TheN-state system dynamics

Shôichirô Sakai

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.

Ángel S. Sanz, Salvador Miret-Artés

Da-Jian Zhang, Chong-Long Liu, D. M. Tong

Many quantum systems of interest are initially correlated with their environments and the reduced dynamics of open systems are an interesting while challenging topic. Affine maps, as an extension of completely positive maps, are a useful tool to describe the reduced dynamics of open systems with initial correlations. However, it is unclear what kind of initial state shares an affine map. In this study, we give a sufficient condition of initial states, in which the reduced dynamics can always be described by an affine map. Our result shows that if the initial states of the combined system constitute a convex set, and if the correspondence between the initial states of the open system and those of the combined system, defined by taking the partial trace, is a bijection, then the reduced dynamics of the open system can be described by an affine map.

Ray LaPierre

Anton Amann

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.

Ray LaPierre