CLASSICAL COVARIANT FIELDS
CONTENTS PART 1: FIELDS INTRODUCTION FUNDAMENTAL AND EFFECTIVE ?ELD THEORIES THE CONTINUUM HYPOTHESIS FORCES STRUCTURAL ELEMENTS OF A DYNAMICAL SYSTEM THE ELECTROMAGNETIC ?ELD MAXWELL’S EQUATIONS 2 1 1 POTENTIALS 2 1 2 GAUGE INVARIANCE 2 1 3 4-VECTORS AND (N + 1)-VECTORS 2 1 4 THE ?ELD STRENGTH 2 1 5 COVARIANT ?ELD EQUATIONS USING Fµ? 2 1 6 TWO INVARIANTS 2 1 7 GAUGE INVARIANCE AND PHYSICAL MOMENTUM 2 1 8 WAVE SOLUTIONS TO MAXWELL’S EQUATIONS CONSERVATION LAWS 2 2 1 CURRENT CONSERVATION 2 2 2 POYNTING’S VECTOR AND ENERGY CONSERVATION ELECTROMAGNETISM IN MATTER 2 3 1 DIELECTRICS FIELD PARAMETERS CHOICE OF PARAMETRIZATION CON?GURATION SPACE 3 2 1 FLAT AND CURVED SPACE 3 2 2 VECTOR EQUATIONS 3 2 3 COORDINATE BASES 3 2 4 EXAMPLE: EUCLIDEAN SPACE 3 2 5 EXAMPLE: MINKOWSKI SPACETIME MOMENTUM SPACE AND WAVES TENSOR TRANSFORMATIONS PROPERTIES EUCLIDEAN AND RIEMANNIAN SPACETIME THE ACTION PRINCIPLE THE ACTION IN NEWTONIAN PARTICLE MECHANICS 4 1 1 VARIATIONAL PRINCIPLE 4 1 2 ?S: EQUATION OF MOTION 4 1 3 THE EULER–LAGRANGE EQUATIONS 4 1 4 ?S: CONTINUITY 4 1 5 RELATIVISTIC POINT PARTICLES FRICTIONAL FORCES AND DISSIPATION FUNCTIONAL DIFFERENTIATION THE ACTION IN COVARIANT ?ELD THEORY 4 4 1 FIELD EQUATIONS AND CONTINUITY 4 4 2 UNIQUENESS OF THE ACTION 4 4 3 LIMITATIONS OF THE ACTION PRINCIPLE 4 4 4 HIGHER DERIVATIVES DYNAMICAL AND NON-DYNAMICAL VARIATIONS 4 5 1 SCALAR ?ELDS 4 5 2 GAUGE AND VECTOR ?ELDS 4 5 3 THE METRIC AND SECOND-RANK TENSORS THE VALUE OF THE ACTION CLASSICAL ?ELD DYNAMICS SOLVING THE ?ELD EQUATIONS 5 1 1 FREE ?ELDS 5 1 2 BOUNDARY CONDITIONS AND CAUSALITY I 5 1 3 POSITIVE AND NEGATIVE ENERGY SOLUTIONS 5 1 4 SOURCES 5 1 5 INTERACTIONS AND MEASUREMENTS GREEN FUNCTIONS AND LINEAR RESPONSE 5 2 1 THE INVERSE PROBLEM 5 2 2 BOUNDARY CONDITIONS AND CAUSALITY II 5 2 3 GREEN FUNCTIONS IN FOURIER MOMENTUM SPACE 5 2 4 LIMITATIONS OF THE GREEN FUNCTION METHOD 5 2 5 GREEN FUNCTIONS AND EIGENFUNCTION METHODS SCALAR ?ELD GREEN FUNCTION 5 3 1 THE WIGHTMAN FUNCTIONS 5 3 2 BOUNDARY CONDITIONS AND POLES IN THE K0 PLANE 5 3 3 RETARDED GREEN FUNCTION 5 3 4 ADVANCED GREEN FUNCTION 5 3 5 FEYNMAN GREEN FUNCTION 5 3 6 COMMENT ON COMPLEX CONTOURS SCALAR GREEN FUNCTIONS IN REAL SPACE 5 4 3 FREQUENCY-DEPENDENT FORM OF GF AND GR IN N = 5 4 4 EUCLIDEAN GREEN FUNCTION IN 2 + 0 DIMENSIONS 