DOCSLIB.ORG

Indecomposability in Field Theory and Applications to Disordered Systems and Geometrical Problems Vasseur Romain

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Indecomposability in Field Theory and Applications to Disordered Systems and Geometrical Problems Vasseur Romain Indecomposability in field theory and applications to disordered systems and geometrical problems Vasseur Romain To cite this version: Vasseur Romain. Indecomposability in field theory and applications to disordered systems andge- ometrical problems. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2013. English. tel-00876155 HAL Id: tel-00876155 https://tel.archives-ouvertes.fr/tel-00876155 Submitted on 23 Oct 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ PIERRE ET MARIE CURIE - PARIS VI et INSTITUT DE PHYSIQUE THEORIQUE´ - CEA/SACLAY Ecole´ Doctorale de Physique de la R´egion Parisienne - ED 107 Th`ese de doctorat Sp´ecialit´e: Physique Th´eorique Indecomposability in field theory and applications to disordered systems and geometrical problems pr´esent´ee par Romain Vasseur pour obtenir le grade de Docteur de l’Universit´ePierre et Marie Curie Th`ese pr´epar´ee sous la direction de Hubert Saleur et de Jesper Jacobsen Soutenue le 27 Septembre 2013 devant le jury compos´ede : Denis Bernard Examinateur Ecole´ Normale Sup´erieure John Cardy Rapporteur Oxford University Matthias Gaberdiel Rapporteur ETH Z¨urich Jesper Jacobsen Membre invit´e(co-directeur) Ecole´ Normale Sup´erieure Nicholas Read Examinateur Yale University Hubert Saleur Directeur de th`ese CEA Saclay Jean-Bernard Zuber Pr´esident du jury Universit´ePierre et Marie Curie ii Indecomposabilit´edans les th´eories des champs et applications aux syst`emes d´esordonn´es et aux probl`emes g´eom´etriques R´esum´e: Les th´eories des champs conformes logarithmiques (LCFTs) sont cruciales pour d´ecrire le comportement critique de syst`emes physiques vari´es : les transitions de phase dans les syst`emes ´electroniques d´esordonn´es sans interaction (comme par exemple la transition entre plateaux dans l’effet Hall quantique entier), les points critiques d´esordonn´es dans les syst`emes statistiques classiques (comme le mod`ele d’Ising avec liens al´eatoires), ou encore les mod`eles g´eom´etriques critiques (comme la percolation ou les marches al´eatoires auto-´evitantes). Les LCFTs d´ecrivent des th´eories non unitaires, qui ne seraient probablement pas pertinentes dans le contexte de la physique des parti- cules, mais qui apparaissent naturellement en mati`ere condens´ee et en physique statis- tique. Sans cette condition d’unitarit´e, toute la puissance alg´ebrique qui a fait le succ`es des th´eories conformes est fortement compromise `acause de “l’ind´ecomposabilit´e” de la th´eorie des repr´esentations sous-jacente. Ceci a pour cons´equence de modifier les fonc- tions de corr´elation alg´ebriques par des corrections logarithmiques, et r´eduit s´ev`erement l’espoir d’une classification g´en´erale. Le but de cette th`ese est d’analyser ces th´eories logarithmiques en ´etudiant leur r´egularisation sur r´eseau, l’id´ee principale ´etant que la plupart des difficult´es alg´ebriques caus´ees par l’ind´ecomposabilit´esont d´ej`apr´esentes dans des syst`emes de taille finie. Notre approche consiste `aconsid´erer des mod`eles statistiques critiques avec matrice de transfert non diagonalisable (ou des chaˆınes de spins critiques avec Hamiltonien non diagonalisable) et d’analyser leur limite thermodynamique `al’aide de diff´erentes m´ethodes num´eriques, alg´ebriques et analytiques. On explique en particulier comment mesurer num´eriquement les param`etres universels qui caract´erisent les repr´esentations ind´ecomposables qui apparaissent `ala limite continue. L’analyse