Universite de Grenoble - Summer School of Theoretical Physics Les Houches 1970

Supported by NATO and the Commissariat a I'Energie Atomique

STATISTICAL MECHANICS AND QUANTUM FIELD THEORY

MECANIQUE STATISTIQUE ET THEORIE QUANTIQUE

DES CHAMPS

edited by C. DeWitt Faculte des Sciences, Grenoble and

University of North Carolina, Chapel Hill

R. Stora Centre d'Etudes Nucleaires de Saclay

Service de Physique theorique

GORDON AND BREACH SCIENCE PUBLISHERS New York London Paris

389

390

H. EPSTEIN CERN, Geneve

J. GINIBRE

LECTURERS

Laboratoire de Physique Theorique et Hautes Energies, Orsay

V. GLASER CERN, Geneve

J. GLIMM New York University, New York

R. GRIFFITHS Carnegie-Mellon University, Pittsburgh

K. HEPP Eidgenossische Technische Hochschule, Zurich

A. JAFFE Harvard University, Cambridge. Massachusetts

O. E. LANFORD University of California. Berkeley

E. H. LIEB Massachussetts Institute of Technology

D. RUELLE Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette

Battelle Consultants

A. Andreotti R.Bott G. Mackey

Participants

ABELLANAS Lorenzo ANDERSON Bo

BROS Jacques CANNON John CHALLIFOUR John

CONNES Alain DILLARD Margaret DIMOCK Jon DOUGLAS Ronald

DUBOIS-VIOLETTE Michel

EZAWA Hiroshi FEDERBUSH Paul

FORTUIN Cornelius FROHLICH Jurg FUCHS Gerard

HUBER Albrecht

IAGOLNITZER Daniel KLAUDER John LAscoux Jean

LENARD Andrew MAHoux Gilbert

PARTICIPANTS

Istituto Matematico Universita, Pisa Mathematics Department, Harvard University Mathematics Department, Harvard University

IHES-Bures-sur-Yvette Institute of Theoretical Physics, University of Lund

391

Service de Physique Theorique, CEN, Saclay The Rockefeller University, New York Department of Physics, Indiana University and Institut flir Theoretische Physik, Universitat Gottingen Ecole Normale Superieure, Paris St-. Augustine's College, Raleigh Department of Physics, Harvard University Department of Mathematics, SUNY, Stony Brook Laboratoire de Physique Theorique, Faculte des Sciences, Orsay Department of Physics, Gakushuin University Department of Mathematics. University of Michigan, Ann Arbor Instituut Lorentz, Leiden Seminar fUr Theoretische Physik, ETH, Zurich Centre de Physique Theorique, Ecole Poly­technique, Paris Institut fUr Theoretische Physik der Universitat Kiel Service de Physique Theorique, CEN, Saclay Bell Telephone Laboratories, Murray Hill Centre de Physique Theorique, Ecole Poly­technique, Paris Department of Mathematics, Indiana University Departement de Physique Theorique, CEN, Saclay

392

MANUCEAU Jerome

MESSAGER Alain NIEMEIJER Theodorus

O'CARROLL Michael

OSTERWALDER Konrad RENOUARD Pierre

ROSEN Lon

SHERMAN Seymour SIMON Barry

SPITZER Frank STREATER Raymond TAVERNIER Jean TESEI Alberto VERTOGEN Gerrit

VESELI¢ Kresimir

VOROS Andre WEHRL Alfred WOOD Peter

PARTICIPANTS

Centre de Physique Theorique, C.N.R.S., Marseille C.N.R.S., Marseille FOM Instituut voor Atoom-en Moleguul fysica, Amsterdam Pontificia Universidade Catolica, Departmento de Fisica, Rio de Janeiro Seminar fUr Theoretische Physik, ETH, ZUrich Centre de Physique, Theorique, Ecole Poly­technique, Paris Courant Institute of Mathematical Science, New York University Mathematics Department, Indiana University Departments of Mathematics and Physics, Princeton University Department of Mathematics, Cornell University Istitute for Theoretical Physics, Lund Faculte des Sciences, Paris Instituto di Fisica, Rome Instituut voor Theoretische Fysica, Ryks Uni­versiteit, Groningen University of Zagreb and Institute "Rudjer Boskovic", Zagreb Division de Physique Theorique, CEN, Saclay CERN, Geneve School of Physics, University of Newcastle upon Tyne

