GRAVITATION AND GAUGE SYMMETRIES
CONTENTS 1 SPACE TIME AND GRAVITATION 1 1 1 RELATIVITY OF SPACE AND TIME 1 HISTORICAL INTRODUCTION 1 RELATIVITY OF MOTION AND THE SPEED OF LIGHT 3 FROM SPACE AND TIME TO SPACETIME 6 1 2 GRAVITATION AND GEOMETRY 9 THE PRINCIPLE OF EQUIVALENCE 9 PHYSICS AND GEOMETRY 11 RELATIVITY COVARIANCE AND MACH S IDEAS 14 PERSPECTIVES OF FURTHER DEVELOPMENTS 18 2 SPACETIME SYMMETRIES 20 2 1 POINCAR E SYMMETRY 21 POINCAR E TRANSFORMATIONS 21 LIE ALGEBRA AND ITS REPRESENTATIONS 22 INVARIANCE OF THE ACTION AND CONSERVATION LAWS 24 2 2 CONFORMAL SYMMETRY 27 CONFORMAL TRANSFORMATIONS AND WEYL RESCALING 27 CONFORMAL ALGEBRA AND NITE TRANSFORMATIONS 29 CONFORMAL SYMMETRY AND CONSERVED CURRENTS 32 CONFORMAL TRANSFORMATIONS IN D 2 35 SPONTANEOUSLY BROKEN SCALE INVARIANCE 37 EXERCISES 39 3 POINCAR E GAUGE THEORY 42 3 1 POINCAR E GAUGE INVARIANCE 43 LOCALIZATION OF POINCAR E SYMMETRY 43 CONSERVATION LAWS AND ELD EQUATIONS 47 ON THE EQUIVALENCE OF DIFFERENT APPROACHES 50 3 2 GEOMETRIC INTERPRETATION 51 RIEMANN CARTAN SPACE U4 51 GEOMETRIC AND GAUGE STRUCTURE OF PGT 61 THE PRINCIPLE OF EQUIVALENCE IN PGT 62 3 3 GRAVITATIONAL DYNAMICS 65 EINSTEIN CARTAN THEORY 65 TELEPARALLEL THEORY 68 GENERAL REMARKS 72 EXERCISES 75 4 WEYL GAUGE THEORY 78 4 1 WEYL GAUGE INVARIANCE 79 LOCALIZATION OF WEYL SYMMETRY 79 CONSERVATION LAWS AND ELD EQUATIONS 83 CONFORMAL VERSUS WEYL GAUGE SYMMETRY 85 4 2 WEYL CARTAN GEOMETRY 86 CONFORMAL TRANSFORMATIONS IN RIEMANN SPACE 86 WEYL SPACE W4 89 WEYL CARTAN SPACE Y4 92 4 3 DYNAMICS 94 WEYL S THEORY OF GRAVITY AND ELECTRODYNAMICS 95 SCALAR ELDS AND THE IMPROVED ENERGY MOMENTUM TENSOR 96 GOLDSTONE BOSONS AS COMPENSATORS 100 GENERAL REMARKS 102 EXERCISES 105 5 HAMILTONIAN DYNAMICS 107 5 1 CONSTRAINED HAMILTONIAN DYNAMICS 108 INTRODUCTION TO DIRAC S THEORY 108 GENERATORS OF GAUGE SYMMETRIES 119 ELECTRODYNAMICS 123 5 2 THE GRAVITATIONAL HAMILTONIAN 125 COVARIANCE AND HAMILTONIAN DYNAMICS 125 PRIMARY CONSTRAINTS 128 THE 3 1 DECOMPOSITION OF SPACETIME 129 CONSTRUCTION OF THE HAMILTONIAN 131 CONSISTENCY OF THE THEORY AND GAUGE CONDITIONS 134 5 3 SPECI C MODELS 136 EINSTEIN CARTAN THEORY 136 THE TELEPARALLEL THEORY 141 EXERCISES 148 6 SYMMETRIES AND CONSERVATION LAWS 152 6 1 GAUGE SYMMETRIES 153 CONSTRAINT ALGEBRA 153 GAUGE GENERATORS 154 6 2 CONSERVATION LAWS EC THEORY 157 ASYMPTOTIC STRUCTURE OF SPACETIME 158 IMPROVING THE POINCAR E GENERATORS 160 ASYMPTOTIC SYMMETRIES AND CONSERVATION LAWS 163 6 3 CONSERVATION LAWS THE TELEPARALLEL THEORY 168 A SIMPLE MODEL 168 THE POINCAR E GAUGE GENERATORS 170 ASYMPTOTIC CONDITIONS 171 THE IMPROVED POINCAR E