ELECTRODYNAMICS, QUANTUM

By Y. DAINM

CONTENTS INTRODUCTION REFERENCES PATH INTEGRALS IN QUANTUM AND STATISTICAL MECHANICS SUMMING OVER ALL PATHS RECALLING QUANTUM MECHANICS FEYNMAN–KAC FORMULA EUCLIDEAN PATH INTEGRAL QUANTUM MECHANICS IN IMAGINARY TIME IMAGINARY-TIME PATH INTEGRAL PATH INTEGRAL IN QUANTUM STATISTICS THERMAL CORRELATION FUNCTIONS THE HARMONIC OSCILLATOR PROBLEMS HIGH-DIMENSIONAL INTEGRALS NUMERICAL ALGORITHMS NEWTON–COTES INTEGRATION METHOD MONTE CARLO INTEGRATION HIT-OR-MISS MONTE CARLO METHOD AND BINOMIAL DISTRIBUTION SUM OF RANDOM NUMBERS AND GAUSSIAN DISTRIBUTION IMPORTANCE SAMPLING SOME BASIC FACTS IN PROBABILITY THEORY PROGRAMS FOR CHAP MONTE CARLO SIMULATIONS IN QUANTUM MECHANICS MARKOV CHAINS FIXED POINTS OF MARKOV CHAINS DETAILED BALANCE ACCEPTANCE RATE METROPOLIS ALGORITHM HEAT BATH ALGORITHM THE ANHARMONIC OSCILLATOR SIMULATING THE ANHARMONIC OSCILLATOR HYBRID-MONTE CARLO ALGORITHM IMPLEMENTING THE HMC-ALGORITHM HMC-ALGORITHM FOR HARMONIC OSCILLATOR SCALAR FIELDS AT ZERO AND FINITE TEMPERATURE QUANTIZATION SCALAR FIELD THEORY AT FINITE TEMPERATURE FREE SCALAR FIELD SCHWINGER FUNCTION AND EFFECTIVE POTENTIAL THE LEGENDRE–FRENCHEL TRANSFORMATION SCALAR FIELD ON A SPACE-TIME LATTICE RANDOM WALK REPRESENTATION OF GREEN’S FUNCTION THERE IS NO LEIBNIZ RULE ON THE LATTICE CLASSICAL SPIN MODELS: AN INTRODUCTION SIMPLE SPIN MODELS FOR (ANTI)FERROMAGNETS ISING MODEL ISING-TYPE SPIN SYSTEMS STANDARD POTTS MODELS THE ZQ MODEL (PLANAR POTTS MODEL, CLOCK MODEL) THE U(1) MODEL O(N) MODELS INTERACTING CONTINUOUS SPINS SPIN SYSTEMS IN THERMAL EQUILIBRIUM VARIATIONAL PRINCIPLES PRINCIPLE FOR GIBBS STATE AND FREE ENERGY FIXED AVERAGE FIELD PROGRAMS FOR THE SIMULATION OF THE ISING CHAIN MEAN FIELD APPROXIMATION APPROXIMATION FOR GENERAL LATTICE MODELS THE ISING MODEL AN ALTERNATIVE DERIVATION CRITICAL EXPONENTS ?,?,?,? SUSCEPTIBILITY MAGNETIZATION AS A FUNCTION OF TEMPERATURE SPECI?C HEAT MAGNETIZATION AS A FUNCTION OF THE MAGNETIC FIELD COMPARISON WITH EXACT AND NUMERICAL RESULTS MEAN-FIELD APPROXIMATION FOR STANDARD POTTS MODELS MEAN-FIELD APPROXIMATION FOR ZQ MODELS LANDAU THEORY AND ORNSTEIN–ZERNIKE EXTENSION CRITICAL EXPONENTS IN LANDAU THEORY TWO-POINT CORRELATION FUNCTION ANTIFERROMAGNETIC SYSTEMS MEAN-FIELD APPROXIMATION FOR LATTICE FIELD THEORIES ?4 AND ?6 SCALAR THEORIES PROGRAM FOR CHAP 7 7 10 PROBLEMS TRANSFER MATRICES, CORRELATION INEQUALITIES AND ROOTS OF PARTITION FUNCTIONS TRANSFER-MATRIX METHOD FOR THE ISING CHAIN TRANSFER MATRIX THE “HAMILTONIAN” THE ANTI-FERROMAGNETIC CHAIN POTTS CHAIN PERRON–FROBENIUS THEOREM THE GENERAL TRANSFER-MATRIX METHOD CONTINUOUS TARGET SPACES EUCLIDEAN QUANTUM MECHANICS REAL SCALAR FIELD CORRELATION INEQUALITIES ROOTS OF THE PARTITION FUNCTION LEE–YANG ZEROES OF ISING CHAIN GENERAL FERROMAGNETIC SYSTEMS HIGH-TEMPERATURE AND LOW-TEMPERATURE EXPANSIONS ISING CHAIN LOW TEMPERATURE HIGH TEMPERATURE HIGH-TEMPERATURE EXPANSIONS