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.
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.
Dynamical evolutions of boson (spin-boson and related) systems based on the widely used canonical commutation relations C* algebra have a shortcoming. The dynamical group is not pointwise norm-continuous and therefore cannot be perturbed by a (bounded self-adjoint) element of the underlying abstract C* algebra. Hence it is necessary to define notions like 'invariant state', 'ground state', 'KMS state', etc. with respect to the perturbed dynamics in suitable Hilbert space representations. Here the original dynamical system is supplemented by an auxiliary pointwise norm-continuous dynamical system in such a way that invariant states of the original system correspond to invariant states of the auxiliary system. This bijective correspondence is sequentially continuous and preserves the KMS (ground state) characterizing conditions. As a typical application it is verified that Spohn's ground states of the spin-boson model (1989) (arising as a temperature to zero limit of thermic equilibrium states) are ground states in the algebraic sense, i.e. are eigenstates of the respective (Gel'fand-Naimark-Segal) Hamiltonian with an eigenvalue at the lower end of the Hamiltonian spectrum.
A dynamics (i.e. a one-parameter group of automorphisms) of a system described by a C *-algebra with a local structure in terms of C *-subalgebras A ( I ) for local domains I of the physical space (a discrete lattice) is normally constructed out of potentials P ( I ), each of which is a self-adjoint element of the subalgebra A ( I ), such that the the first time derivative of the dynamical change of any local observable A is i times the convergent sum of the commutator [ P ( I ), A ] over all finite regions I . We will invert this relation under the assumption (obviously assumed in the usual approach) that local observables all have the first time derivative, i.e. we prove the existence of potentials for any given dynamics satisfying the above-stated condition. Furthermore, by imposing a further condition for the potential P ( I ) to be chosen for each I that it does not have a portion which can be shifted to potentials for any proper subset of I , we also show (1) the existence, (2) uniqueness, (3) an automatic convergence property for the sum over I , and (4) a quite convenient property for the chosen potential. The so-obtained properties (3) and (4) are not assumed and are very useful, though they were never noticed nor used before. We consider a system of finite kinds of finite spins and fermions on a discrete lattice, local regions being all finite subsets of the lattice and all local algebras being full matrix algebras of finite dimensions. For all dynamics for which all elements of any local algebra is once time differentiable, we prove that there exist a system of potentials which describe the time derivative of the given dynamics by a convergent sum stated above. The fundamental technique for finding such potentials is a non-commutative expectation which is defined on the basis of a product state of the algebra. For each choice of a product state, we obtain one expectation which produces one set of potentials, all of which satisfy what we call the standardness condition and the convergence condition. We call this family of potentials standard potentials (corresponding to any specific choice of the product state). The standard potentials corresponding to different product states are different but produces the same time derivative, known as equivalent potentials.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.
Dynamical C*-systems without norm-continuity properties (e.g., the dynamics of a boson-type system) are studied. The respective dynamics cannot be perturbed by (bounded, self-adjoint) elements of the underlying abstract C*-algebra. Hence, it is necessary to define notions like ‘‘invariant state,’’ ‘‘ground state,’’ ‘‘KMS-state,’’ etc. with respect to the perturbed dynamics in suitable Hilbert space representations. Here, the original dynamical system is supplemented by an auxiliary pointwise norm-continuous dynamical system in such a way that invariant states of the original system correspond to certain invariant states of the auxiliary system. This bijective correspondence is sequentially continuous and preserves the KMS- (ground state) characterizing conditions. As a typical application it is verified that Spohn’s ground states of the spin-boson model (arising as a temperature to zero limit of thermic equilibrium states) are ground states in the algebraic sense, i.e., are eigenstates of the respective (Gel’fand–Naimark–Segal) Hamiltonian with an eigenvalue at the lower end of the Hamiltonian’s spectrum.
We consider the dynamics of systems of lattice bosons with infinitely many degrees of freedom. We show that their dynamics defines a group of automorphisms on a C∗--algebra introduced by Buchholz, which extends the resolvent algebra of local field operators. For states that admit uniform bounds on moments of the local particle number, we derive propagation bounds of Lieb--Robinson type. Using these bounds, we show that the dynamics of local observables gives rise to a strongly continuous unitary group in the GNS representation. Moreover, accumulation points of finite-volume Gibbs states satisfy the KMS condition with respect to this group. This, in particular, proves the existence of KMS states.
We presents a unified and concise exposition of key topics in the mathematical theory of open quantum systems, developed within the framework of operator algebras. The manuscript consolidates and extends a series of invited articles originally prepared for the Modern Encyclopedia of Mathematical Physics, combining foundational material with modern perspectives on non-equilibrium quantum statistical mechanics. After introducing the C*- and W*-algebraic formulation of quantum mechanics, the paper reviews quantum dynamical systems, KMS states, and Tomita-Takesaki modular theory, as well as CCR and CAR algebras for bosonic and fermionic systems. Particular emphasis is placed on infinite systems, non-equilibrium steady states, entropy production, and linear response theory. The later sections develop a systematic treatment of small systems coupled to reservoirs, open lattice quantum spin systems, culminating in a detailed discussion of competing notions of quantum entropy production. The presentation highlights structural insights, conceptual clarity, and connections between equilibrium and non-equilibrium phenomena, providing a self-contained reference for researchers and graduate students in mathematical physics.