5 4 5 MASSIVE CASE SCHR¨ODINGER GREEN FUNCTION DIRAC GREEN FUNCTIONS PHOTON GREEN FUNCTIONS PRINCIPAL VALUES AND KRAMERS–KRONIG RELATIONS REPRESENTATION OF BOUND STATES IN ?ELD THEORY STATISTICAL INTERPRETATION OF THE ?ELD FLUCTUATIONS AND VIRTUAL PROCESSES 6 1 1 FLUCTUATION GENERATORS: GF(X, X?) AND GE(X, X?) 6 1 2 CORRELATION FUNCTIONS AND GENERATING FUNCTIONALS 6 1 3 SYMMETRY AND CAUSAL BOUNDARY CONDITIONS 6 1 4 WORK AND DISSIPATION AT STEADY STATE 6 1 5 FLUCTUATIONS 6 1 6 DIVERGENT ?UCTUATIONS: TRANSPORT 6 1 7 FLUCTUATION DISSIPATION THEOREM SPONTANEOUS SYMMETRY BREAKING EXAMPLES AND APPLICATIONS FREE PARTICLES 7 1 1 VELOCITY DISTRIBUTIONS FIELDS OF BOUND PARTICLES INTERACTION BETWEEN MATTER AND RADIATION 7 3 1 MAXWELL’S EQUATIONS 7 3 2 ELECTROMAGNETIC WAVES 7 3 3 DISPERSION AND THE FARADAY EFFECT 7 3 4 RADIATION FROM MOVING CHARGES IN N = 3: RETARDATION RESONANCE PHENOMENA AND DAMPENING ?ELDS 7 4 1 CHERENKOV RADIATION 7 4 2 CYCLOTRON RADIATION 7 4 3 LANDAU DAMPING 7 4 4 LASER COOLING HYDRODYNAMICS 7 5 1 NAVIER–STOKES EQUATIONS 7 5 2 DIFFUSION 7 5 3 FORCED BROWNIAN MOTION VORTEX ?ELDS IN 2 + 1 DIMENSIONS 7 6 1 A VORTEX MODEL 7 6 2 GREEN FUNCTIONS 7 6 3 RELATIONSHIP BETWEEN ?(R ?R?) AND G(R ?R?) 7 6 4 SINGULAR NATURE OF ?(R ?R?) PART 2: GROUPS AND ?ELDS FIELD TRANSFORMATIONS GROUP THEORY 8 1 1 DE?NITION OF A GROUP 8 1 2 GROUP TRANSFORMATIONS 8 1 3 USE OF VARIABLES WHICH TRANSFORM LIKE GROUP VECTORS COSETS AND THE FACTOR GROUP 8 2 1 COSETS 8 2 2 CONJUGACY AND INVARIANT SUB-GROUPS 8 2 3 SCHUR’S LEMMA AND THE CENTRE OF A GROUP 8 2 4 THE FACTOR GROUP G/H GROUP REPRESENTATIONS 8 3 1 DE?NITION OF A REPRESENTATION G R 8 3 2 IN?NITESIMAL GROUP GENERATORS 8 3 3 PROPER GROUP ELEMENTS 8 3 4 CONJUGATE REPRESENTATIONS 8 3 5 CONGRUENT REPRESENTATIONS REDUCIBLE AND IRREDUCIBLE REPRESENTATIONS 8 4 1 INVARIANT SUB-SPACES 8 4 2 REDUCIBILITY LIE GROUPS AND LIE ALGEBRAS 8 5 1 NORMALIZATION OF THE GENERATORS 8 5 2 ADJOINT TRANSFORMATIONS AND UNITARITY 8 5 3 CASIMIR INVARIANTS 8 5 4 SUB-ALGEBRA 8 5 5 THE CARTAN SUB-ALGEBRA 8 5 6 EXAMPLE OF DIAGONALIZATION 8 5 7 ROOTS AND WEIGHTS 8 5 8 THE CARTAN–WEYL BASIS 8 5 9 GROUP VECTORS AND DIRAC NOTATION 8 5 10 EXAMPLE: ROTATIONAL EIGENVALUES IN THREE DIMENSIONS EXAMPLES OF DISCRETE AND CONTINUOUS GROUPS 8 6 1 GL(N, C): THE GENERAL LINEAR GROUP 8 6 2 U(N): UNITARY