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Journal of Mathematical Physics

The Heisenberg dynamics of spin systems: A quasi*-algebras approach

Fabio Bagarello, Camillo Trapanı

Published Sep 1, 1996• 32 citations
Publisher

Abstract

The problem of the existence of the thermodynamical limit of the algebraic dynamics for a class of spin systems is considered in the framework of a generalized algebraic approach in terms of a special class of quasi*-algebras, called CQ*-algebras. Physical applications to (almost) mean-field models and to bubble models are discussed.
DOI
10.1063/1.531652

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Cambridge University Press eBooks•Aug 30, 1991•0 citations
Following the work of P. Bona [J. Math. Phys. 29, 2223 (1988)], a class of infinite mean-field quantum lattice systems is considered within the framework of algebraic quantum mechanics, and the following question is discussed: Which are the possible dynamical descriptions of the system, i.e., in which representations π of the system’s quasilocal C*-algebra 𝒜 do the local time evolutions converge in the thermodynamic limit to an automorphism group of a certain enlarged C*-algebra 𝒞π containing π(𝒜) and (some of) the classical observables of the system in π. To this end we associate a ‘‘classical state space’’ Eπ to any representation π (which is a subset of the phase-space E⊇RL defined by Bona for the largest possible representation), and show how it is characterized by the π-normal states, thereby obtaining a way to determine Eπ explicitly in many cases. The answer to the question posed then reads: The limiting (Heisenberg) dynamics exists in π on 𝒞π if and only if the set Eπ is invariant under the flow φQt on E which describes the time evolution of classical observables. Conversely, to any invariant, closed subset B of E, there exists a dynamical description of the quantum system with classical state space B.
An infinite quantum system with correctly defined dynamics τQ as an automorphism group of a C*-algebra 𝒞 of observables is determined by any continuous unitary representation U(G) of a connected Lie group G, as well as by an arbitrary differentiable real function Q on the dual space 𝔤* to the Lie algebra 𝔤 of G with the canonically defined Poisson flow φQ on 𝔤*. For specific choices of Q and G, the system can be obtained as the thermodynamic limit of a net of finite lattice systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)]. A simple nontrivial model of this type is the quasispin BCS model of superconductivity in the strong coupling limit, or a corresponding model of the Josephson junction. A peculiar feature of the considered models is τQ noninvariance of the usually considered C*-algebra 𝒜 of quasilocal observables, as well as an important role of classical dynamics of a set of macroscopic (intensive) observables Qφ in the description of τQ. The work is restricted here to norm-continuous representations U(G), in which case 𝒞 is isomorphic to the tensor product 𝒜⊗𝒩, where 𝒩 is the commutative algebra of classical (intensive) observables of the considered infinite quantum system.
Abstract The microscopic dynamics for a class of long range interacting multi-lattice quantum systems is constructed in the thermodynamical limit by means of operator algebraic concepts. By direct estimations the existence of the limiting Schrödinger dynamics is proven for a set of states, which comprises also globally non-equilibrium situations. The expectation values of the classical observables in the pure phase states are shown to satisfy a set of coupled non-linear differential equations. The limiting Heisenberg dynamics is derived as a W*-automorphism group in the partially universal von Neumann algebra which corresponds to the selected set of states; it is in general, however, not σ-weakly continuous in the time parameter.
Introduction In this chapter, we shall discuss a unified axiomatic treatment of quantum lattice systems and quasi-free dynamics in Fermion field theory within the framework of C*-dynamical systems. In these systems, time evolution, equilibrium states (KMS states), ground states, stability under bounded perturbations and phase transitions are important physical notions. Here we shall present an abstract treatment of these notions within the theory of C*-dynamical systems. About time evolution, we shall emphasize the approximate innerness property for the corresponding time automorphism group in a given C*-dynamical system. In fact, the approximate innerness assures the existence of a ground state, and the existence of a KMS state at each inverse temperature under the assumption of ‘the existence of a tracial state’, which is always satisfied in quantum lattice systems and in canonical anticommutation relation algebras. We shall also discuss in detail one of the most important open problems (the Powers-Sakai conjecture) in C*-dynamical systems. In each section we shall explain relations between C*-dynamical systems and physical systems. For simplicity we shall assume that every C*-algebra has an identity (unless otherwise stated). Approximately inner C*-dynamics A quantum lattice system consists of a set of particles confined to a lattice and interacting at distance. There are two physical interpretations of these models. One is a lattice gas and the other is a spin system. The lattice gas views each point of the lattice as a possible site for a finite number of AT-particles, i.e. each point of the configuration space can be empty or occupied by 1, 2, …, N -particles.