Symmetry breakdown in statistical mechanics
Lecture given at the Ecole d'Eté de Physique Théorique. Cargèse, Corsica, 1969. Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical quantum lattice systems.
Some Remarks on the ground state of infinite systems in statistical mechanics
Abstract : We investigate the ground states of infinite quantum lattice systems. It is shown in particular that a positive energy operator is associated with these states.
Statistical mechanics of a one-dimensional lattice gas
Abstract : We study the statistical mechanics of an infinite one-dimensional classical lattice gas. Extending a result of Van Hove we show that, for a large class of interactions, such a system has no phase transition. The equilibrium state of the system is represented by a measure which is invariant under the effect of lattice translations. The dynamical system defined by this invariant measure is shown to be a K-system.
Almost periodic states on C*-algebras *)
Abstract : Given a C*-algebra O with a group of automorphisms, we define and study almost periodic states on O. A natural decomposition of such states is introduced and discussed. *) These notes are a result of discussions between S. Dpolicher, G. Gallavotti and D. Ruelle, they are not intended for publication.
A Variational formulation of equilibrium statistical mechanics and the Gibbs phase rule
Abstract : It is shown that for an infinite lattice system, thermodynamic equilibrium is the solution of a variational problem involving a mean entropy of states introduced earlier [2]. As an application, a version of the Gibbs phase rule is proved.
Mean entropy of States in classical statistical mechanics
Abstract : The equilibrium states for an infinite system of classical mechanics may be represented by states over Abelian C* algebras. We consider here continuous and lattice systems and define a mean entropy for their states. The properties of this mean entropy are investigated : linearity, upper semi-continuity, integral representations. In the lattice case, it is found that our mean entropy coincides with the KOLMOGOROV-SINAI invariant of ergodic theory.
Integral représentations of invariant states on B*-algebras
Abstract : Let ?? be a B* algebra with a group G of automorphisms and K be the set of G?invariant states on ??. We discuss conditions under which a G?invariant state has a unique integral representation in terms of extremal points of K, i.e., extremal invariant states.
The States of classical statistical mechanics
Abstract : a state of an infinite system in classical statistical mechanics is usually described by its correlation functions. We discuss here other description in particular as 1) a state on a B*-algebras, 2) a collection of density distributions, 3) a field theory 4) a mesure on a space of configurations of infitely many particles. We consier the relations between these various descriptions and prove under very general conditions an integral representation of a state as superposition of extremal invariant states corresponding to pure thermodynamical phases.
Abstract : A number of results are derived which are pertinent to the description of physical systems by states on C*-algebras invariants under a symmetry group. In particular an integral decomposition relevant to the study of lower symmetry is obtained which is occur in equilibrium statistical mechanics as existence of crystals, ferromagnetic states, etc... A characterization is given of strongly clustering euclidean invariant states, and it is shown that they cannot be decomposed into states of lower symmetry.
Abstract : It is shown that a quantal or classical system of N particles of distinct species ?,? = 1, 2, … ? interacting through pair potentials ???(r) are stable, in the sense that the total energy is always bounded below by ?NB, provided ???(r) exceeds some ???(2)(r) whose Fourier transform ????(p)????(p) corresponds to a positive semidefinite ? × ? matrix for all p. This result is applied to discuss charged systems and stability is proved for Coulomb interactions if the charges are somewhat smeared rather than concentrated at points. For a large class of potentials it is shown that classical instability implies quantum instability in the case of bosons and, in three or more dimensions, also of fermions. Quantum systems with Coulomb interactions (point charges) are discussed and it is shown in particular that their stability cannot depend on the ratios between the masses of the particles.
Seiberg-Witten prepotential from instanton counting
Abstract : Direct evaluation of the Seiberg-Witten prepotential is accomplished following the localization programme suggested some time ago. Our results agree with all low-instanton calculations available in the literature. We present a two-parameter generalization of the Seiberg-Witten prepotential, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy. To Arkady Vainshtein on his 60th anniversary
Target space symmetries in topological theories I.
Abstract : We study realization of the target space diffeomorphisms in the type C topological string. We found that the charges, which generate transformations of the boundary observables, form an algebra, which differs from that of bulk charges by the contribution of the bubbled disks. We discuss applications to noncommutative field theories.
Abstract : We find a new gauge in which U(1) noncommutative instantons are explicitly non-singular on noncommutative R^4. We also present a pedagogical introduction to the noncommutative gauge theories.