LES HOUCHES SESSION XXVIII

28 Juillet - 6 Septembre 1975

METHODES EN THEORIE DES CHAMPS

METHODS IN FIELD THEORY

Edite par

ROGER BALIAN

et

JEAN ZINN-JUSTIN

Centre de Sac/a)'

I 1976

NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM. NEW YORK· OXFORD

393

394

LES HOUCHES, ECOLE D'ETE DE PHYSIQUE THEORIQUE SESSION XXVIII

ORGANISME D'INTERET COMMUN DE L'UNNERSITE SCIENTIFIQUE ET MEDICALE DE GRENOBLE ET DE L'INSTITUT NATIONAL

POLYTECHNIQUE DE GRENOBLE

INSTITUT D'ETUDES AVANCEES DE LA DNISION DES AFF AIRES SCIENTIFIQUES DE L'OT AN

AIDE PAR LE COMMISSARIAT A L'ENERGIE ATOMIQUE

Membres du conseil: M. Soutif, president, L. Neel, vice-president, P. Averbuch, M.-Y. Bernard, 1.-F. Detoeuf, C. DeWitt, 1.-M. Dolique, B. Dreyfus, 1. Gregory, S. Haroche, M. Jacob, Y. Rocard, D. Thoulouze, 1. Yoccoz, 1. Yvon

Directeur: Roger Balian, Service de Physique Theorique, Centre de -Saclay, BP no. 2, 91190 Gif-sur-Yvette

Directeur scientifique de /a session: Jean Zinn-Justin, Service de Physique Theorique, Centre de Saday, BP no. 2,91190 Gif-sur-Yvette

L.D. FADDEEV, C.G. CALLAN, B.W. LEE, DJ. GROSS,

CONFERENCIERS

Introduction to functional methods Introduction to renormalization theory Gauge theories Applications of the renormalization group to high-energy physics

L.D. F ADDEEV, S. COLEMAN, T .D. LEE, Lectures on the soliton approach to particle structure (seminars)

K.G. WILSON, G. 't HOOFT, Lectures on quark confinement (seminars) M.l.G. VELTMAN, Quantum theory of gravitation E. BREZIN, Applications of the renormalization group to critical

phenomena

395

PARTICIPANTS

Alvarez-Estrada, Ramon, Departamento de Fisica Teorica, Facultad de Ciencias Fisicas, Universidad Complutense, Madrid, Spain.

Banks, Thomas, Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Israel.

Bernard, Claude, Department of Physics, Jefferson Laboratory, Harvard Uni­versity, Cambridge, Mass. 02138, USA.

Bervillier, Claude, Service de Physique Theorique, CEN-Saclay, 91190 Gif-sur­Yvette, France.

Collecott, Peter, Department of Applied Mathematics and Theoretical Physics, Cambridge, England.

Coullet, Pie"e, Laboratoire de Physique de la Matiere Condensee, Equipe Mecanique Statistique, Universite de Nice, 06000 Nice, France.

De Witt, Bryce, University of Texas, Austin, Texas 78712, USA. Dolan, Louise Ann, MIT 6-415, Cambridge, Mass. 02134, USA. Duff, Michael, Mathematics Department, King's College, London, England. Epstein, Geoffrey, Department of Physics, Michigan State University, East

Lansing, Michigan, USA. Fayet, Pie"e, Laboratoire de Physique Theorique, Ecole Normale Superieure,

75231 Paris Cede x 05, France. Galic, Hrvoje, Institute Ruder Boskovi9, Zagreb, Yugoslavia. Girardi, Georges, Service de Physique Theorique, CEN-Saclay, 91190 Gif-sur­

