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

Franco Strocchi

Jean‐Pierre Gazeau

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.

Hervé Bergeron, J.P. Gazeau

Hervé Bergeron, Evaldo M. F. Curado, Jean‐Pierre Gazeau +1 more

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.

R. Fresneda, Jean‐Pierre Gazeau

We present quantizations of functions (or distributions) on a measure space viewed as a classical set. They are based on positive operator-valued measures. To illustrate the large range of potentialities of the method, we develop examples where the classical sets are the group SU(2), the (projective) Weyl-Heisenberg group, and the affine group. Applications to quantum cosmology and to the quantization of constraints are outlined.