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 four dimensions, it is possible for a scalar field to have a vacuum expectation value that would be forbidden if the vacuum were invariant under some continuous transformation group, even though this group is a symmetry group in the sense that the associated local currents are conserved. This is the Goldstone phenomenon, and Goldstone's theorem states that this phenomenon is always accompanied by the appearance of massless scalar bosons. The purpose of this note is to show that in two dimensions the Goldstone phenomenon can not occur; Goldstone's theorem does not end with two alternatives (either manifest symmetry or Goldstone bosons) but with only one (manifest symmetry).

/pdf/there-are-no-goldstone-bosons-in-two-dimensions-9i0recriy6.pdf

There are no Goldstone bosons in two-dimensions

Classical mechanics was first envisaged by Newton, formed into a powerful tool by Euler, and brought to perfection by Lagrange and Laplace. It has served as the paradigm of science ever since. Even the great revolutions of 19th century phys icsnamely, the FaradayMaxwell electro-magnetic theory and the kinetic t h e o r y w e r e viewed as further support for the complete adequacy of the mechanistic world view. The physicist at the end of the 19th century had a coherent conceptual scheme which, in principle at least, answered all his questions about the world. The only work left to be done was the computing of the next decimal. This consensus began to unravel at the beginning of the 20th century. The work of Planck, Einstein, and Bohr simply could not be made to fit. The series of ad hoc moves by Bohr, Eherenfest, et al., now called the old quantum theory, was viewed by all as, at best, a stopgap. In the period 1925-27 a new synthesis was formed by Heisenberg, Schr6dinger, Dirac and others. This new synthesis was so successful that even today, fifty years later, physicists still teach quantum mechanics as it was formulated by these men. Nevertheless, two foundational tasks remained: that of providing a rigorous mathematical formulation of the theory, and that of providing a systematic comparison with classical mechanics so that the full ramifications of the quantum revolution could be clearly revealed. These tasks are, of course, related, and a possible fringe benefit of the second task might be the pointing of the way 'beyond quantum theory'. These tasks were taken up by von Neumann as a consequence of a seminar on the foundations of quantum mechanics conducted by Hilbert in the fall of 1926. In papers published in 1927 and in his book, The Mathemat ical Foundations of Quantum Mechanics, von Neumann provided the first completely rigorous

The mathematical foundations of quantum mechanics

This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

The Theory of Quantum Information

In order to provide a mathmatical framework for the process of making repeated measurements on continuous observables in a statistical system we make a mathematical definition of an instrument, a concept which generalises that of an observable and that of an operation. It is then possible to develop such notions as joint and conditional probabilities without any of the commutation conditions needed in the approach via observables. One of the crucial notions is that of repeatability which we show is implicitly assumed in most of the axiomatic treatments of quantum mechanics, but whose abandonment leads to a much more flexible approach to measurement theory.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103842336

An operational approach to quantum probability

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.

An Algebraic Approach to 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.

/pdf/stability-and-equilibrium-states-3i4oz7wenr.pdf

Stability and equilibrium states

Starting from aC*-algebra$\mathfrak{A}$ and a locally compact groupT of automorphisms of$\mathfrak{A}$ we construct a “covariance algebra”$\mathfrak{A}_1^T $ with the property that the corresponding *-representations are in one-to-one correspondence with covariant representations of$\mathfrak{A}$ i.e. *-representations of$\mathfrak{A}$ in which the automorphisms are continuously unitarily implemented. We further construct for relativistic field theory an algebra$\mathfrak{A}_1^T \left( V \right)$ yielding the *-representations of$\mathfrak{A}$ in which the space time translations have their spectrum contained inV. The problem of denumerable occurence of superselection sectors is formulated as a condition on the spectrum of$\mathfrak{A}_1^T \left( V \right)$. Finally we consider the covariance algebra$\mathfrak{A}_1^T $ built with space translations alone and show its relevance for the discussion of equilibrium states in statistical mechanics, namely we restore in this framework the equivalence of uniqueness of the vacuum, irreducibility and a weak clustering property.

/pdf/covariance-algebras-in-field-theory-and-statistical-5d89lotduf.pdf

Covariance algebras in field theory and statistical mechanics

The concept of locality in quantum physics leads to mathematical structures in which the basic object is an operator algebra with a net of distinguished subalgebras (the “local” subalgebras). Such nets provide a classification of the states of this algebra in equivalence classes determined by local or asymptotic properties. The corresponding equivalence relations are natural generalizations of the (more stringent) standard quasiequivalence relation (they are also useful for classifying states by their properties with respect to automorphism groups). After discussing general nets from this point of view we investigate in the last section more specialized nets (funnels of von Neumann algebras) with special emphasis on their locally normal states.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1103842071

Nets of C *-algebras and classification of states

Mathematically we consider aC*-algebra
 
$$\mathfrak{A}$$

 , acted upon by the groupT of space-translations, which has an asymptotic abelian property. We analyse invariant states over
 
$$\mathfrak{A}$$

 . Physically this programme can be considered as a kinematical study of equilibrium states in statistical mechanics. Each invariant state can be uniquely decomposed into elementary invariant states (E-states). These elementary states have, amongst other characteristics, the physical property that space-averages of local observables are constants in the corresponding representations. In anE-state the discrete spectrum S
 D
 of space-translations is additive which gives rise to the classificationE
I,E
II, andE
III corresponding to the three possibilities that S
 D
 contains one point, a lattice of points, or a set with accumulation points. AnE
II-state can be uniquely decomposed into states (L-states) having a symmetry with respect to a closed subgroupT

 L
 of (S
 D
 and T
 L
 are reciprocal lattices).L-states have properties with respect toT

 L
 analogous to the properties ofE
I-states with respect toT. The decomposition intoL-states is the inverse process of ‘homogenizing’ a lattice state by smearing it over a lattice distance. The mathematical methods which we employ have more general applications.

/pdf/invariant-states-in-statistical-mechanics-mws2qldhpr.pdf

Daniel Kastler

Papers

An Algebraic Approach to Quantum Field Theory

Stability and equilibrium states

Covariance algebras in field theory and statistical mechanics

Nets of C *-algebras and classification of states

Invariant states in statistical mechanics