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.

General properties of entropy

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.

On revolutionizing quantum field theory with Tomita’s modular theory

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.

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Algebraic Quantum Field Theory

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.

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Stability and equilibrium states

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Entropy for automorphisms of II1 von neumann algebras

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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R. V. Kadison

Papers

Asymptotically abelian systems