affine integral quantization is studied and applied to the motion of a particle in a punctured plane R^2_\ast=R^2\{0}, for which the phase space is R^2_\ast=R^2\{0}X R^2. We examine the consequences of different quantizer operators built from weight functions on this phase space. To illustrate the procedure, we examine two examples of weights. The first one corresponds to 2-D coherent state families, while the second one corresponds to the affine inversion in the punctured plane. The later yields the usual canonical quantization and a quasi-probability distribution (2-D affine Wigner function) which is real, marginal in both position and momentum.