Integral quantizations with two basic examples

TL;DR

This paper explores integral quantization methods, emphasizing covariance and probabilistic aspects, with applications to the Weyl-Heisenberg and affine groups demonstrating how different quantizations can recover canonical quantum structures and regularize dilation issues.

Contribution

It introduces a generalized integral quantization framework that incorporates group representation theory and probabilistic features, with two novel applications to important groups in quantum mechanics.

Findings

Existence of quantizations yielding canonical commutation relations.

Regularization of dilation origin in affine group quantization.

Generalization of coherent state quantization to broader contexts.

Abstract

The paper is devoted to integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical-quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl-Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that there exists a world of quantizations which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space.