d´etaill´ee d’une vaste classe de mod`eles sur r´eseau nous permet ´egalement de conjecturer une classification de toutes les LCFTs chirales pertinentes physiquement, pour lesquelles la seule sym´etrie est donn´ee par l’alg`ebre de Virasoro. Cette approche est aussi partiellement ´etendue aux th´eories non chirales, avec une attention particuli`ere port´ee au probl`eme bien connu de la formulation d’une th´eorie des champs coh´erente qui d´ecrirait la percolation en deux dimensions. On montre que les mod`eles sur r´eseaux p´eriodiques ou avec bords peuvent ˆetre reli´es alg´ebriquement seulement dans le cas des mod`eles minimaux, impliquant des cons´equences int´eressantes pour les th´eories des champs sous-jacentes. Un certain nombre d’applications aux syst`emes d´esordonn´es et aux mod`eles g´eom´etriques sont ´egalement abord´ees, avec en particulier une discussion d´etaill´ee des observables avec comportement logarithmique au point critique dans le mod`ele de Potts en dimension arbitraire. Mots clefs : Th´eories des Champs Conformes Logarithmiques, Repr´esentations inde- composables, Mod`eles sur r´eseau, Syst`emes d´esordonn´es, Probl`emes g´eom´etriques. iii Indecomposability in field theory and applications to disordered systems and geometrical problems Abstract: Logarithmic Conformal Field Theories (LCFTs) are crucial for describing the crit- ical behavior of a variety of physical systems. These include phase transitions in dis- ordered non-interacting electronic systems (such as the transition between plateaus in the integer quantum Hall effect), disordered critical points in classical statistical mod- els (such as the random bond Ising model), or critical geometrical models (such as polymers and percolation). LCFTs appear when one has to give up the unitarity con- dition, which is natural in particle physics applications, but not in statistical mechanics and condensed matter physics. Without unitarity, the powerful algebraic approach of conformal invariance encounters formidable technical difficulties due to ‘indecompos- ability’. This in turn yields logarithmic corrections to the power-law correlations at the critical point, and prevents the use of general classification techniques that have proven so powerful in the unitary case. The goal of this thesis is to understand LCFTs by studying their lattice regular- izations, the crucial point being that most algebraic complications due to indecompos- ability occur in finite size systems as well. Our approach is thus to consider critical statistical models with non-diagonalizable transfer matrices, or gapless quantum spin chains with non-diagonalizable hamiltonians, and to study their scaling limit by uti- lizing a variety of algebraic, numerical and integrable techniques. We show how to measure numerically universal parameters that characterize the indecomposable rep- resentations of the Virasoro algebra which emerge in the thermodynamic limit. An extensive understanding of a wide class of lattice models allows us to conjecture a ten- tative classification of all possible (chiral) LCFTs with Virasoro symmetry only. This approach is partially extended to the bulk case, for which we discuss how the long- standing bulk CFT formulation of percolation can be tackled along these lines. We also argue that boundary and periodic lattice models can be related algebraically only in the case of minimal models, and we work out the consequences for the underlying boundary and bulk field theories. Several concrete applications to disordered systems and geometrical problems are discussed, and we uncover a large class of geometrical observables in the Potts model that behave logarithmically at the critical point. Key words: Logarithmic Conformal Field Theory, Indecomposable