Heisenberg Group Algebra and Strict Weyl Quantization

Jean‐Pierre Gazeau

Franco Strocchi

Yingfei Gu

This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.

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.

Reinhard Honegger, Alfred Rieckers

Starting from a (possibly infinite dimensional) pre-symplectic space (E, σ) , we study a class of modified Weyl quantizations. For each value of the real Planck parameter ℏ , we have a C*-Weyl algebra \mathcal W(E,ℏ_σ) , which altogether constitute a continuous field of C*-algebras, as discussed in previous works. For ℏ = 0 , we construct a Fréchet–Poisson algebra, densely contained in \mathcal W(E, 0) , as the classical observables to be quantized. The quantized Weyl elements are decorated by so-called quantization factors, indicating the kind of normal ordering in specific cases. Under some assumptions on the quantization factors, the quantization map may be extended to the Fréchet–Poisson algebra. It is demonstrated to constitute a strict and continuous deformation quantization, equivalent to the Weyl quantization, in the sense of Rieffel and Landsman. Realizing the C*-algebraic quantization maps in regular and faithful Hilbert space representations leads to quantizations of the unbounded field expressions.