Titre :	Introduction to statistical field theory
Type de document :	texte imprimé
Auteurs :	Brézin, Édouard
Editeur :	Cambridge : Cambridge university press
Année de publication :	2010
Importance :	1 vol (166 p.)
Format :	24 cm
ISBN/ISSN/EAN :	978-0-521-19303-0
Note générale :	978-0-521-19303-0
Langues :	Anglais (eng)
Catégories :	Mathématique
Mots-clés :	Théorie quantique des champs
Index. décimale :	519 Mathématiques appliquées, probabilités (statistiques mathématiques)
Résumé :	La connaissance du groupe de renormalisation et de la théorie des champs est un élément clé de la physique et est essentielle en physique de la matière condensée et des particules. Écrit pour les étudiants de premier cycle et débutants, ce manuel fournit une introduction concise à ce sujet. Le manuel traite directement de l’expansion en boucle de l’énergie libre, également appelée méthode du champ d’arrière-plan. C'est une méthode puissante, en particulier lorsqu'il s'agit de symétries et de mécanismes statistiques. En se concentrant sur l'énergie libre, l'auteur évite de longs développements sur les techniques de la théorie des champs. La nécessité de renormaliser suit ensuite
Note de contenu :	Sommaire A few well-known basic results 1 1.1 The Boltzmann law 1 1.1.1 The classical canonical ensemble 1 1.1.2 The quantum canonical ensemble 2 1.1.3 The grand canonical ensemble 3 1.2 Thermodynamics from statistical physics 3 1.2.1 The thermodynamic limit 3 1.3 Gaussian integrals and Wick’s theorem 4 1.4 Functional derivatives 6 1.5 d-dimensional integrals 6 Additional references 8 2 Introduction: order parameters, broken symmetries 9 2.1 Can statistical mechanics be used to describe phase transitions? 9 2.2 The order–disorder competition 10 2.3 Order parameter, symmetry and broken symmetry 12 2.4 More general symmetries 16 2.5 Characterization of a phase transition through correlations 18 2.6 Phase coexistence, critical points, critical exponents 19 3 Examples of physical situations modelled by the Ising model 22 3.1 Heisenberg’s exchange forces 22 3.2 Heisenberg and Ising Hamiltonians 24 3.3 Lattice gas 26 3.4 More examples 28 3.5 A first connection with field theory 29 v vi Contents 4 A few results for the Ising model 32 4.1 One-dimensional Ising model: transfer matrix 32 4.2 One-dimensional Ising model: correlation functions 35 4.3 Absence of phase transition in one dimension 37 4.4 A glance at the two-dimensional Ising model 38 4.5 Proof of broken symmetry in two dimensions (and more) 38 4.6 Correlation inequalities 42 4.7 Lower critical dimension: heuristic approach 44 4.8 Digression: Feynman path integrals, the transfer matrix and the Schrödinger equation 47 5 High-temperature and low-temperature expansions 52 5.1 High-temperature expansion for the Ising model 52 5.1.1 Continuous symmetry 55 5.2 Low-temperature expansion 56 5.2.1 Kramers–Wannier duality 57 5.3 Low-temperature expansion for a continuous symmetry group 58 6 Some geometric problems related to phase transitions 60 6.1 Polymers and self-avoiding walks 60 6.2 Potts model and percolation 64 7 Phenomenological description of critical behaviour 68 7.1 Landau theory 68 7.2 Landau theory near the critical point: homogeneous case 71 7.3 Landau theory and spatial correlations 75 7.4 Transitions without symmetry breaking: the liquid–gas transition 78 7.5 Thermodynamic meaning of {m} 79 7.6 Universality 80 7.7 Scaling laws 82 8 Mean field theory 85 8.1 Weiss ‘molecular field’ 85 8.2 Mean field theory: the variational method 87 8.3 A simpler alternative approach 92 9 Beyond the mean field theory 95 9.1 The first correction to the mean-field free energy 95 9.2 Physical consequences 97 10 Introduction to the renormalization group 100 10.1 Renormalized theories and critical points 101 10.2 Kadanoff block spins 101 10.3 Examples of real space renormalization groups: ‘decimation’ 103 10.4 Structure of the renormalization group equations 109 Contents vii 11 Renormalization group for the Ï•4 theory 113 11.1 Renormalization group . . . without renormalization 114 11.2 Study of the renormalization group flow in dimension four 116 11.3 Critical behaviour of the susceptibility in dimension four 118 11.4 Multi-component order parameters 120 11.5 Epsilon expansion 122 11.6 An exercise on the renormalization group: the cubic fixed point 125 12 Renormalized theory 128 12.1 The meaning of renormalizability 128 12.2 Renormalization of the massless theory 132 12.3 The renormalized critical free energy (at one-loop order) 134 12.4 Away from Tc 136 13 Goldstone modes 138 13.1 Broken symmetries and massless modes 138 13.2 Linear and non-linear O(n) sigma models 142 13.3 Regularization and renormalization of the O(n) non-linear sigma model in two dimensions 144 13.3.1 Regularization 144 13.3.2 Perturbation expansion and renormalization 148 13.4 Renormalization group equations for the O(n) non-linear sigma model and the (d − 2) expansion 150 13.4.1 Integration of RG equations and scaling 151 13.5 Extensions to other non-linear sigma models 153 14 Large n 156 14.1 The linear O(n) model 156 14.2 O(n) sigma model
Côte titre :	Fs/13517-13518

Introduction to statistical field theory [texte imprimé] / Brézin, Édouard . - Cambridge : Cambridge university press, 2010 . - 1 vol (166 p.) ; 24 cm.
