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Integral Quantization for the Discrete Cylinder

Authors :
Jean-Pierre Gazeau
Romain Murenzi
Source :
Quantum Reports, Vol 4, Iss 4, Pp 362-379 (2022)
Publication Year :
2022
Publisher :
MDPI AG, 2022.

Abstract

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.

Details

Language :
English
ISSN :
2624960X
Volume :
4
Issue :
4
Database :
Directory of Open Access Journals
Journal :
Quantum Reports
Publication Type :
Academic Journal
Accession number :
edsdoj.269b9e88ccc145048a3b3268db7e834e
Document Type :
article
Full Text :
https://doi.org/10.3390/quantum4040026