On a dynamical quantization

Edward James Anderson

Nguyen T. Thao, Martin Vetterli

A. O. Bolivar

We have worked out a quantization method directly from classical dynamics without using Hamiltonian and Lagrangian functions; we call it dynamical quantization. The present article compares such a method with the Dirac and Feynman quantization procedures and also verifies the logical consistence of the dynamical quantization calculating the classical limit of a Brownian particle, for example. PACS Nos.: 03.65.–w, 05.30.–d, 05.40.+j, 52.65.Ff

Vasily Evgenevich Tarasov

In this paper a generalization of Weyl quantization which maps a dynamical operator in a function space to a dynamical superoperator in an operator space is suggested. Quantization of dynamical operator, which cannot be represented as Poisson bracket with some function, is considered. The usual Weyl quantization of observables can be derived as a specific case of suggested quantization if dynamical operator is an operator of multiplication on a function. This approach allows to define consistent Weyl quantization of non-Hamiltonian and dissipative systems. Examples of the harmonic oscillator with friction and a system which evolves by Fokker-Planck-type equation are considered.

Maurice A. de Gosson

Franco Gallone, Antonio Sparzani

Segal quantization, usually thought of and used as a tool for quantizing kinematical frameworks, is extended to (finite-dimensional) dynamical systems, i.e., to kinematical frameworks plus dynamical motions applied to them. Such a procedure allows us to classify quantum dynamical systems, and to understand how physical inequivalence appears in spite of von Neumann unitary equivalence, for the former is also grounded on the evolution operators of the systems being different functions of the labeled observables of the systems. Second quantization is also examined and is shown to be just one possible procedure of quantization, which can only be used in a particular class of cases.