Algebraic quantization

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.

GIOVANNI GIACHETTA, Luigi Mangiarotti, Gennadi Aleksandrovich Sardanashvily

We provide geometric quantization of the vertical cotangent bundle V*Q→Q→R, equipped with the canonical Poisson structure and treated as a momentum phase space of nonrelativistic time-dependent mechanics. We show that this quantization is equivalent to fiberwise quantization of symplectic fibers of V*Q→R and that the quantum algebra of time-dependent mechanics is an instantwise algebra. Quantization of the classical evolution equation defines a connection on this instantwise algebra and describes quantum evolution in time-dependent mechanics as a parallel transport.

Stephen D. Bartlett

A theory of quantization is given, for a model with symmetry, as the inverse of dequantization. Dequanrization is defined as tin� process of extracting the classical mechanics of a model from its quantal counterpart. It is shown that classical dynamics is embedded in quantum dynamics as a constrained subsystem. Thus, classical mechanics is obtained from quantum mechanics without taking limits of large-dimensional representations or letting h �? 0. The embedding shows that, under appropriate conditions, a quantal density can be identified with a classical state. The results are consistent with the conjecture that an algebraic model is quantized by constructing a unitary irrep of its spectrum generating algebra. Quantization of operators not in this algebra is determined by compatibility with dequantization: this method resolves the Dirac problem in a way that avoids the ambiguities of canonical quantization. Geometric quantization is used to express the vector coherent state inducing construction as a theory of quantization. The result provides physically useful quantum models for systems with an algebraic structure and intrinsic degrees of freedom. The results are illustrated by application to the S i (3) model of a rotor. An algebraic model of a rigid rotor is presented, anti the techniques of vector coherent state quantization are applied to give a quantization, including intrinsic degrees of freedom, that is compatible with the geometric structure of the corresponding classical model. An explicit dequantization of the asymmetric top, leading to Euler�s equations, illustrates the consistency of this quantization.

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau