W*-Dynamics of Infinite Dissipative Quantum Systems

TL;DR

This paper develops a mathematical framework for describing the dynamics of infinite open dissipative quantum systems using W*-algebras, extending previous models from conservative to dissipative cases.

Contribution

It introduces a formulation of the dynamics of infinite dissipative quantum systems within the W*-algebra framework, generalizing earlier conservative system models.

Findings

Established that the system's dynamics are given by a semigroup of completely positive transformations.

Extended previous formulations to include open dissipative quantum systems.

Provided a rigorous mathematical foundation for infinite volume dissipative quantum dynamics.

Abstract

We formulate the dynamics of an infinitely extended open dissipative quantum system, ${\Sigma]$,in the Schroedinger picture.The generic model on which this is based comprises a C*-algebra,$[\cal A}$,of observables, a folium, ${\cal F}$, of states on this algebra and a one parameter semigroup,${\tau}$, of linear transformations of ${\cal F}$ that represents its dynamics and is given by a natural infinite volume limit of the corresponding semigroup for a finite system. On this basis, we establish that the dynamic of ${\Sigma}$ is given by a one parameter semigroup of completely positive transformations of the W*-star algebra dual to ${\cal F}$. This result serves to extend our earlier formulation [1[ of infinitely extended conservative systems to open dissipative ones.