Quantum-classical correspondence. Dynamical quantization and the classical limit.
type: Item
instance of: scholarly article
title: Quantum-classical correspondence. Dynamical quantization and the classical limit.
zbMATH Open document ID: 1074.81002
author: A. O. Bolivar
published in: The Frontiers Collection
publication date: 2004-11-11T00:00:00Z
review text: The book is devoted to the quantization problem, i.e. how to do a quantum description starting from a classical one, and to dequantization, i.e. how to recover the the classical system, once the the quantum dynamics has been obtained. The author is focusing his attention on the open systems, i.e. physical systems interacting with the environment. The interaction is considered using the probabilistic approach. As a model, the author considers the Brownian motion of a particle immersed in a fluid that has thermodynamic and hydrodynamic properties, in terms of which both the friction and the diffusion coefficients occurring in the Langevin equation and the Fokker-Planck equation can be expressed. The main point of the book is the description of the Brownian motion in quantum physics without using the the Hamiltonian or the Lagrange formalism. The original part of the book is extracted from the articles of Bolivar devoted to the so called dynamic quantization, a method of quantizing open systems without using the Hamilton or the Lagrangians formalisms [\textit{A. O. Bolivar}, Quantization of non-Hamiltonian systems, Phys. Rev. A 58, 4330--4335 (1998); Phys. Lett., A 307, No. 4, 229--232 (2003; Zbl 1006.82019); Physica A 315, No. 3--4, 601--615 (2002; Zbl 1001.82020); Physica A 301, No. 1--4, 219--240 (2001; Zbl 1022.70016); J. Math. Phys. 42, No. 9, 4020--4030 (2001; Zbl 1011.81031); Random Oper. Stoch. Equ. 9, No. 3, 275--286 (2001; Zbl 0974.81031)]. A. O. Bolivar has developed a procedure to take the classical limit \(\hbar\rightarrow 0\) of conservative and nonconservative quantum dynamics. In Chapter 2 the author presents some elements of the theory of stochastic processes, such as the probability space, conditional probability, conditional density function. Chapter 3 is devoted to the classical physics of dissipation and fluctuation phenomena. The contributions of Einstein, Smoluchowski, Rayleigh, Langevin, Klein and Kramers are reviewed. Standard methods in classi…
reviewed by: Stefan Berceanu
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 82B31
Mathematics Subject Classification ID: 82C10
Mathematics Subject Classification ID: 82C70
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 81S20
Mathematics Subject Classification ID: 82C31
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 81S30
Mathematics Subject Classification ID: 81P20
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 2115116
zbMATH Keywords: stochastic processes
zbMATH Keywords: dissipation and fluctuations
zbMATH Keywords: quantization and dequantization
zbMATH Keywords: open system
zbMATH Keywords: Markovian Fokker-Plank equation
zbMATH Keywords: dynamic quantization
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2115116