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THE KOLMOGOROV-LAX-PHILLIPS SYSTEMS AS BRANCH SYSTEMS OF THE REICHEMBACH MODEL

Fisica del Espacio

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Based in some mathematical theorems, related to Lax-Phillips scattering theory, a unified (classical-quantum) formalism is introduced to explain the appearance of irreversibility in some class of systems (KLPS-systems). These systems are endowed with a unitary evolution that maps densities into densities, and they have a fixed subsystem in which this evolution is strongly contracting, yielding a Lyapounov variable and a unique equilibrium state. A global model of these systems is also studied.

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Cambridge University Press eBooks•Apr 16, 2009•0 citations
Based in some mathematical theorems, related to Lax-Phillips scattering theory, a unified (classical-quantum) formalism is introduced to explain the appearance of irreversibility in some class of systems (KLPS-systems). These systems are endowed with a unitary evolution that maps densities into densities, and they have a fixed subsystem in which this evolution is strongly contracting, yielding a Lyapounov variable and a unique equilibrium state. A global model of these systems is also studied.
It has been shown by Flesia and Piron that the scattering theory of Lax and Phillips, developed primarily for the treatment of hyperbolic systems (e.g., classical scattering of electromagnetic waves on a finite target), can be applied to the quantum theory. In order to construct the correspondence, the continuously infinite family of (isomorphic) Hilbert spaces associated with each value of the time (are taken to be a direct integral representation (with a choice of measure) of a larger Hilbert space which includes the variable t in its measure space. We show that this theory, for which the evolution contracted to the subspace orthogonal to the incoming and outgoing subspaces is a contractive semigroup, provides a natural description of the irreversible motion of an unstable quantum system
The scattering theory of Lax and Phillips, designed primarily for hyperbolic systems, such as electromagnetic or acoustic waves, is described. This theory provides a realization of the theorem of Foias and Nagy; there is a subspace of the Hilbert space in which the unitary evolution of the system, restricted to this subspace, is realized as a semigroup. The embedding of the quantum theory into this structure, carried out by Flesia and Piron, is reviewed. We show how the density matrix for an effectively pure state can evolve to an effectively mixed state (decoherence) in this framework. Necessary conditions are given for the realization of the relation between the spectrum of the generator of the semigroup and the singularities of the SSS-matrix (in energy representation). It is shown that these conditions may be met in the Liouville space formulation of quantum evolution, and in the Hilbert space of relativistic quantum theory.
These notes are a short introduction to the mathematical theory of open quantum systems. They are meant to serve as an entry point into a broad research area which has applications across the quantum sciences dealing with systems subjected to external noise. The guiding idea is to let the key structures of the theory emerge from a concrete model. By working through the dissipative Jaynes-Cummings model the reader will dis- cover explicitly how irreversible dynamics arises from a unitary system-reservoir evolution. The notions of the continuous mode limit, correlation functions, spectral density appear in a natural manner and lead to the evolution equation of the open system in form of a master equation. This sets the stage for the more general analysis of completely positive, trace preserving (CPTP) maps and the study of quantum dynamical semigroups. We motivate and prove the Kraus representation theorem, the dilation theorem and the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) theorem. Working through the exercises (for which full solutions are supplied) will reinforce the ideas introduced in the main text.
Abstract The relations between the Schrödinger and wave evolutions are investigated. A generalization of the Lax‐Phillips scattering theory for the case of the Schrödinger equation is constructed. An analog of the wave decay operator is obtained. The general ideas are illustrated by an explicitly solvable model of multichannel scattering on the halfaxis. A Lax‐Phillips scattering theory is developed for the model operator obtained by a self‐adjoint perturbation of the background operator. © 1993 John Wiley & Sons, Int.
Intrinsic theory of irreversibility In connection with the previous chapter on Wigner–Weisskopf, quantum irreversibility, Gamov states and the Lax–Phillips theory of decay, we shall examine the recent program in statistical mechanics carried forward by Ilya Prigogine and his colleagues since his early book in 1962 (Prigogine, 1962). This discussion is somewhat out of the focus of the present book, since the work to be discussed has principally been devoted to isolated quantum systems, not those in interaction with “reservoirs.” In addition, Prigogine's work is very classical in its content. We shall not attempt the task of reviewing the many changes that have taken place between 1962 and the present. Fine texts describing some of this work are those of Balescu (1963, 1975). A recent critical overview of the “modern view” is in the unpublished thesis of B. C. Bishop of the University of Texas Philosophy Department (Bishop, 1999). In many ways this is related to the use of Gel'fand triplets (Gel'fand and Vilenkin, 1968), mentioned in Chapter 17.6, to describe irreversible quantum states, i.e. Gamov states. See Chapter 17, the book of Bohm and Gadella (1989) and also the more recent review article by Bohm and colleagues (Bohm et al. , 1997). The object in statistical mechanics is to extend the range of the Liouville operator (classical and quantum) such that its eigenvalues are complex and intrinsically irreversible. Statistical mechanics has always been focused on many particle systems and the appearance of a continuum spectrum in the thermodynamic limit, already used in many places in this book. In the classical case Antoniou and Tasaki carried out the adaptation of the Gel'fand triplet approach to the Liouville operator, but this is, apparently, not possible quantum mechanically (Antoniou and Tasaki, 1993). Thus, another route must be taken in what Prigogine has termed, for emphasis, “large Poincaré systems” which have a continuum spectrum (Prigogine, 1997).