Yvette, France. Incoul, Roland, Universite d'Etat de Mons, Faculte des Sciences, B 7000

Mons, Belgium. Jevicki, Antal, City College of New York, Department of Physics, New York,

NY 10031, USA. Joglekar, Satish, School of Natural Sciences, Institute for Advanced Study,

Princeton, NJ 08540, USA. Julia, Bernard, Physics Department, Princeton University, Princeton,

NJ 08540, USA. Karczmarczuk, Jerzy, Department of Theoretical Physics, Jagellonian Univer-

sity, 30059 Cracow, Poland. Kunszt, Zoltan, Department of Atomic Physics, Eotvos University, 1088 Buda-

pest, Hungary.

396 Participants

Lang, Wolfdieter, Lyman Laboratory, Harvard University, Cambridge, Mass.

02138, USA. Magnen, Jacques, Ecole Poly technique, Centre de Physique Theorique,

91120 Palaiseau, France. Mason, Anthony, Physics Department, Imperial College, London SW7, England. Mc Brian, Olivier, Department of Mathematics, Rockefeller University, New

York, NY 10021, USA. Meyer, Steffen, Institut flir Theoretische Physik, Freie Universitat Berlin,

D 1000 Berlin-33, Germany. Nappi, Chiara Rosanna, Istituto di Fisica Teorica, Mostra d'OItremare,

80125 Napoli, Italy. Nienhuis, Bernard, Instituut voor Theoretische Fysica, Utrecht, The Nether-

lands. Otten, Christ, Instituut voor Theoretische Fysica, Utrecht, The Netherlands. Passarino, Giampiero, Istituto di Fisica Teorica, 110125 Torino,ltaly. Pi, So-Young, Department of Physics, Rockefeller University, New York,

NY 10021, USA. Rayski, Jacek, Department of Theoretical Physics, J agellonian University,

30059 Cracow, Poland. Shankar, Ramamurti, Physics Department, Harvard University, Cambridge,

Mass. 02138, USA. Soyeur, Madeleine, Service de Physique Theorique, (,EN-Saclay, 91190 Gif­

sur-Yvette, France. Stelle, Kellogg, Department of Physics, Brandeis University, Waltham, Mass.

02154, USA. Strocchi, Franco, Scuola Normale Superiore, 56100 Pisa, Italy. Talon, Michel, Universite Paris VI, 75231 Paris Cedex OS, France. Thierry-Mieg, Jean, Groupe d' Astrophysique Relativiste, Observatoire de

Meudon, 92 Meudon, France. West, Peter, Theory Group, Physics Department, Imperial College, London

SW7, England. Windey, Paul, Service de Physique Theorique et Mathematique, Universite

Libre de Bruxelles, 1050 Bruxelles, Belgium. Witten, Edward, Jadwin Hall, Princeton, NJ 08540, USA. Zee, Anthony, Department of Physics, Princeton University, Princeton,

NJ 08540, USA. Zuber, Jean-Bernard, Service de Physique Theorique, CEN-Saclay, 91190 Gif­

sur-Yvette, France.

Index ADE

gauge symmetry 151 (AdS)4 x S7 271, 273, 275 Action functional

classical 113 Angle-action variables 164,166-167

quantum 170 Antipode 78-80 Asymptotic freedom 381-382 Automorphic forms 139 Backward Heisenberg equation 4-5,

12, 15 Ballistic wires 179-180, 201-202, 207 BCS theory 16 Bethe ansatz 162, 227

algebraic 171, 173 equations 232

Bethe equation 171 B.F. gauge theory 248

two-dimensional 250 Binary mixture 224 Black brane 277 Bloch equation 5-6, 12 Bogomolny

bound 278, 283-284, 286, 295 equation 249 formulation 250

Bogomolny-Prasad-Sommerfeld limit 255

Bose gas 162 BPS state 21, 28, 30-31 Bracket identities 127 Brane probe 313 Branes

intersecting 296, 307, 309-310, 312

Breathers 166 Canonical systems

standard 129 Causal functions 2 Characteristic classes 40, 42 Chern-Simons (C-S) 204, 263, 269,