representations, Temperley-Lieb algebra, Lattice models, Disordered systems, Geometrical problems. iv Remerciements Je voudrais commencer par remercier John Cardy et Matthias Gaberdiel, qui ont accept´ela p´enible tache de rapporteur, ainsi que les examinateurs, Denis Bernard, Nicholas Read et Jean-Bernard Zuber. Merci pour votre aide et vos conseils pour am´eliorer mon manuscrit. Je suis tr`es conscient du travail qu’ˆetre membre d’un comit´e de th`ese repr´esente, et je vous suis extrˆemement reconnaissant d’avoir accept´ede braver des heures de transport pour venir assister `ama soutenance. Je remercie aussi Hubert et Jesper, qui en plus de leur qualit´es scientifiques indis- cutables, m’ont beaucoup appris et aid´edans bien des domaines, en particulier dans mes moments de doute, ou pendant la p´eriode difficile des recherches de postdocs. Merci ´egalement de m’avoir soutenu durant mon changement de th´ematique apr`es ma premi`ere moiti´ede th`ese, mˆeme si cela repr´esentait un certain nombre de risques et de difficult´es pour vous aussi. Je sais que j’ai pu rˆaler `acertains moments, mais je suis vraiment ravi d’avoir fait ma th`ese avec vous deux, vous avez toujours ´et´edisponibles et `al’´ecoute en cas de probl`eme : travailler avec vous durant ces trois ann´ees a toujours ´et´eun plaisir. Je tiens ´egalement `aremercier tout particuli`erement Azat Gainutdinov qui a ´et´epostdoc `aSaclay durant toute ma th`ese : j’ai beaucoup appr´eci´etravailler avec toi sur tous ces projets, ¸ca a ´et´etr`es stimulant et b´en´efique pour moi. J’ai eu la chance d’avoir des bureaux dans deux laboratoires, au LPTENS et `a l’IPhT au CEA, ce qui me semble ˆetre un luxe `al’heure ou l’espace disponible pour les th´esards est chaque ann´ee plus restreint. Merci `atous les gens qui font la vie de ces
Load more

Recommended publications
  • Arxiv:1910.05796V2 [Math-Ph] 26 Oct 2020 5
    Towards a conformal field theory for Schramm-Loewner evolutions Eveliina Peltola [email protected] Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany We discuss the partition function point of view for chordal Schramm-Loewner evolutions and their relation- ship with correlation functions in conformal field theory. Both are closely related to crossing probabilities and interfaces in critical models in two-dimensional statistical mechanics. We gather and supplement previous results with different perspectives, point out remaining difficulties, and suggest directions for future studies. 1. Introduction 2 2. Schramm-Loewner evolution in statistical mechanics and conformal field theory4 A. Schramm-Loewner evolution (SLE) 5 B. SLE in critical models – the Ising model6 C. Conformal field theory (CFT) 7 D. Martingale observables for interfaces 9 3. Multiple SLE partition functions and applications 12 A. Multiple SLEs and their partition functions 12 B. Definition of the multiple SLE partition functions 13 C. Properties of the multiple SLE partition functions 16 D. Relation to Schramm-Loewner evolutions and critical models 18 4. Fusion and operator product expansion 20 A. Fusion and operator product expansion in conformal field theory 21 B. Fusion: analytic approach 22 C. Fusion: systematic algebraic approach 23 D. Operator product expansion for multiple SLE partition functions 26 arXiv:1910.05796v2 [math-ph] 26 Oct 2020 5. From OPE structure to products of random distributions? 27 A. Conformal bootstrap 27 B. Random tempered distributions 28 C. Bootstrap construction? 29 Appendix A. Representation theory of the Virasoro algebra 31 Appendix B. Probabilistic construction of the pure partition functions 32 Appendix C.
    [Show full text]
  • Notes on Statistical Field Theory
    Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual)..........................