ISBN : 978-0-521-19303-0
978-0-521-19303-0
Langues : Anglais (eng)

Catégories :	Mathématique
Mots-clés :	Théorie quantique des champs
Index. décimale :	519 Mathématiques appliquées, probabilités (statistiques mathématiques)
Résumé :	La connaissance du groupe de renormalisation et de la théorie des champs est un élément clé de la physique et est essentielle en physique de la matière condensée et des particules. Écrit pour les étudiants de premier cycle et débutants, ce manuel fournit une introduction concise à ce sujet. Le manuel traite directement de l’expansion en boucle de l’énergie libre, également appelée méthode du champ d’arrière-plan. C'est une méthode puissante, en particulier lorsqu'il s'agit de symétries et de mécanismes statistiques. En se concentrant sur l'énergie libre, l'auteur évite de longs développements sur les techniques de la théorie des champs. La nécessité de renormaliser suit ensuite
Note de contenu :	Sommaire A few well-known basic results 1 1.1 The Boltzmann law 1 1.1.1 The classical canonical ensemble 1 1.1.2 The quantum canonical ensemble 2 1.1.3 The grand canonical ensemble 3 1.2 Thermodynamics from statistical physics 3 1.2.1 The thermodynamic limit 3 1.3 Gaussian integrals and Wick’s theorem 4 1.4 Functional derivatives 6 1.5 d-dimensional integrals 6 Additional references 8 2 Introduction: order parameters, broken symmetries 9 2.1 Can statistical mechanics be used to describe phase transitions? 9 2.2 The order–disorder competition 10 2.3 Order parameter, symmetry and broken symmetry 12 2.4 More general symmetries 16 2.5 Characterization of a phase transition through correlations 18 2.6 Phase coexistence, critical points, critical exponents 19 3 Examples of physical situations modelled by the Ising model 22 3.1 Heisenberg’s exchange forces 22 3.2 Heisenberg and Ising Hamiltonians 24 3.3 Lattice gas 26 3.4 More examples 28 3.5 A first connection with field theory 29 v vi Contents 4 A few results for the Ising model 32 4.1 One-dimensional Ising model: transfer matrix 32 4.2 One-dimensional Ising model: correlation functions 35 4.3 Absence of phase transition in one dimension 37 4.4 A glance at the two-dimensional Ising model 38 4.5 Proof of broken symmetry in two dimensions (and more) 38 4.6 Correlation inequalities 42 4.7 Lower critical dimension: heuristic approach 44 4.8 Digression: Feynman path integrals, the transfer matrix and the Schrödinger equation 47 5 High-temperature and low-temperature expansions 52 5.1 High-temperature expansion for the Ising model 52 5.1.1 Continuous symmetry 55 5.2 Low-temperature expansion 56 5.2.1 Kramers–Wannier duality 57 5.3 Low-temperature expansion for a continuous symmetry group 58 6 Some geometric problems related to phase transitions 60 6.1 Polymers and self-avoiding walks 60 6.2 Potts model and percolation 64 7 Phenomenological description of critical behaviour 68 7.1 Landau theory 68 7.2 Landau theory near the critical point: homogeneous case 71 7.3 Landau theory and spatial correlations 75 7.4 Transitions without symmetry breaking: the liquid–gas transition 78 7.5 Thermodynamic meaning of {m} 79 7.6 Universality 80 7.7 Scaling laws 82 8 Mean field theory 85 8.1 Weiss ‘molecular field’ 85 8.2 Mean field theory: the variational method 87 8.3 A simpler alternative approach 92 9 Beyond the mean field theory 95 9.1 The first correction to the mean-field free energy 95 9.2 Physical consequences 97 10 Introduction to the renormalization group 100 10.1 Renormalized theories and critical points 101 10.2 Kadanoff block spins 101 10.3 Examples of real space renormalization groups: ‘decimation’ 103 10.4 Structure of the renormalization group equations 109 Contents vii 11 Renormalization group for the Ï•4 theory 113 11.1 Renormalization group . . . without renormalization 114 11.2 Study of the renormalization group flow in dimension four 116 11.3 Critical behaviour of the susceptibility in dimension four 118 11.4 Multi-component order parameters 120 11.5 Epsilon expansion 122 11.6 An exercise on the renormalization group: the cubic fixed point 125 12 Renormalized theory 128 12.1 The meaning of renormalizability 128 12.2 Renormalization of the massless theory 132 12.3 The renormalized critical free energy (at one-loop order) 134 12.4 Away from Tc 136 13 Goldstone modes 138 13.1 Broken symmetries and massless modes 138 13.2 Linear and non-linear O(n) sigma models 142 13.3 Regularization and renormalization of the O(n) non-linear sigma model in two dimensions 144 13.3.1 Regularization 144 13.3.2 Perturbation expansion and renormalization 148 13.4 Renormalization group equations for the O(n) non-linear sigma model and the (d − 2) expansion 150 13.4.1 Integration of RG equations and scaling 151 13.5 Extensions to other non-linear sigma models 153 14 Large n 156 14.1 The linear O(n) model 156 14.2 O(n) sigma model
Côte titre :	Fs/13517-13518