288, 303-304, 369 dimensional reduction of 247 finite temperature theory 254 kinetic term 245 three-dimensional 250, 254

Chiral Bose field 251 kinetic term 252 two-form 139

Chiral tensor field free quantum 146

Coadjoint orbits 168 Configuration space 112 Connes-Moscovici weights 88 Coproduct 76-79, 92-93 Correlations 2, 15 Counterterm

local 70-71, 79, 86, 356 Critical phenomena 382-383, 385 Cut

admissible 74-76 elementary 74 full 78 normal 78

D brane 21, 24 Differential algebra 353, 358

399

400

Differential geometry 179-181, 211 Differentiation

left 113 right 113

Dimensional reduction 287-289, 293-294,296-298,301,311,317, 324-325, 341

(Scherk-Schwarz) 300, 305, 307 Dirac quantization condition 324-

325, 327, 329-331, 336, 339 Dirichlet brane 21-22 Divergences 375-379

overlapping 103 Domains 224 Domain wall 224, 300, 306-307 Duality 318, 320-321, 328, 330, 333,

337-339, 346 Duistermaat-Heckmann theorem 53-

54 Effective action 260 Effective charge 379-380 Effective field theory 375, 382, 385 Entropy 7, 221

maximum 6, 11 Equivalence

of Peierls and Poisson bracket 129

Equivariant cohomology 35,37,39-40,47,51,54

FCR see Fundamental Commuta-tion Relations

Factorized scattering 162 Faddeev Popov quantization 359 Feshbach map 197-198 Feynman functional integral 135 Feynman integral

renormalized 86

Fiber projection 364-365 Field

components 112 histories 113

Fine tuning problem 377, 386 Fivebrane 139

M-heory 139 partition function 140, 143

Flows 114 Fluctuations 2, 15 Forest formula 86 Functional

derivative 113 Fourier transform 135

Fundamental Commutation Rela-tions (FCR) 170

GSa 140 Galileo boost 251 Gauge

algebra 114-115 fixing form 364 form 365 group 28, 115-116 potential 258, 260, 301, 322, 327

Gauge theories 245 infinitesimal gauge transforma­

tions 246 large gauge transformations 246

Gauss' law 247-248, 251 Generating

functional 3, 7, 11 operator 3-5

Global structure 273, 298 Grand canonical equilibrium 2, 13 Green's functions 119

advanced 119, 120 left 120

reciprocity telations for 121 retarded 119-120 right 120 variation of 122

Group of formal diffeomorphisms 68

Ji R and JiT

relation between 88 Harmonic map 341, 343-344 Hartree-Fock 13-14, 16

(time-dependent) 14, 16 Heavy ion collisions 15 Heisenberg

ferromagnet 167 picture 4

History 112 Hochschild cohomology 93 Holographic principle 29 Homotopy 37-4.0 Hopf algebra 59, 79, 82, 94-95, 102,

173 Ji R 73, 93 JiT 60, 88, 102 of rooted trees 68, 79-80, 83,

86, 90, 94, 106 Instanton 255 Invariance

Galilean 246, 250 Lorentz 246, 255

Invariants absolute 116 conditional 116

Inverse scattering method 162 classical 161, 166 quantum 168, 171, 173

Jacobi field operator 114, 119, 129 zero-eigenvalue eigenfunction of

118 Jacobi fields 117 K3 27, 30, 151 KdV 161,163

401

Kaluza-Klein 31,261,287-291,293-294, 301-302, 305-306, 310-311, 318

Klein-Gordon form 251 Kolmogorov equations 4 Landau ghost 380 Lax

approach 167 approach on lattice 170 equation 163

Lie algebra of formal vector fields 61, 101 on C} 95

Linear response 15 Liouville

equation 248 -von Neumann equation 12, 15

Lorentzian lattice 155 Magnetic chains 162 Many-electron system 179, 188 Massive bosons on the 2-torus 149 Matrix Theory 23-24, 26, 29-33 Matter field 246 Mean field

approximation 13 theory 16, 382

Modular functions 139 groups S L( n, Z) 139

Monopole 255, 310-311 Monodromy 169

group 235 M-theory 19, 23, 31-32

402

consistent quantization 158 NSE see Nonliear 8chrodinger equa­

tion Noncommutative Geometry 33, 59,

179-180, 210-211 Nonlinear Schrodinger equation (NSE)

