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Journal of Mathematical Physics

Phase-spaces and dynamical descriptions of infinite mean-field quantum systems

Thomas Unnerstall

Published Mar 1, 1990• 12 citations
Publisher

Abstract

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.

DOI
10.1063/1.528904

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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.
The general structure and existence of equilibrium and ground states of infinite spin systems with the mean-field type interaction of Hepp and Lieb [Helv. Phys. Acta 46, 573 (1973)] is investigated. The form and the role of the ‘‘Bogoliubov–Haag’’ Hamiltonians is cleared. The analysis is based on a recent mathematically correct definition of the corresponding time automorphism group τQ (specified by a classical Hamiltonian Q) of an appropriately defined ‘‘physical’’ C*-algebra of observables 𝒞 containing an abelian subalgebra 𝒩 of classical macroscopic quantities as well as the conventional ‘‘bare’’ quasilocal (sub-) algebra 𝒜, 𝒞=𝒜⊗𝒩 [P. Bóna, J. Math. Phys. 29, 2223 (1988)]. A Lie group G of ‘‘hidden symmetries’’ acting nontrivially (also) on 𝒩 plays an essential role in the presented theory. The applications to the quasispin strong-coupling version of the BCS model of superconductivity, and to the corresponding model of the Josephson junction illustrate possibilities of the developed formalism as well as an easy manipulation with it. A simple mathematical description of ‘‘symmetry breaking’’ is sketched for the considered class of mathematical models of large quantal systems.
Local observables of an infinite system are described by a quasilocal C*-algebra A. Traditionally classical observables are introduced as parameters that can be assigned to certain states on the local observables. As fundamental theories should be essentially free of parameters this can hardly be regarded as satisfactory. In the present paper classical observables are introduced explicitly as operators by a transitive system of imprimitivity of a kinematical group in the center of the W*-algebra π(A)″ belonging to a representation π of A. The states of the system are represented by linear functionals on π(A)″, which are dispersion-free on the center of π(A)″. For classical observables related to a kinematical group a W*-description—with algebra π(A)″ and its associated linear functionals that are dispersion-free on the center and normal on the noncentral part of π(A)″—is equivalent to a C*-description—with algebra A and a distinguished family of linear functionals on A, determined uniquely by π. The implications of these results on the interpretation of quantum mechanics of infinite systems are discussed.
. In this paper we extend the theory of mean-field-dynamical semigroups given in [DW1,Du1] to treat the irreversible mean-field dynamics of quasi-local mean-field observables. These are observables which are site averaged except within a region of tagged sites. In the thermodynamic limit the tagged sites absorb the whole lattice, but also become negligible in proportion to the bulk. We develop the theory in detail for a class of interactions which contains the mean-field versions of quantum lattice interactions with infinite range. For this class we obtain an explicit form of the dynamics in the thermodynamic limit. We show that the evolution of the bulk is governed by a flow on the one-particle state space, whereas the evolution of local perturbations in the tagged region factorizes over sites, and is governed by a cocycle of completely positive maps. We obtain an H-theorem which suggests that local disturbances typically become completely delocalized for large times, and we show this...
The algebraic formulation of quantum statistical mechanics is extended so as to include local unbounded observables. We start, in a usual way, with a C*-algebra U of quasilocal bounded observables in Fock space HF, an arbitrary, locally normal state φ on U, and a corresponding Gel'fand-Naimark-Segal (GNS) representation Rφ of U in a Hilbert space Hφ. We then construct a set QL of local, closed operators whose domains are dense in HF, such that QL includes all the local observables of the system. The representation Rφ is then extended so as to provide a *-homomorphism of HL into the closed, densely defined operators in Hφ. Correspondingly, a number of results previously established for the local bounded observables are extended to the unbounded ones. For appropriate classes of locally normal states, these extended results include the Kubo-Martin-Schwinger boundary conditions, the spatially asymptotic and ergodic properties of space-correlation functions, and the temporally ergodic properties of time-correlation functions. It is also shown that, for locally normal Gibbs states, the time correlations between elements of a specified subset of QL are thermodynamical limits of the corresponding correlations for finite systems.
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.