Covariant Affine Integral Quantization(s)

TL;DR

This paper explores covariant affine integral quantization on the half-plane, applying it to particle motion on the half-line, and investigates different quantizer operators derived from weight functions, including thermal and affine inversion cases.

Contribution

It introduces a systematic study of covariant affine integral quantization with various weight functions and demonstrates their effects on quantum representations and distributions.

Findings

Thermal density operators produce temperature-dependent distributions.

Affine inversion yields canonical quantization and a real affine Wigner function.

The approach provides new insights into quantization on the half-plane.

Abstract

Covariant affine integral quantization of the half-plane is studied and applied to the motion of a particle on the half-line. We examine the consequences of different quantizer operators built from weight functions on the half-plane. To illustrate the procedure, we examine two particular choices of the weight function, yielding thermal density operators and affine inversion respectively. The former gives rise to a temperature-dependent probability distribution on the half-plane whereas the later yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both momentum p and position q.