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Physical Examples of integral quantizations with Weyl-Heisenberg and affine groups

Jean‐Pierre Gazeau

Published Apr 14, 2014• 0 citations
Publisher

Abstract

In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.

DOI
10.22323/1.200.0003

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In this paper we describe and apply integral quantization, a procedure based on operator-valued measures and the resolution of the identity.We explore its covariance properties in the context of group representation theory.Three applications based on group representations are carried out.The first one concerns the covariant integral quantization based on the spin one-half irreducible representation of SU(2).In this case we show that the quantization of both the quadratic Hopf variables and the Euler angles reproduces the SU(2) Lie algebra.The second example revisits integral quantization based on the Weyl-Heisenberg group.By completing previous results, we show the universality of the canonical commutation relation in such quantizations.In the last example we revisit and enrich the integral quantization based on the affine group of the real line and we give a short account of a relevant application.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.
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.
We explain the powerful role that operator-valued measures can play in quantizing any set equipped with a measure, for instance a group (resp. group coset) with its invariant (resp. quasi-invariant) measure. Coherent state quantization is a particular case. Such integral quantizations are illustrated with two examples based on the Weyl-Heisenberg group and on the affine group respectively. An interesting application of the affine quantization in quantum cosmology is mentioned, and we sketch a construction of new coherent states for the hydrogen atom.
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 latter yields the usual canonical quantization and a quasi-probability distribution (affine Wigner function) which is real, marginal in both momentum p and position q.