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Journal of Mathematical Physics

Perturbation theory of C*-systems without norm-continuity properties

Anton Amann

Published Oct 1, 1991• 3 citations
Publisher

Abstract

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.

DOI
10.1063/1.529079

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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.
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.
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.
We consider the problem of constructing quantum dynamics for symmetric Hamiltonian operators that have no self-adjoint extensions. For an earlier studied model, it was found that an elliptic self-adjoint regularization of a symmetric Hamiltonian operator allows one to construct quantum dynamics for vector states on certain C *-subalgebras of the algebra of bounded operators in a Hilbert space. In the present study, we prove that one can extend the dynamics to arbitrary states on these C *-subalgebras while preserving the continuity and convexity. We show that the obtained extension of the dynamics of the set of states on C *-subalgebras is the limit of a sequence of regularized dynamics under removal of the elliptic regularization. We also analyze the properties of the limit dynamics of the set of states on the C *-subalgebras.
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.
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∗C^*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.
This paper deals with perturbation theory for discrete spectra of linear operators. To simplify exposition we consider here self-adjoint operators. This theory is based on the Feshbach-Schur map and it has advantages with respect to the standard perturbation theory in three aspects: (a) it readily produces rigorous estimates on eigenvalues and eigenfunctions with explicit constants; (b) it is compact and elementary (it uses properties of norms and the fundamental theorem of algebra about solutions of polynomial equations); and (c) it is based on a self-contained formulation of a fixed point problem for the eigenvalues and eigenfunctions, allowing for easy iterations. We apply our abstract results to obtain rigorous bounds on the ground states of Helium-type ions.