Trieste lectures on solitons in noncommutative Gauge theories
Abstract : We present a pedagogical introduction into noncommutative gauge theories, their stringy origin, and non-perturbative effects, including monopole and instanton solutions.
Lectures on open strings, and noncommutative Gauge theories
Abstract :The background independent formulation of the gauge theories on D-branes in flat space-time is considered, some examples of the solutions of their equations of motion are presented, the solutions of Dirac equation in these backgrounds are analyzed, and the generalizations to the curved spaces, like orbifolds, conifolds, and K3 surfaces are discussed.
Milne Universe, Tachyons, and quantum group
Abstract : We analyze the spectrum of the bosonic and superstring on the orbifold of the space-time by a boost, leading to the cosmological singularity. We show that the modular invariance leads to the spectrum where the twisted sector tachyon, together with all other twisted sector fields, present in the Euclidean version of the orbifold, is absent. This makes impossible to resolve the singularity by a marginal deformation of the worldsheet CFT. We also establish a relation between the resolution of rotational orbifolds in Euclidean and Lorentzian setups, and quantum groups. The analysis confirms the impossibility of resolving the cosmological orbifold singularity.
Abstract : We review the generalization of field theory to space-time with noncommuting coordinates, starting with the basics and covering most of the active directions of research. Such theories are now known to emerge from limits of M theory and string theory, and to describe quantum Hall states. In the last few years they have been studied intensively, and many qualitatively new phenomena have been discovered, both on the classical and quantum level. To appear in Reviews of Modern Physics.
Un Problème de topologie générale : le problème de Souslin
Ce manuscrit est un anti-résumé de la communication qui sera faite par l'auteur à la rencontre des 17 et 18 novembre 1967, du groupe d'étude du problème à 2 corps
The Present Status of CP, T and CPT Invariance
Introduction : Last time I came to Sweden, cars were drivent on the left side of the roads. This week most cars I saw were driven on the right side. I am very grateful to the Nobel Foundation for this opportunity to observe this P violation. I may have found the cause of this P violation by looking at buses or trucks with the same inscription on both sides ; they are not symmetrically written. I have also made a couter-experiment to verify this theory. In Tokyo I found an equal probability for cars to be driven on the left side or the right side of the streets and I did observe that identical texts were placed symmetrically on buses and trucks : the first character is near the front part on both sides so the same text must be read from left to right on one side and from right to left on the other side. Since I have never been yet in a car undergoing a spontaneous CP transformation, I will be able to give you today my report on CP, T and CPT invariance.
Applications of Group Theory to Quantum Physics : Algebraic Apects
Lectures given at the 1969 Battelle Summer Rencontres in Mathematics and Physics. Seattle - Washington - USA
Breaking of the SU(3) x SU(3) Symmetry in Hadronic Physics
Abstract : This is a preliminary version of chapter II and II of a paper on internal symmetries wich we are preparing. It contains a coordinate free formulation of SU(3) invariant algebras on the octet space. We calculate it now in the hope that it might be of help to the readers of the accompanying paper on SU(3)xSU(13) where the same formalism is used to discuss some physically interesting problems. We believe that the geometrical approach presented in these papers will help clarifying the relations and the properties of the fundamental interactions. It might be used for example to understand the geometrical basis of some recent attempts to compute Cabibbo's angle.
Properties of the breaking of hadronic internal symmetry
Abstract : The directions of breaking of the hadronic internal symmetry by the electromagnetic, semileptonic—and nonleptonic—weak and CP violating interactions are characterized by remarkable mathematical properties. These directions correspond to idempotents or nilpotents of an algebra and they are critical, i.e., every invariant function for the symmetry group, e.g., (SU(3) × SU(3)) × (1, P, C, PC) has an extremum on these directions.
Covariant symmetric non-associative algebras on group representations
Abstract : When the vector space ? carries a representation p of the group G, if the decomposition of the symmetrized tensor product representation contains p, there exists a symmetrical non-associative algebra on ? with G a group of automorphisms. We give explicitly for dimension 2 (two maximal algebras). We also give several examples of th use of these algebras in physics and in mathematics. This paper will appear in part two of Symmetry in Nature, a volume in honour of Luigi Radicati di Brozolo, Sculoa Normale Superiore, Pisa, 1989
Les Concepts fondamentaux de la cristallographie
Résumé : Nous introduisons un nouvel ensemble de définitions des concepts de la cristallographie en dimension n. Il est basé sur les propriétés générales des actions de groupe. Enfin nous précison comment on détermine le systèmes cristallographique et la classe de Bravais d'un groupe cristallographique. Abstract : We introduce a new set of definitions of the fundamental concepts of n-dimensional crystallography. It is based on general properties of group actions. We also show how to determine the crystallographic system and the Bravais class of a cristallographic group.