MATRICES 8 6 3 SU(N): THE SPECIAL UNITARY GROUP 8 6 4 SU(2) 8 6 5 U(1): THE SET OF NUMBERS Z : |Z|2 = Z N: THE NTH ROOTS OF UNITY O(N): THE ORTHOGONAL GROUP 8 6 8 SO(3): THE THREE-DIMENSIONAL ROTATION GROUP 8 6 9 SO(2): THE TWO-DIMENSIONAL ROTATION GROUP UNIVERSAL COVER GROUPS AND CENTRES 8 7 1 CENTRE OF SU(N) IS Z N 8 7 2 CONGRUENT ALGEBRAS: N-ALITY 8 7 3 SIMPLE AND SEMI-SIMPLE LIE ALGEBRAS SUMMARY SPACETIME TRANSFORMATIONS PARITY AND TIME REVERSAL TRANSLATIONAL INVARIANCE 9 2 1 GROUP REPRESENTATIONS ON COORDINATE SPACE 9 2 2 BLOCH’S THEOREM: GROUP REPRESENTATIONS ON ?ELD SPACE 9 2 3 SPATIAL TOPOLOGY AND BOUNDARY CONDITIONS ROTATIONAL INVARIANCE: SO(N) 9 3 1 GROUP REPRESENTATIONS ON COORDINATE SPACE 9 3 2 EIGENFUNCTIONS: CIRCULAR AND SPHERICAL HARMONICS LORENTZ INVARIANCE 9 4 1 PHYSICAL BASIS 9 4 2 LORENTZ BOOSTS AND ROTATIONS 9 4 3 THE HOMOGENEOUS LORENTZ GROUP: SO(1, N) 9 4 4 DIFFERENT REPRESENTATIONS OF THE LORENTZ GROUP IN 3 + DIMENSIONS 9 4 5 OTHER SPACETIME DIMENSIONS 9 4 6 FACTORIZATION OF PROPER LORENTZ TRANSFORMATIONS 9 4 7 THE INHOMOGENEOUS LORENTZ GROUP OR POINCAR´E GROUP IN 3 + 1 DIMENSIONS 9 4 8 CURVED SPACETIME: KILLING’S EQUATION GALILEAN INVARIANCE 9 5 1 PHYSICAL BASIS 9 5 2 RETARDATION AND BOOSTS 9 5 3 GENERATOR ALGEBRA CONFORMAL INVARIANCE 9 6 1 SCALAR ?ELDS IN N + 1 DIMENSIONS 9 6 2 THE MAXWELL ?ELD IN N + 1 DIMENSIONS SCALE INVARIANCE BREAKING SPACETIME SYMMETRY EXAMPLE: NAVIER–STOKES EQUATIONS KINEMATICAL AND DYNAMICAL TRANSFORMATIONS GLOBAL OR RIGID SYMMETRIES LOCAL SYMMETRIES DERIVATIVES WITH A PHYSICAL INTERPRETATION 10 3 1 HERMITICITY 10 3 2 COMMUTATIVITY WITH TRANSFORMATIONS 10 3 3 FORM-INVARIANT DERIVATIVES CHARGE CONJUGATION TCP INVARIANCE EXAMPLES 10 6 1 GAUGE INVARIANCE: ELECTROMAGNETISM 10 6 2 LORENTZ INVARIANCE: GRAVITY 10 6 3 THE TWO-LEVEL ATOM IN A STRONG RADIATION ?ELD GLOBAL SYMMETRY BREAKING LOCAL SYMMETRY BREAKING DYNAMICAL SYMMETRY BREAKING MECHANISMS POSITION AND MOMENTUM POSITION, ENERGY AND MOMENTUM PARTICLES AND POSITION 11 2 1 SCHR¨ODINGER ?ELD 11 2 2 KLEIN–GORDON ?ELD 11 2 3 DIRAC ?ELD 11 2 4 SPIN S ?ELDS IN 3 + 1 DIMENSIONS THE ENERGY–MOMENTUM TENSOR ?µ? 11 3 1 EXAMPLE: CLASSICAL PARTICLE MECHANICS 11 3 2 EXAMPLE: THE COMPLEX SCALAR ?ELD 11 3 3 EXAMPLE: CONSERVATION SPACETIME INVARIANCE AND SYMMETRY ON INDICES ?