Asymptotically abelian systems
TL;DR: In this article, the authors studied a class of systems which are asymptotically abelian with respect to their invariant states (i.e., they are invariant under a vector cyclic).
Abstract: We study pairs {\(\mathfrak{A}\), α} for which\(\mathfrak{A}\) is aC*-algebra and α is a homomorphism of a locally compact, non-compact groupG into the group of *-automorphisms of\(\mathfrak{A}\). We examine, especially, those systems {\(\mathfrak{A}\), α} which are (weakly) asymptotically abelian with respect to their invariant states (i.e. 〈Φ |A αg(B) — αg(B)A〉 → 0 asg → ∞ for those states Φ such that Φ(αg(A)) = Φ(A) for allg inG andA in\(\mathfrak{A}\)). For concrete systems (those with\(\mathfrak{A}\)-acting on a Hilbert space andg → αg implemented by a unitary representationg →Ug on this space) we prove, among other results, that the operators commuting with\(\mathfrak{A}\) and {Ug} form a commuting family when there is a vector cyclic under\(\mathfrak{A}\) and invariant under {Ug}. We characterize the extremal invariant states, in this case, in terms of “weak clustering” properties and also in terms of “factor” and “irreducibility” properties of {\(\mathfrak{A}\),Ug}. Specializing to amenable groups, we describe “operator means” arising from invariant group means; and we study systems which are “asymptotically abelian in mean”. Our interest in these structures resides in their appearance in the “infinite system” approach to quantum statistical mechanics.
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TL;DR: This paper discusses properties of entropy, as well as related concepts such as relative entropy, skew entropy, dynamical entropy, etc, in detail with reference to their implications in statistical mechanics, to get a glimpse of systems with infinitely many degrees of freedom.
Abstract: It is rather paradoxical that, although entropy is one of the most important quantities in physics, its main properties are rarely listed in the usual textbooks on statistical mechanics. In this paper we try to fill this gap by discussing these properties, as, for instance, invariance, additivity, concavity, subadditivity, strong subadditivity, continuity, etc., in detail, with reference to their implications in statistical mechanics. In addition, we consider related concepts such as relative entropy, skew entropy, dynamical entropy, etc. Taking into account that statistical mechanics deals with large, essentially infinite systems, we finally will get a glimpse of systems with infinitely many degrees of freedom.
1,712 citations
TL;DR: In this paper, the authors present a review of the results obtained up to this point and some problems for the future will be discussed at the end of the book, including some problems that are closely connected with the modular theory and that should be treated in the future.
Abstract: In the book of Haag [Local Quantum Physics (Springer Verlag, Berlin, 1992)] about local quantum field theory the main results are obtained by the older methods of C * - and W * -algebra theory. A great advance, especially in the theory of W * -algebras, is due to Tomita’s discovery of the theory of modular Hilbert algebras [Quasi-standard von Neumann algebras, Preprint (1967)]. Because of the abstract nature of the underlying concepts, this theory became (except for some sporadic results) a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the half-sided translations and the half-sided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Sec. IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Sec. VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.
227 citations
01 Feb 2006
TL;DR: Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools as mentioned in this paper.
Abstract: Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools -- the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is a survey of AQFT, with an orientation towards foundational topics. In addition to covering the basics of the theory, we discuss issues related to nonlocality, the particle concept, the field concept, and inequivalent representations. We also provide a detailed account of the analysis of superselection rules by Doplicher, Haag, and Roberts (DHR); and we give an alternative proof of Doplicher and Robert's reconstruction of fields and gauge group from the category of physical representations of the observable algebra. The latter is based on unpublished ideas due to J. E. Roberts and the abstract duality theorem for symmetric tensor *-categories, a self-contained proof of which is given in the appendix.
206 citations
TL;DR: In this paper, the authors propose to characterize equilibrium states by the three properties of stationarity, stability for local perturbations of the dynamics, and relative purity, and show that a state with these properties either gives rise to a one-sided energy spectrum or is a KMS- (i.e. essentially a limit Gibbs-) state.
Abstract: For an infinite dynamical quantum system idealized as aC*-algebra acted upon by time-translations automorphisms in an asymptotically abelian way, we propose to characterize equilibrium states by the three properties of stationarity, stability for local perturbations of the dynamics, and relative purity. We show that a state with these properties either gives rise to a one-sided energy spectrum or is a KMS- (i.e. essentially a limit Gibbs-) state.
189 citations
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Book•
01 Jan 1948
TL;DR: The theory of semi-groups has been studied extensively in the literature, see as discussed by the authors for a survey of some of the main applications of semi groups in the context of functional analysis.
Abstract: Part One. Functional Analysis: Abstract spaces Linear transformations Vector-valued functions Banach algebras General properties Analysis in a Banach algebra Laplace integrals and binomial series Part Two. Basic Properties of Semi-Groups: Subadditive functions Semi-modules Addition theorem in a Banach algebra Semi-groups in the strong topology Generator and resolvent Generation of semi-groups Part Three. Advanced Analytical Theory of Semi-Groups: Perturbation theory Adjoint theory Operational calculus Spectral theory Holomorphic semi-groups Applications to ergodic theory Part Four. Special Semi-groups and Applications: Translations and powers Trigonometric semi-groups Semi-groups in $L_p(-\infty,\infty)$ Semi-groups in Hilbert space Miscellaneous applications Part Five. Extensions of the theory: Notes on Banach algebras Lie semi-groups Functions on vectors to vectors Bibliography Index.
3,462 citations
Book•
01 Jan 1957
1,184 citations
TL;DR: In this article, it was shown that two quantum theories dealing in the Hilbert spaces of state vectors H1 and H2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not.
Abstract: It is shown that two quantum theories dealing, respectively, in the Hilbert spaces of state vectors H1 and H2 are physically equivalent whenever we have a faithful representation of the same abstract algebra of observables in both spaces, no matter whether the representations are unitarily equivalent or not. This allows a purely algebraic formulation of the theory. The framework of an algebraic version of quantum field theory is discussed and compared to the customary operator approach. It is pointed out that one reason (and possibly the only one) for the existence of unitarily inequivalent faithful, irreducible representations in quantum field theory is the (physically irrelevant) behavior of the states with respect to observations made infinitely far away. The separation between such ``global'' features and the local ones is studied. An application of this point of view to superselection rules shows that, for example, in electrodynamics the Hilbert space of states with charge zero carries already all the relevant physical information.
1,160 citations
"Asymptotically abelian systems" refers background in this paper
...In the general frame of quantum mechanics the physical observables are described as self-adjoint operators on a Hilbert space ~ and t he bounded observables (corresponding to bounded operators) therefore generate a C*-algebra acting on ~. The algebraic approach to field theory [ 1 , 2] proposes to consider as physical only the local observables...
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TL;DR: In this article, the authors studied the representation of the C*-algebra of observables corresponding to thermal equilibrium of a system at given temperature T and chemical potential μ and showed that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant.
Abstract: Representations of theC*-algebra\(\mathfrak{A}\) of observables corresponding to thermal equilibrium of a system at given temperatureT and chemical potential μ are studied. Both for finite and for infinite systems it is shown that the representation is reducible and that there exists a conjugation in the representation space, which maps the von Neumann algebra spanned by the representative of\(\mathfrak{A}\) onto its commutant. This means that there is an equivalent anti-linear representation of\(\mathfrak{A}\) in the commutant. The relation of these properties with the Kubo-Martin-Schwinger boundary condition is discussed.
763 citations
"Asymptotically abelian systems" refers background in this paper
...A further specialization of particular interest in the study of ground states at finite temperature [ 14 ] has equivalent formulations stated in the...
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