Quantum-classical correspondence. Dynamical quantization and the classical limit.
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 classical physics such as the formulation of Newton, Hamilton, Lagrange and Jacobi are shortly mentioned, but also the Caldeira-Leggett approach. Chapter 4 is devoted to the quantum physics. The quantizations of Planck-Bohr, Heisenberg, Schrödinger, Dirac, Feynmann, Nelson and Olavo are recalled. The original contribution included in the book is presented in \S 4.9, where the open systems are quantized without Hamiltonians or Lagrangians. This method is called by Bolivar dynamic quantization. As a typical example, the author applies his method to Brownian motion and derives the Caldeira-Leggett master equation, valid for a particular class of stochastic processes, the so called Markovian processes. Also a non-Markovian master equation is obtained. Chapter 5 approaches the problem of dequantization. The standard methods, as the Bohr correspondence principle, the Ehrenfest theorem, the WKB method, the Feynman method, and the decoherenece method are critically reviewed. More precisely, Bolivar starts with the Langevin and Fokker-Planck equations describing open systems in phase space. By means of the Fourier transform, the author introduces the so called Wigner representation of the classical mechanics, characterized by an arbitrary parameter \({l}\), and quantization is defined when \(l\rightarrow \hbar\) in the time evolution of the classical Wigner function. The author obtains in the von Neumann representation the quantum master equations (e.g. the Caldeira-Leggett equation) without using the Hamiltonian formalism. Dequantization is obtained by taking a Fourier transform on the von Neumann function. \S 6 deals with a summary and some open questions. The book contains some Appendices some of a general nature, and some facts extracted from Bolivar's articles. The paper is based on 272 references. The author insists more on the physical and philosophical aspects than on the mathematical rigor. However, the mathematician will find a nice presentation of some classical and open problems in the theory of quantization and dequantization of open systems and a lot of references to classical articles.
scientific article; zbMATH DE number 2115116