CLASSICAL THEORY OF GAUGE FIELDS
CONTENTS PART I GAUGE PRINCIPLE IN ELECTRODYNAMICS ELECTROMAGNETIC-?ELD ACTION IN VACUUM GAUGE INVARIANCE GENERAL SOLUTION OF MAXWELL’S EQUATIONS IN VACUUM CHOICE OF GAUGE SCALAR AND VECTOR FIELDS SYSTEM OF UNITS ?= C = 1 SCALAR ?ELD ACTION MASSIVE VECTOR ?ELD COMPLEX SCALAR ?ELD DEGREES OF FREEDOM INTERACTION OF ?ELDS WITH EXTERNAL SOURCES INTERACTING ?ELDS GAUGE-INVARIANT INTERACTION IN SCALAR ELECTRODYNAMICS NOETHER’S THEOREM ELEMENTS OF THE THEORY OF LIE GROUPS AND ALGEBRAS GROUPS LIE GROUPS AND ALGEBRAS REPRESENTATIONS OF LIE GROUPS AND LIE ALGEBRAS COMPACT LIE GROUPS AND ALGEBRAS NON-ABELIAN GAUGE FIELDS NON-ABELIAN GLOBAL SYMMETRIES NON-ABELIAN GAUGE INVARIANCE AND GAUGE ?ELDS: THE GROUP SU(2) GENERALIZATION TO OTHER GROUPS FIELD EQUATIONS CAUCHY PROBLEM AND GAUGE CONDITIONS SPONTANEOUS BREAKING OF GLOBAL SYMMETRY SPONTANEOUS BREAKING OF DISCRETE SYMMETRY SPONTANEOUS BREAKING OF GLOBAL U(1) SYMMETRY NAMBU– GOLDSTONE BOSONS PARTIAL SYMMETRY BREAKING: THE SO(3) MODEL GENERAL CASE GOLDSTONE’S THEOREM HIGGS MECHANISM EXAMPLE OF AN ABELIAN MODEL NON-ABELIAN CASE: MODEL WITH COMPLETE BREAKING OF SU(2) SYMMETRY EXAMPLE OF PARTIAL BREAKING OF GAUGE SYMMETRY: BOSONIC SECTOR OF STANDARD ELECTROWEAK THEORY SUPPLEMENTARY PROBLEMS FOR PART I PART II THE SIMPLEST TOPOLOGICAL SOLITONS KINK SCALE TRANSFORMATIONS AND THEOREMS ON THE ABSENCE OF SOLITONS THE VORTEX SOLITON IN A MODEL OF N-?ELD IN (2 + 1)-DIMENSIONAL SPACE–TIME ELEMENTS OF HOMOTOPY THEORY HOMOTOPY OF MAPPINGS THE FUNDAMENTAL GROUP HOMOTOPY GROUPS FIBER BUNDLES AND HOMOTOPY GROUPS SUMMARY OF THE RESULTS MAGNETIC MONOPOLES THE SOLITON IN A MODEL WITH GAUGE GROUP SU(2) MAGNETIC CHARGE GENERALIZATION TO OTHER MODELS DYONS 10 NON-TOPOLOGICAL SOLITONS 11 TUNNELING AND EUCLIDEAN CLASSICAL SOLUTIONS IN QUANTUM MECHANICS 11 1 DECAY OF A METASTABLE STATE IN QUANTUM MECHANICS OF ONE VARIABLE 11 2 GENERALIZATION TO THE CASE OF MANY VARIABLES 11 3 TUNNELING IN POTENTIALS WITH CLASSICAL DEGENERACY 12 DECAY OF A FALSE VACUUM IN SCALAR FIELD THEORY 12 1 PRELIMINARY CONSIDERATIONS 12 2 DECAY PROBABILITY: EUCLIDEAN BUBBLE (BOUNCE) 12 3 THIN-WALL APPROXIMATION 13 INSTANTONS AND SPHALERONS IN GAUGE THEORIES 13 1 EUCLIDEAN GAUGE THEORIES 13 2 INSTANTONS IN YANG–MILLS THEORY 13 3 CLASSICAL VACUA AND ?-VACUA 13 4 SPHALERONS IN FOUR-DIMENSIONAL MODELS WITH THE HIGGS MECHANISM SUPPLEMENTARY PROBLEMS FOR PART II PART III 14 FERMIONS IN BACKGROUND FIELDS 14 1 FREE DIRAC EQUATION 14 2 SOLUTIONS OF THE FREE DIRAC EQUATION DIRAC SEA 14 3 FERMIONS IN BACKGROUND BOSONIC ?ELDS 14 4 FERMIONIC SECTOR OF THE STANDARD MODEL 15 FERMIONS AND TOPOLOGICAL EXTERNAL FIELDS IN TWO-DIMENSIONAL MODELS 15 1 CHARGE FRACTIONALIZATION 15 2 LEVEL CROSSING AND NON-CONSERVATION OF FERMION QUANTUM NUMBERS 16 FERMIONS IN BACKGROUND FIELDS OF SOLITONS AND STRINGS IN FOUR-DIMENSIONAL SPACE–TIME 16 1 FERMIONS IN A MONOPOLE BACKGROUND ?ELD: INTEGER ANGULAR MOMENTUM AND FERMION NUMBER FRACTIONALIZATION 16 2 SCATTERING OF FERMIONS O?A MONOPOLE: NON-CONSERVATION OF FERMION NUMBERS 16 3 ZERO MODES IN A BACKGROUND ?ELD OF A VORTEX: SUPERCONDUCTING STRINGS 17 NON-CONSERVATION OF FERMION QUANTUM NUMBERS IN FOUR-DIMENSIONAL NON-ABELIAN THEORIES 17 1 LEVEL CROSSING AND EUCLIDEAN FERMION ZERO MODES 17 2 FERMION ZERO MODE IN AN INSTANTON ?ELD 17 3 SELECTION RULES 17 4 ELECTROWEAK NON-CONSERVATION OF BARYON AND LEPTON NUMBERS AT HIGH TEMPERATURES