    [Show full text]
  • Arxiv:0807.3472V1 [Cond-Mat.Stat-Mech] 22 Jul 2008 Conformal Field Theory and Statistical Mechanics∗
    Conformal Field Theory and Statistical Mechanics∗ John Cardy July 2008 arXiv:0807.3472v1 [cond-mat.stat-mech] 22 Jul 2008 ∗Lectures given at the Summer School on Exact methods in low-dimensional statistical physics and quantum computing, les Houches, July 2008. 1 CFT and Statistical Mechanics 2 Contents 1 Introduction 3 2 Scale invariance and conformal invariance in critical behaviour 3 2.1 Scaleinvariance ................................. 3 2.2 Conformalmappingsingeneral . .. 6 3 The role of the stress tensor 7 3.0.1 An example - free (gaussian) scalar field . ... 8 3.1 ConformalWardidentity. 9 3.2 An application - entanglement entropy . ..... 12 4 Radial quantisation and the Virasoro algebra 14 4.1 Radialquantisation.............................. 14 4.2 TheVirasoroalgebra .............................. 15 4.3 NullstatesandtheKacformula . .. 16 4.4 Fusionrules ................................... 17 5 CFT on the cylinder and torus 18 5.1 CFTonthecylinder .............................. 18 5.2 Modularinvarianceonthetorus . ... 20 5.2.1 Modular invariance for the minimal models . .... 22 6 Height models, loop models and Coulomb gas methods 24 6.1 Heightmodelsandloopmodels . 24 6.2 Coulombgasmethods ............................. 27 6.3 Identification with minimal models . .... 28 7 Boundary conformal field theory 29 7.1 Conformal boundary conditions and Ward identities . ......... 29 7.2 CFT on the annulus and classification of boundary states . ......... 30 7.2.1 Example................................. 34 CFT and Statistical Mechanics 3 7.3
    [Show full text]
  • Statistical Field Theory University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] –1– Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very di↵erent perspectives. They have titles like “Critical Phenomena”, “Phase Transitions”, “Renormalisation Group” or, less helpfully, “Advanced Statistical Mechanics”. Here are some that I particularly like Nigel Goldenfeld, Phase Transitions and the Renormalization Group • Agreatbook,coveringthebasicmaterialthatwe’llneedanddelvingdeeperinplaces. Mehran Kardar, Statistical Physics of Fields • The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. John Cardy, Scaling and Renormalisation in Statistical Physics • Abeautifullittlebookfromoneofthemastersofconformalfieldtheory.Itcoversthe material from a slightly di↵erent perspective than these lectures, with more focus on renormalisation in real space. Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter • Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content. Shankar covers modern methods of QFT, with an easygoing style suitable for bedtime reading. Anumberofexcellentlecturenotesareavailableontheweb.Linkscanbefoundon the course webpage: http://www.damtp.cam.ac.uk/user/tong/sft.html.
    [Show full text]
  • Euclidean Field Theory
    February 2, 2011 Euclidean Field Theory Kasper Peeters & Marija Zamaklar Lecture notes for the M.Sc. in Elementary Particle Theory at Durham University. Copyright c 2009-2011 Kasper Peeters & Marija Zamaklar Department of Mathematical Sciences University of Durham South Road Durham DH1 3LE United Kingdom http://maths.dur.ac.uk/users/kasper.peeters/eft.html [email protected] [email protected] 1 Introducing Euclidean Field Theory5 1.1 Executive summary and recommended literature............5 1.2 Path integrals and partition sums.....................7 1.3 Finite temperature quantum field theory.................8 1.4 Phase transitions and critical phenomena................ 10 2 Discrete models 15 2.1 Ising models................................. 15 2.1.1 One-dimensional Ising model................... 15 2.1.2 Two-dimensional Ising model................... 18 2.1.3 Kramers-Wannier duality and dual lattices........... 19 2.2 Kosterlitz-Thouless model......................... 21 3 Effective descriptions 25 3.1 Characterising phase transitions..................... 25 3.2 Mean field theory.............................. 27 3.3 Landau-Ginzburg.............................. 28 4 Universality and renormalisation 31 4.1 Kadanoff block scaling........................... 31 4.2 Renormalisation group flow........................ 33 4.2.1 Scaling near the critical point................... 33 4.2.2 Critical surface and universality................. 34 4.3 Momentum space flow and quantum field theory........... 35 5 Lattice gauge theory 37 5.1 Lattice action................................. 37 5.2 Wilson loops................................. 38 5.3 Strong coupling expansion and confinement.............. 39 5.4 Monopole condensation.......................... 39 3 4 1 Introducing Euclidean Field Theory 1.1. Executive summary and recommended literature This course is all about the close relation between two subjects which at first sight have very little to do with each other: quantum field theory and the theory of criti- cal statistical phenomena.