GENERATORS 173 CONSERVED CHARGES 176 6 4 CHERN SIMONS GAUGE THEORY IN D 3 179 CHERN SIMONS ACTION 180 CANONICAL ANALYSIS 183 SYMMETRIES AT THE BOUNDARY 187 EXERCISES 191 7 GRAVITY IN AT SPACETIME 195 7 1 THEORIES OF LONG RANGE FORCES 196 SCALAR ELD 196 VECTOR ELD 197 THE SYMMETRIC TENSOR ELD 202 THE SIGN OF THE STATIC INTERACTION 206 7 2 ATTEMPTS TO BUILD A REALISTIC THEORY 207 SCALAR GRAVITATIONAL ELD 207 SYMMETRIC TENSOR GRAVITATIONAL ELD 210 CAN THE GRAVITON HAVE A MASS 214 THE CONSISTENCY PROBLEM 219 EXERCISES 220 8 NONLINEAR EFFECTS IN GRAVITY 222 8 1 NONLINEAR EFFECTS IN YANG MILLS THEORY 222 NON ABELIAN YANG MILLS THEORY 222 SCALAR ELECTRODYNAMICS 226 8 2 SCALAR THEORY OF GRAVITY 228 8 3 TENSOR THEORY OF GRAVITY 231 THE ITERATIVE PROCEDURE 231 FORMULATION OF A COMPLETE THEORY 232 8 4 THE RST ORDER FORMALISM 237 YANG MILLS THEORY 237 EINSTEIN S THEORY 239 EXERCISES 243 9 SUPERSYMMETRY AND SUPERGRAVITY 245 9 1 SUPERSYMMETRY 246 FERMI BOSE SYMMETRY 246 SUPERSYMMETRIC EXTENSION OF THE POINCAR E ALGEBRA 251 THE FREE WESS ZUMINO MODEL 255 SUPERSYMMETRIC ELECTRODYNAMICS 257 9 2 REPRESENTATIONS OF SUPERSYMMETRY 259 INVARIANTS OF THE SUPER POINCAR E ALGEBRA 259 MASSLESS STATES 261 MASSIVE STATES 264 SUPERMULTIPLETS OF ELDS 267 TENSOR CALCULUS AND INVARIANTS 269 THE INTERACTING WESS ZUMINO MODEL 271 9 3 SUPERGRAVITY 273 THE RARITA SCHWINGER ELD 273 LINEARIZED THEORY 276 COMPLETE SUPERGRAVITY 278 ALGEBRA OF LOCAL SUPERSYMMETRIES 281 AUXILIARY ELDS 283 GENERAL REMARKS 286 EXERCISES 290 10 KALUZA KLEIN THEORY 293 10 1 BASIC IDEAS 294 GRAVITY IN VE DIMENSIONS 294 GROUND STATE AND STABILITY 298 10 2 FIVE DIMENSIONAL KK THEORY 303 FIVE DIMENSIONAL GRAVITY AND EFFECTIVE THEORY 303 CHOOSING DYNAMICAL VARIABLES 307 THE MASSLESS SECTOR OF THE EFFECTIVE THEORY 310 DYNAMICS OF MATTER AND THE FTH DIMENSION 312 SYMMETRIES AND THE PARTICLE SPECTRUM 315 10 3 HIGHER DIMENSIONAL KK THEORY 320 GENERAL STRUCTURE OF HIGHER DIMENSIONAL GRAVITY 320 THE MASSLESS SECTOR OF THE EFFECTIVE THEORY 325 SPONTANEOUS COMPACTI CATION 329 GENERAL REMARKS 331 EXERCISES 335 11 STRING THEORY 338 11 1 CLASSICAL BOSONIC STRINGS 339 THE RELATIVISTIC POINT PARTICLE 339 ACTION PRINCIPLE FOR THE STRING 341 HAMILTONIAN FORMALISM AND SYMMETRIES 344 11 2 OSCILLATOR FORMALISM 346 OPEN STRING 347 CLOSED STRINGS 349 CLASSICAL VIRASORO ALGEBRA 350 11 3 FIRST QUANTIZATION 353 QUANTUM MECHANICS OF THE STRING 353 QUANTUM VIRASORO ALGEBRA 355 FOCK SPACE OF STATES 356 11 4 COVARIANT ELD THEORY 358 GAUGE SYMMETRIES 358 THE ACTION FOR THE FREE STRING ELD 361 ELECTRODYNAMICS 362 GRAVITY 364 11 5 GENERAL REMARKS 366 EXERCISES 369