The authors conclude a research programme developed by various authors in several papers. The basic idea of this programme is to explain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacting with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In their approach they study the limits of quantities related to the whole compound system. As a corollary they obtain an explanation of the physical origins of the quantum Brownian motion. They study the low-density limit of a system coupled to a quasi-free boson reservoir in the equilibrium state at inverse temperature beta and fugacity z, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir.
The authors conclude a research programme developed by various authors in several papers. The basic idea of this programme is to explain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacting with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In their approach they study the limits of quantities related to the whole compound system. As a corollary they obtain an explanation of the physical origins of the quantum Brownian motion. They study the low-density limit of a system coupled to a quasi-free boson reservoir in the equilibrium state at inverse temperature beta and fugacity z, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir.
A rigorous derivation of quantum Langevin equation from microscopic dynamics in the low density limit is given. We consider a quantum model of a microscopic system (test particle) coupled with a reservoir (gas of light Bose particles) via interaction of scattering type. We formulate a mathematical procedure (the so-called stochastic golden rule) which allows us to determine the quantum Langevin equation in the limit of large time and small density of particles of the reservoir. The quantum Langevin equation describes not only dynamics of the system but also the reservoir. We show that the generator of the corresponding master equation has the Lindblad form of most general generators of completely positive semigroups.
The present article is devoted to the explanation of the irreversible behavior of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system, whose Hamiltonian has a continuous spectrum, interacting with a reservoir and studies the limits of quantities related to the whole compound system. A macroscopic equation is obtained for the limit of the compound system, which is a quantum stochastic differential equation of Poisson type on some Hilbert module (no longer a space) and whose coefficients are uniquely determined by the one-particle Hamiltonian of the original system and whose driving noises are the creation, annihilation, and number (or gauge) processes living on the Fock module over this module.
We complete the program started in [4] by proving that, in the weak coupling limit, the matrix elements, in the collective coherent vectors, of the Heisenberg evolved of an observable of a system coupled to a quasi-free reservoir through a laser type interaction, converge to the matrix elements of a quantum stochastic process satisfying a quantum Langevin equation driven by a quantum Brownian motion. Our results apply to an arbitrary quasi-free reservoir so, in particular, the finite temperature case is included.
In only 150 pages, not counting appendices, references, or the index, this book is one author's perspective of the massive theoretical and philosophical hurdles in the no-man's-land separating the classical and quantum domains of physics. It ends with him emphasizing his own theoretical contribution to this area. In his own words, he has attempted to answer: 1. 'How can we obtain the quantum dynamics of open systems initially described by the equations of motion of classical physics (quantization process)? 2. 'How can we retrieve classical dynamics from the quantum mechanical equations of motion by means of a classical limiting process (dequantization process)?' However, this monograph seems overly ambitious. Although the publisher's description refers to this book as 'an accessible entrée', we find that this author scrambles too hastily over the peaks of information that are contained in his large collection of 272 references. Introductory motivating discussions are lacking. Profound ideas are glossed over superficially and shoddily. Equations morph. But no new convincing understanding of the physical world results. The author takes the viewpoint that physical systems are always in interaction with their environment and are thus not isolated and, therefore, not Hamiltonian. This impels him to produce a method of quantization of these stochastic systems without the need of a Hamiltonian. He also has interest in obtaining the classical limit of the quantized results. However, this reviewer does not understand why one needs to consider open systems to understand 'quantum-classical correspondence'. The author demonstrates his method using various examples of the Smoluchowski form of the Fokker--Planck equation. He then renders these equations in a Wigner representation, uses what he terms 'an infinitesimality condition', and associates with a constant having the dimensions of an action. He thereby claims to develop master equations, such as the Caldeira--Leggett equation, without use of a Hamiltonian. However, his procedure is seriously flawed. For example, he considers a case of anomalous Brownian motion using a Kolmogorov stochastic equation, and two cases of Brownian motion, the first with a constant force and the second with a linear force. In each of these three cases, his alleged solutions for a function F(x,p,t), which he wishes to interpret as a probability density, are not in fact solutions of their respective evolution equations. The author, in order to interpret these as probability densities, appears to have obtained the correct solutions of these equations, but then has normalized each of them with a time-dependent function, so that their integrals over the whole phase space remains unity for all time. However, an integration of the original equations over phase space demonstrates that an integral of their solutions over the whole phase space must be time-dependent. Despite the poor editing of this monograph, in which errors in equations abound, and in which the dimensionality of quantities is occasionally incorrect, this reviewer's interest in this subject was not extinguished. We are thus thankful for its extensive bibliography.
The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, focusing in particular on the correlations and the squeezing of the probability distribution induced by the environment.
The white noise approach to the investigation of the dynamics of a quantum particle interacting with a dilute and in general non-equilibrium gaseous environment in the low density limit is outlined. The low density limit is the kinetic Markovian regime when only pair collisions (i.e., collisions of the test particle with one particle of the gas at one time moment) contribute to the dynamics. In the white noise approach one first proves that the appropriate operators describing the gas converge in the sense of appropriate matrix elements to certain operators of quantum white noise. Then these white noise operators are used to derive quantum white noise and quantum stochastic equations describing the approximate dynamics of the total system consisting of the particle and the gas. The derivation is given ab initio, starting from the exact microscopic quantum dynamics. The limiting dynamics is described by a quantum stochastic equation driven by a quantum Poisson process. This equation then applied to the derivation of quantum Langevin equation and linear Boltzmann equation for the reduced density matrix of the test particle. The first part of the paper describes the approach which was developed by L. Accardi, I.V. Volovich and the author and uses the Fock-antiFock (or GNS) representation for the CCR algebra of the gas. The second part presents the approach to the derivation of the limiting equations directly in terms of the correlation functions, without use of the Fock-antiFock representation. This approach simplifies the derivation and allows to express the strength of the quantum number process directly in terms of the one-particle S-matrix.