Symmetry and classification of energy bands in crystals
To appear in Proceedings XVth International Colloquium on Group Theretical Methods in Physics (Varna, June 1987) Abstract : An energy band in a solid contains an infinite number of states which transform linearly as a space group representation induced from a finite dimensional representation of the isotropy group of a point in space. A band representation is elementary if it cannot be decomposed as a direct sum of band representations; it describes a single band. We give a complete classification of the inequivalent elementary band representations.
Structure and classification of band representations
Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.
Abstract : An historical survey of our knowledge of crystal structure and of the mathematical problem of paving space with one or a few types of stones show that a crystal has not necessarily a translation lattice. The recently discovered crystals of alloys Al with 14% Mn have an ecosahedral symetry (incompatible qith translation lattice). Their electron micrographs show a structure similar to a Penrose tiling. It seems that long range order characterizing crystals can be obtained from projection of a thin slab of genuine crystal in higher dimension so that crystal physical properties can be described by quasi-periodic functions (as generalizing periodic ones) or may be almost periodic functions.
Symmetry and conservation laws in particle physics in the fifties
Introduction to spontaneous symmetry breaking : some examples
Covariant vector fields and renormalization groups
Invited lecture given at th XIIIth international Colloquium on group theoretical methods on Physics
Mixing angles, B-meson life-time in the six quark model based on S0(10) × U(1)PQ
Abstract : We show that a recently proposed multigenerational, grand unified model with three parameters (top-quark mass and two phases of Higgs vacuum expectation values) yields results in good agreement with the most recent phenomenological limits placed on the weak mixing angles of the quarks. These limits take into account the experimental value for B-meson life-time.
Charge conjugation : a Contribution to the history of this internal quantum number of particle physics
Conference given at the First International Meeting on the History of Scientific Ideas Symmetrie in Phyiscs, Sant Feliu de Guixols, Catalonia, Spain, September 20-26, 1983
Une Nouvelle et abondante source d'énergie ?
Article demandé pour le volume jubilaire du cinquantenaire de la découverte de la radioactivité articficielle
Renormalization group fixed points of general n-vector models
Abstract : We make a general study of symmetry and stability of the fixed points of the quartic Hamiltonian of an n-component field (or order parameter} for n&4. Simple proofs of known results are given. Among new results, we shou that when it exists the stable fixed point is unique; we give some precision on its symmetry and on its attractor basin.
Zeros of covariant vector fields for the point groups : Invariant formulation
Abstract. 2014 All finite as well as infinite (matrix) point subgroups of full orthogonal groups in two and three dimensions are considered. For each point group a polynomial integrity basis for invariants and the basic polynomial vector fields are first given. Then, the strata are defined via equations and inequalities involving the integrity basis. Finally, equations for zeros of a covariant vector field are given on each stratum in terms of the integrity basis, which appears via coefficients in the expansion of the vector field on the vector-field basis. All the results are tabulated and an illustration using the cubic group is presented. Mathematical background sufficient for extensions of the results is also given. Résumé. - Pour tous les groupes ponctuels (sous-groupes fermés) finis ou continus des groupes orthogonaux 0(2) et 0(3) nous donnons une base d’intégrité e?(x) pour les polynômes invariants et pour les champs de vecteurs polynomiaux, nous donnons les équations et inégalités définissant les strates (union des orbites de même type). Finalement, nous écrivons des équations donnant les zéros d’un champ de vecteurs covariants sur une strate donnée; ces équations sont linéaires dans les composantes (invariantes) du champ de vecteurs sur les e?(x), les coefficients de ces termes étant eux-mêmes des invariants. Tous ces résultats sont résumés dans des tables et illustrés par un exemple d’application physique de symétrie cubique. La méthode mathématique pour obtenir ces résultats est expliquée de façon à permettre au lecteur de l’appliquer à d’autres groupes.
Landau theory of second order phase transitions and invariant theory
Abstract : We give some basic concepts on group actions and theorems in invariant theory useful for the contrcution and the study of Landau polynomials. We apply them to the study of the isotropy groups of extrema of Landau polynomials. We explain some technique for finding new extrema from old ones and we state some recent results concerning Landau theory of second order phase transitions.