It is proved that the C*-algebra of observables of an infinite classical system is isomorphic to the group algebra on the test function space 𝒟. The physical dynamical system consisting of infinitely many noninteracting particles is studied. A particular class of states, called the quasifree states, is exhibited and their properties are studied. Some results on the spectral properties of monoparticle evolutions are obtained. Finally we give explicitly a solution of the classical KMS condition for these evolutions.
It is proved that the C*-algebra of observables of an infinite classical system is isomorphic to the group algebra on the test function space 𝒟. The physical dynamical system consisting of infinitely many noninteracting particles is studied. A particular class of states, called the quasifree states, is exhibited and their properties are studied. Some results on the spectral properties of monoparticle evolutions are obtained. Finally we give explicitly a solution of the classical KMS condition for these evolutions.
In this paper, infinite translation-invariant and periodic systems of classical particles are treated directly—not as limits of finite systems. A formalism of classical creation and annihilation operators that create and destroy classical identical particles at points of phase space is employed. By use of this formalism, classical dynamical variables are expressed without reference to the canonical coordinates and momenta of individual particles, or to the number of particles. Different physical situations are described by different representations of the algebra of creation and annihilation operators. The concept of thermal equilibrium is generalized so as to be meaningful in infinite systems. Stationary states and thermal-equilibrium states for an infinite system of noninteracting particles (possibly in a periodic external potential) are exhibited explicitly.
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.
A particular type of continuous quantum system with infinitely many particles is analyzed, and the existence of dynamics is proven in the GNS representations of certain states. The dynamics is not a group of automorphisms on the original algebra, so equilibrium states are defined in terms of the KMS condition in the representations of the states. The basic theorems about KMS states do not apply here. Nevertheless, for a special class of interactions it is proven that the central decomposition of an equilibrium state is concentrated on a Borel set of equilibrium factor states and that such factor states are precisely the extremal equilibrium states. Furthermore, the equilibrium factor states are in one-to-one correspondence with sets of functions satisfying a certain system of trace equations. This explicit correspondence is then used to show that there are no phase transitions for high temperature, and an example of a phase transition is constructed for low temperature. The phase transition also provides an example of continuous symmetry breaking.
The time irreversible evolution of quantum systems with infinite number of particles (QSINP) was studied within a newly constructed algebraic framework. The QSINP could be described by the quantum infinite lattice field. The *-Algebra R is the set of all dynamic variables of the QSINP, in which a special addition and multiplication operations were defined. The full pure state vector (FPSV) ρ, the pure state vector {ρ_f} and the equivalent relations within them are also well defined. The set of all pure state vectors, which are equivalent to the FPSV ρ, is a Hilbert Space {H_ρ}, if the addition, the inner product and the norm were defined in it. The set of all linear transformation {N_ρ} on {H_ρ} is isomorphic to R, and {{N_ρ},{H_ρ}} is the representation of R, associated with ρ. The *-Algebra R has infinitely many non-equivalent irreducible GNS constructions (representations), associated with different nonequivalent FPSVs respectively. It is proved that, the dynamical motions of the QSINP is totally within the GNS construction associated with the initial pure state vector; However, the time reversal transformation makes the initial FPSV and its corresponding dynamical evolution into another non-equivalent GNS construction. Therefore, within the GNS construction associated with the FPSV, the dynamics of the QSINP is time irreversible. Finally, due to the Liouville operator has real continuous spectrum within the GNS construction, the longtime asymptotic solution of Liouville equation in the GNS construction could be treated by the formal scattering theory. The Master equation with dissipative term was obtained, which is formally irrelevant of the initial state and the corresponding GNS construction. This master equation could be regarded as the evolution equation of the QSINP on R.
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.
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.