    [Show full text]
  • String Theory and Related Subjects
    String theory and related subjects Marios Petropoulos Centre de Physique Théorique, Ecole Polytechnique ICFP Theoretical Physics What is string theory? • Worldwide scientific activity • Branch of mathematical/theoretical physics • high-energy particle physics • gravity/cosmology • statistical mechanics/condensed matter • geormetry and more mathematical topics A few words on particle physics ... • Since 1931 (Berkeley) particles are produced in accelerators and colliders • at high energy: relativistic c • at microscopic scales: quantum ħ • Particles and interactions studied with relativistic quantum mechanics: Quantum Field Theory • Culminated in the 70’s with a gauge theory: the Standard Model Note: QFT is a tool used in statistical mechanics and condensed matter ... and gravity • Described since 1916 by a classical field theory named General Relativity • Verified with high precision in the solar system and beyond: mercury perihelion precession, Shapiro effect, pulsars, GPS • Successfully applied to cosmology: Big-Bang and further evolution • Hard to reconcile with ħ: non-renormalizable QFT String theory historically • Prehistory : dual models (1967-1974) • History : theory of gravity (1974-1984) • Modern ages : theory of everything (1984-2000) • Maturity : toolbox like and beyond QFT (2000-) String theory ideally • The point-like object is abandoned • The techniques include QFT • The output is the possibility for various particles and interactions to emerge ... • ... in particular gauge theories & quantum gravity defined in 10 space-time dimensions String theory practically • Quantum gravity • supergravity • various aspects of black holes such as black-hole thermodynamics ... • Grand-unified theory • analyse compactifications: geometry • comply with SM: phenomenology, supersymmetry String theory practically • Cosmology • initial singularity • cosmological evolution • More generally • develop techniques beyond QFT, dualities ... • understand (and prove?) holography: integrability • apply holographic correspondence to QCD, condensed-matter systems, superfluids ..
    [Show full text]
  • Statistical Field Theory University of Cambridge Part III Mathematical Tripos
    Preprint typeset in JHEP style - HYPER VERSION Michaelmas Term, 2017 Statistical Field Theory University of Cambridge Part III Mathematical Tripos David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/sft.html [email protected] { 1 { Recommended Books and Resources There are a large number of books which cover the material in these lectures, although often from very different perspectives. They have titles like \Critical Phenomena", \Phase Transitions", \Renormalisation Group" or, less helpfully, \Advanced Statistical Mechanics". Here are some that I particularly like • Nigel Goldenfeld, Phase Transitions and the Renormalization Group A great book, covering the basic material that we'll need and delving deeper in places. • Mehran Kardar, Statistical Physics of Fields The second of two volumes on statistical mechanics. It cuts a concise path through the subject, at the expense of being a little telegraphic in places. It is based on lecture notes which you can find on the web; a link is given on the course website. • John Cardy, Scaling and Renormalisation in Statistical Physics A beautiful little book from one of the masters of conformal field theory. It covers the material from a slightly different perspective than these lectures, with more focus on renormalisation in real space. • Chaikin and Lubensky, Principles of Condensed Matter Physics • Shankar, Quantum Field Theory and Condensed Matter Both of these are more all-round condensed matter books, but with substantial sections on critical phenomena and the renormalisation group. Chaikin and Lubensky is more traditional, and packed full of content.