163, 165, 168-169, 250, 252 chiral 251 "gauged" 247 modified 251

Nonlocal Lagrangian 249 Orientifold 21 P-brane 262, 266, 270, 275, 377-

278, 280-282, 284, 293-296, 299,310,314,322-323,326, 330-331, 339, 346

Particle trajectories 228 Partition function 2, 151

fivebrane 140, 143 of a chiral boson 142 world-sheet 158 world-volume 158

Pauli's quantum theory of the elec-tron 183

Peierls bracket 126-128 Perturbative unitarity 143 Phason 221-222 Physical observables 116

disturbances in 124 reciprocity relation for 124

Poisson bracket 131 Quantization

(charge) 318, 322-324 Faddeev Popov 359

Quantum group 173 Quantum Hall

edge states 252

effect 179-180, 201-202, 206-207 fluid 369

Quantum theory of space-time 179, 208

Quasiclassical quantization 166, 168 Quasicrystalline phase 222, 240 Quasicrystals 218, 240 Quasiperiodic 219 RPA see Random phase approxima­

tion Random matrices 367-368

universality in 367 Random phase approximation (RPA)

15-16 (backward dual) 15

Random tHings 221, 223, 240 Reciprocity relations 121 Renormalizable 382, 385-386

theory 378 Renormalization 59, 68, 81, 357,

375,378 BPHZ 71 group 194, 201, 375, 379, 383-

386 group analysis 199, 201 map 81,199 (perturbative) 353-354, 356

Response functions 2 Riemann sheets 235 Riemann theta function 143 Roots 233 Ruelle Sullivan form 363 a-model 188, 260, 313, 315-317,319,

321, 341-342, 344-345 8L(6, Z) 150 SchrOdinger picture 4-5 Schwinger variational principle 132,

134 S-dua1156 Section

of a fibre bundle 112 Self-dual theories 247 Sine-Gordon equation 165 Skew fields 115 Slavnov Taylor identity 353, 361-

362 Slavnov symmetry 361-362, 365 Small disturbances 117

homogeneous equation of 117 inhomogeneous equation of 117

Solitons 161, 164 Soliton solutions 248

single soliton at rest 250 multi-soliton 250 quantization 250

Space of field histories 113

Spin manifold 142 Stability of matter 189, 191 Standard model 375, 381 Strings 260

heterotic 20, 157 type I 20 type II 19, 151, 157

Structure functions 116 Supercommutator 131

function 126, 130 Super

Jacobi identity 128 Lie bracket 115

Supergravity 259,263,269,278,287, 296, 299, 301-302, 305-307, 313-314, 318-321, 329, 333, 338-339, 341, 346

403

Supermembrane 271, 278, 286, 314 Supersymmetry 258, 263, 277, 283-

286,295,302,308,311-313, 315-319, 326, 344, 377

algebra 188, 258, 280, 283, 286 infinitesimal parameters 114 local transformations 114 spectral data 185, 188, 210-211

Supertranspose 114 Supplementary condition 118 Symplectic Hamiltonian procedure

248 T2 x CP2 151 T-duality symmetry group 155 Toda

equation 248 lattice 168

Transfer matrix 162 Transition amplitude 132 Tree

rooted 73 Triviality problem 386 Truncation 289

consistent 262-263, 287, 291 Two-dimensional lattice models 162,

169 U ni versali ty 383 Vacuum

of supergravity 271, 274 Vacuum energy 147

regularized 147 Variational principles 8, 10 Winding number 246 XXZ model 173 Yang-Baxter relation 172 Yang-Mills/Higgs model 255 Zero-curvature condition 167