Integral Quantization for the Discrete Cylinder

Abstract

Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalised positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is to allow 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 $\mathbb{Z}\times\,\mathbb{S}^1$, 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 $\mathbb{Z}\times\,\mathrm{SO}(2)$. 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 $L^2(\mathbb{S}^1)$, 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 Bi\`evre-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 Fej\'er kernels. Applications to stellar representations are in progress.