    [Show full text]
  • A Statistical Field Theory for Classical Particles, Foundations and Applica- Tions in Cosmological Structure Formation, © January 2015 ABSTRACT
    Dissertation submitted to the Combined Faculties for the Natural Sciences and for Mathematics of the Ruperta-Carola University of Heidelberg for the degree of Doctor of Natural Sciences Put forward by Dipl.-Phys. Felix Fabis born in Heidelberg Oral Examination: 15.04.2015 ASTATISTICALFIELDTHEORYFORCLASSICALPARTICLES foundations and applications in cosmological structure formation Refereed by: Prof. Dr. Matthias Bartelmann Prof. Dr. Manfred Salmhofer January 2015 – Final Version Felix Fabis: A Statistical Field Theory for Classical Particles, Foundations and Applica- tions in Cosmological Structure Formation, © January 2015 ABSTRACT We develop a field-theoretical version of kinetic theory based on the pioneering work in [39, 40, 22]. A canonical generating functional for correlators of macroscopic ob- servables is derived using the language of path integrals with the microscopic phase- space coordinates as the fundamental variables determining the dynamics. The find- ings of the original works are generalised to systems with correlated initial conditions. An exact initial phase-space distribution for a homogeneous and isotropic Gaussian random field is calculated, translated into a diagram language and factorised into connected contributions. The grand canonical generating functional is derived for systems obeying statistical homogeneity and isotropy. Using this, a perturbation the- ory in terms of coupled integral equations is developed and shown to suffer from inconsistencies due to the presence of initial correlations. We discover hints at a pos- sible cure by using the first order solution for two-point cumulants to reorganise one- loop diagrams. Applied to cosmological structure formation the first order solution for the density power spectrum is found to be the familiar linear growth solution.
    [Show full text]
  • String Theory, Superstring Theory and Beyond
    Cambridge University Press 0521672287 - String Theory: Superstring Theory and Beyond, Volume II Joseph Polchinski Frontmatter More information String Theory, Superstring Theory and Beyond The two volumes that comprise String Theory provide an up-to-date, comprehensive, and pedagogic introduction to string theory. Volume I, An Introduction to the Bosonic String, provides a thorough introduction to the bosonic string, based on the Polyakov path integral and conformal field theory. The first four chapters introduce the central ideas of string theory, the tools of conformal field theory and of the Polyakov path integral, and the covariant quantization of the string. The next three chapters treat string interactions: the general formalism, and detailed treatments of the tree-level and one loop amplitudes. Chapter eight covers toroidal compactification and many important aspects of string physics, such as T-duality and D-branes. Chapter nine treats higher-order amplitudes, including an analysis of the finiteness and unitarity, and various nonperturbative ideas. An appendix giving a short course on path integral methods is also included. Volume II, Superstring Theory and Beyond, begins with an introduction to supersym- metric string theories and goes on to a broad presentation of the important advances of recent years. The first three chapters introduce the type I, type II, and heterotic superstring theories and their interactions. The next two chapters present important recent discoveries about strongly coupled strings, beginning with a detailed treatment of D-branes and their dynamics, and covering string duality, M-theory, and black hole entropy. A following chapter collects many classic results in conformal field theory. The final four chapters are concerned with four-dimensional string theories, and have two goals: to show how some of the simplest string models connect with previous ideas for unifying the Standard Model; and to collect many important and beautiful general results on world-sheet and spacetime symmetries.
    [Show full text]
  • The Connection Between Statistical Mechanics and Quantum Field Theory
    ITP-SB-94-07 HEPTH 9403084 The Connection Between Statistical Mechanics and Quantum Field Theory Barry M. McCoy 1 Institute for Theoretical Physics State University of New York Stony Brook, NY 11794-3840 Abstract A four part series of lectures on the connection of statistical mechanics and quantum field theory. The general principles relating statistical mechanics and the path integral formulation of quantum field theory are presented in the first lecture. These principles are arXiv:hep-th/9403084v2 18 Mar 1994 then illustrated in lecture 2 by a presentation of the theory of the Ising model for H = 0, where both the homogeneous and randomly inhomogeneous models are treated and the scaling theory and the relation with Fredholm determinants and Painlev´eequations is presented. In lecture 3 we consider the Ising model with H 6= 0, where the relation with gauge theory is used to discuss the phenomenon of confinement. We conclude in the last lecture with a discussion of quantum spin diffusion in one dimensional chains and a presentation of the chiral Potts model which illustrates the physical effects that can occur when the Euclidean and Minkowski regions are not connected by an analytic continuation. 3/94 1 [email protected] Part I. Introduction Abstract The connection between the Euclidean path integral formulation of quantum field theory and classical statistical mechanics is surveyed in terms of the theory of critical phenomena and the concept of renormalization. Quantum statistical mechanics is surveyed with an emphasis on diffusive phenomena. The particle interpretation of quantum field theory is discussed in the context of nonperturbative statistical mechanics.
