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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