µ? FOR GAUGE THEORIES ANOTHER ENERGY–MOMENTUM TENSOR Tµ? 11 6 1 VARIATIONAL DE?NITION 11 6 2 THE TRACE OF THE ENERGY–MOMENTUM TENSOR Tµ? 11 6 3 THE CONFORMALLY IMPROVED Tµ? ANGULAR MOMENTUM AND SPIN 11 7 1 ALGEBRA OF ORBITAL MOTION IN 3 + 1 DIMENSIONS 11 7 2 THE NATURE OF ANGULAR MOMENTUM IN N + 1 DIMENSIONS 11 7 3 COVARIANT DESCRIPTION IN 3 + 1 DIMENSIONS 11 7 4 INTRINSIC SPIN OF TENSOR ?ELDS IN 3 + 1 DIMENSIONS 11 7 5 HELICITY VERSUS SPIN 11 7 6 FRACTIONAL SPIN IN 2 + 1 DIMENSIONS WORK, FORCE AND TRANSPORT IN OPEN SYSTEMS 11 8 1 THE GENERALIZED FORCE F? = ?µT µ? 11 8 2 WORK AND POWER 11 8 3 HYDRODYNAMIC ?OW AND ENTROPY 11 8 4 THERMODYNAMICAL ENERGY CONSERVATION 11 8 5 KUBO FORMULAE FOR TRANSPORT COEF?CIENTS EXAMPLE: RADIATION PRESSURE CHARGE AND CURRENT CONSERVED CURRENT AND NOETHER’S THEOREM ELECTRIC CURRENT Jµ FOR POINT CHARGES ELECTRIC CURRENT FOR ?ELDS REQUIREMENTS FOR A CONSERVED PROBABILITY REAL ?ELDS SUPER-CONDUCTIVITY DUALITY, POINT CHARGES AND MONOPOLES THE NON-RELATIVISTIC LIMIT PARTICLES AND ANTI-PARTICLES KLEIN–GORDON ?ELD 13 2 1 THE FREE SCALAR ?ELD 13 2 2 NON-RELATIVISTIC LIMIT OF GF(X, X?) DIRAC ?ELD 13 3 1 THE FREE DIRAC ?ELD 13 3 2 THE DIRAC GREEN FUNCTION 13 3 3 SPINOR ELECTRODYNAMICS THERMAL AND EUCLIDEAN GREEN FUNCTIONS ENERGY CONSERVATION RESIDUAL CURVATURE AND CONSTRAINTS UNI?ED KINEMATICS AND DYNAMICS CLASSICAL HAMILTONIAN PARTICLE DYNAMICS 14 1 1 HAMILTON’S EQUATIONS OF MOTION 14 1 2 SYMMETRY AND CONSERVATION 14 1 3 SYMPLECTIC TRANSFORMATIONS 14 1 4 POISSON BRACKETS 14 1 5 GENERAL CANONICAL TRANSFORMATIONS 14 1 6 VARIATIONS OF DYNAMICAL VARIABLES AND POISSON BRACKETS 14 1 7 DERIVATION OF GENERATORS FROM THE ACTION 14 1 8 CONJUGATE VARIABLES AND DYNAMICAL COMPLETENESS 14 1 9 THE JACOBI IDENTITY AND GROUP ALGEBRA CLASSICAL LAGRANGIAN ?ELD DYNAMICS 14 2 1 SPACETIME CONTINUUM 14 2 2 POISSON BRACKETS OF ?ELDS CLASSICAL STATISTICAL MECHANICS 14 3 1 ENSEMBLES AND ERGODICITY 14 3 2 EXPECTATION VALUES AND CORRELATIONS 14 3 3 LIOUVILLE’S THEOREM 14 3 4 AVERAGED DYNAMICAL VARIATIONS QUANTUM MECHANICS 14 4 1 NON-RELATIVISTIC QUANTUM MECHANICS IN TERMS OF GROUPS AND OPERATORS 14 4 2 QUANTUM MECHANICAL ACTION PRINCIPLE 14 4 3 RELATIVISTIC QUANTUM MECHANICS CANONICALLY COMPLETE THEORIES EPILOGUE: QUANTUM ?ELD THEORY CLASSICAL LOOSE ENDS QUANTUM ACTION PRINCIPLE 15 2 1 OPERATOR VARIATIONS 15 2 2 EXAMPLE: OPERATOR EQUATIONS OF MOTION PATH INTEGRAL FORMULATION POSTSCRIPT PART 3: REFERENCE: A COMPENDIUM OF ?ELDS GALLERY OF DE?NITIONS UNITS CONSTANTS ENGINEERING DIMENSIONS ORDERS OF MAGNITUDE 16 4 1 SIZES 16 4 2 DENSITIES AND PRESSURE 16 4 3 TEMPERATURES 16 4 4 ENERGIES 16 4 5 WAVELENGTHS 16 4 6 VELOCITIES 16 4 7 ELECTRIC ?ELDS 16 4 8 MAGNETIC ?ELDS 16 4 9 CURRENTS SUMMATION CONVENTION SYMBOLS AND SIGNS 16 6 1 BASIS NOTATION 16 6 2 VOLUME ELEMENTS 16 6 3 SYMMETRICAL AND ANTI-SYMMETRICAL COMBINATIONS 16 6 4 DERIVATIVES 16 6 5 MOMENTA 16 6 6 POSITION, VELOCITY AND ACCELERATION LIMITS THE SCHR¨ODINGER ?ELD THE ACTION FIELD EQUATIONS AND CONTINUITY FREE-?ELD SOLUTIONS EXPRESSION FOR THE GREEN FUNCTION FORMAL SOLUTION BY GREEN FUNCTIONS CONSERVED NORM AND PROBABILITY ENERGY–MOMENTUM TENSOR THE REAL KLEIN–GORDON ?ELD REALITY OF SOLUTIONS NORMALIZATION ALL THE GREEN FUNCTIONS THE ENERGY–MOMENTUM TENSOR THE COMPLEX KLEIN–GORDON ?ELD FORMULATION AS A TWO-COMPONENT REAL ?ELD THE DIRAC ?ELD THE ? -MATRICES 20 2 1 EXAMPLE: N + 1 = 20 2 2 EXAMPLE: N + 1 = TRANSFORMATION PROPERTIES OF THE DIRAC EQUATION 20 3 1 ROTATIONS 20 3 2 BOOSTS 20 3 3 PARITY AND TIME REVERSAL 20 3 4 CHARGE CONJUGATION CHIRALITY IN 3 + 1 DIMENSIONS FIELD CONTINUITY FREE-?ELD SOLUTIONS IN N = INVARIANT NORMALIZATION IN P-SPACE 20 10 EXPRESSIONS FOR THE GREEN FUNCTIONS 20 11 THE ENERGY–MOMENTUM TENSOR 20 12 SPINOR ELECTRODYNAMICS THE MAXWELL RADIATION ?ELD CHARGES IN A VACUUM 21 1 1 THE ACTION 21 1 2 FIELD EQUATIONS AND CONTINUITY 21 1 3 THE JACOBI–BIANCHI IDENTITY 21 1 4 FORMAL SOLUTION BY GREEN FUNCTIONS 21 1 5 LORENTZ GAUGE 21 1 6 COULOMB/RADIATION GAUGE 21 1 7 RETARDED GREEN FUNCTION IN N = 21 1 8 THE ENERGY–MOMENTUM TENSOR EFFECTIVE THEORY OF DIELECTRIC AND MAGNETIC MEDIA 21 2 1 THE MAXWELL ACTION AND HAMILTONIAN IN A MEDIUM 21 2 2 FIELD EQUATIONS AND CONTINUITY 21 2 3 REINSTATING COVARIANCE WITH C ?C/N 21 2 4 GREEN FUNCTION THE MASSIVE PROCA ?ELD ACTION AND ?ELD EQUATIONS NON-ABELIAN ?ELDS LIE GROUPS AND ALGEBRAS CONSTRUCTION EQUATIONS OF MOTION AND CONTINUITY MULTIPLE REPRESENTATIONS THE ADJOINT REPRESENTATION COMMONLY USED GENERATORS 23 8 1 SU(2) HERMITIAN FUNDAMENTAL REPRESENTATION 23 8 2 SU(2) HERMITIAN ADJOINT REPRESENTATION 23 8 3 SU(3) HERMITIAN FUNDAMENTAL REPRESENTATION 23 8 4 SU(3) HERMITIAN ADJOINT REPRESENTATION