    [Show full text]
  • Curriculum Vitae of German Sierra (Dated: December 2014)
    Curriculum Vitae of German Sierra (Dated: December 2014) Personal Data Name : Germ´an Surname: Sierra Rodero Nationality: Spanish Date of Birth: 13 May 1955 Professional data Organism : Consejo Superior de Investigaciones Cientificas (Spain) Centre: Instituto de Fisica Teorica CSIC-UAM Address: C/ Nicolas Cabrera 13-15, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain. Phone: : (34) 91 299 9875 Email: [email protected] Professional Category: Full professor. Academic Career Graduate in Physics, Univ. Complutense de Madrid, Spain (1978). PhD in Physics, Univ. Complutense de Madrid, Spain (1981). Postdoc at l'Ecole Normale Superieure, Paris, France (1981 -1983). Titular professor at Univ. Complutense de Madrid, Spain (1984 - 1987). Fellow at CERN, Geneva, Switzerland (1987 - 1989). Scientific researcher, CSIC, Spain (1989 - 2005). Full professor, CSIC, Spain (2005 - today). Publications From Web of Science: 136 articles, 3.871 citations and h-index: 31. Articles [1] "Causality of a wave equation and invariance of its hyperbolicity conditions", A. F. Ra~naday G. Sierra, Phys. Rev. D22 (1980) 385. [2] "A theory of the Rarita-Schwinger field without superluminality", A. F.Ra~nada,G. Sierra, Phys. Rev.D22 (1980) 2416. [3] "Classical and Quantum fields with secondary constraints", G. Sierra, Phys.Rev.D26 (1982) 2730. [4] "Hyperkahler supersymmetric σ-model in six dimensions", G. Sierra, P. K. Townsend, Phys.Lett. 124B (1983) 497. Preprint LPTENS 83/7. [5] "Supersymmetry in six dimensions", P. S. Howe, G. Sierra, P. K. Townsend, Nucl.Phys. B221 (1983) 331. [6] "Chiral anomalies and constraints on the gauge group in higher dimensional supersymmetric Yang-Mills theories", P. K. Townsend, G.
    [Show full text]
  • Mean Field Theory for Sigmoid Belief Networks
    Journal of Arti cial Intelligence Research 4 1996 61|76 Submitted 11/95; published 3/96 Mean Field Theory for Sigmoid Belief Networks Lawrence K. Saul [email protected] Tommi Jaakkola [email protected] Michael I. Jordan [email protected] Center for Biological and Computational Learning Massachusetts Institute of Technology 79 Amherst Street, E10-243 Cambridge, MA 02139 Abstract We develop a mean eld theory for sigmoid b elief networks based on ideas from statistical mechanics. Our mean eld theory provides a tractable approximation to the true probability dis- tribution in these networks; it also yields a lower b ound on the likeliho o d of evidence. We demon- strate the utility of this framework on a b enchmark problem in statistical pattern recognition|the classi cation of handwritten digits. 1. Intro duction Bayesian b elief networks Pearl, 1988; Lauritzen & Spiegelhalter, 1988 provide a rich graphical representation of probabilistic mo dels. The no des in these networks represent random variables, while the links represent causal in uences. These asso ciations endow directed acyclic graphs DAGs with a precise probabilistic semantics. The ease of interpretation a orded by this semantics explains the growing app eal of b elief networks, now widely used as mo dels of planning, reasoning, and uncertainty. Inference and learning in b elief networks are p ossible insofar as one can eciently compute or approximate the likeliho o d of observed patterns of evidence Buntine, 1994; Russell, Binder, Koller, & Kanazawa, 1995. There exist provably ecient algorithms for computing likeliho o ds in b elief networks with tree or chain-like architectures.
    [Show full text]