FOR ISING MODELS GENERAL RESULTS AND TWO-DIMENSIONAL MODEL THREE-DIMENSIONAL MODEL LOW-TEMPERATURE EXPANSION OF ISING MODELS FREE ENERGY AND MAGNETIZATION OF TWO-DIMENSIONAL MODEL IMPROVED SERIES STUDIES FOR ISING-TYPE MODELS HIGH-TEMPERATURE EXPANSIONS OF O(N) SIGMA MODELS EXPANSIONS OF PARTITION FUNCTION AND FREE ENERGY POLYMERS AND SELF-AVOIDING WALKS PEIERLS ARGUMENT AND DUALITY TRANSFORMATIONS 10 1 PEIERLS’ ARGUMENT 10 1 1 EXTENSION TO HIGHER DIMENSIONS 10 2 DUALITY TRANSFORMATION OF TWO-DIMENSIONAL ISING MODEL 10 2 1 AN ALGEBRAIC DERIVATION 10 2 2 TWO-POINT FUNCTION 10 2 3 POTTS MODELS 10 2 4 CURL AND DIVERGENCE ON A LATTICE 10 3 DUALITY TRANSFORMATION OF THREE-DIMENSIONAL ISING MODEL 10 3 1 LOCAL GAUGE TRANSFORMATIONS 10 4 DUALITY TRANSFORMATION OF THREE-DIMENSIONAL ZN GAUGE MODEL 10 4 1 WILSON LOOPS 10 4 2 DUALITY TRANSFORMATION OF U(1) GAUGE MODEL 10 5 DUALITY TRANSFORMATION OF FOUR-DIMENSIONAL ZN GAUGE MODEL 10 6 PROBLEMS RENORMALIZATION GROUP ON THE LATTICE 11 1 DECIMATION OF SPINS 11 1 1 ISING CHAIN 11 1 2 THE TWO-DIMENSIONAL ISING MODEL 11 2 FIXED POINTS 11 2 1 THE VICINITY OF A FIXED POINT 11 2 2 DERIVATION OF SCALING LAWS 11 3 BLOCK-SPIN TRANSFORMATION 11 4 CONTINUUM LIMIT OF NON-INTERACTING SCALAR FIELDS 11 4 1 CORRELATION LENGTH FOR INTERACTING SYSTEMS 11 5 CONTINUUM LIMIT OF SPIN MODELS 11 6 PROGRAMS FOR CHAP 11 11 7 PROBLEMS FUNCTIONAL RENORMALIZATION GROUP 12 1 SCALE-DEPENDENT FUNCTIONALS 12 2 DERIVATION OF THE FLOW EQUATION CONTENTS 12 3 FUNCTIONAL RENORMALIZATION APPLIED TO QUANTUM MECHANICS 12 3 1 PROJECTION ONTO POLYNOMIALS OF ORDER 12 12 3 2 CHANGING THE REGULATOR FUNCTION 12 3 3 SOLVING THE FLOW EQUATION FOR NON-CONVEX POTENTIALS 12 4 SCALAR FIELD THEORY 12 4 1 FIXED POINTS 12 4 2 CRITICAL EXPONENTS 12 5 LINEAR O(N) MODELS 12 5 1 LARGE N LIMIT 12 5 2 EXACT SOLUTION OF THE FLOW EQUATION 12 6 WAVE FUNCTION RENORMALIZATION 12 6 1 RG EQUATION FOR WAVE FUNCTION RENORMALIZATION 12 7 OUTLOOK 12 8 PROGRAMS FOR CHAP 12 12 9 PROBLEMS LATTICE GAUGE THEORIES 13 1 CONTINUUM GAUGE THEORIES 13 1 1 PARALLEL TRANSPORT 13 2 GAUGE-INVARIANT FORMULATION OF LATTICE HIGGS MODELS 13 2 1 WILSON ACTION OF PURE GAUGE THEORIES 13 2 2 WEAK AND STRONG COUPLING LIMITS OF HIGGS MODELS 13 3 MEAN FIELD APPROXIMATION 13 3 1 Z2 GAUGE MODEL 13 3 2 U(1) GAUGE THEORY 13 3 3 SU(N) GAUGE THEORIES 13 3 4 HIGGS MODEL 13 4 EXPECTED PHASE DIAGRAMS AT ZERO TEMPERATURE 13 5 ELITZUR’S THEOREM 13 5 1 PROOF FOR PURE Z2 GAUGE THEORY 13 5 2 GENERAL ARGUMENT 13 6 OBSERVABLES IN PURE GAUGE THEORIES 13 6 1 STRING TENSION 13 6 2 STRONG COUPLING EXPANSION FOR PURE GAUGE THEORIES 13 6 3 GLUEBALLS 13 7 GAUGE THEORIES AT FINITE TEMPERATURE 13 7 1 CENTER SYMMETRY 13 7 2 G2 GAUGE THEORY 13 8 PROBLEMS TWO-DIMENSIONAL LATTICE GAUGE THEORIES AND GROUP INTEGRALS 14 1 ABELIAN GAUGE THEORIES ON THE TORUS 14 1 1 Z2 GAUGE THEORY 14 1 2 U(1) GAUGE THEORY CONTENTS 14 2 NON-ABELIAN LATTICE GAUGE THEORIES ON THE 2D TORUS 14 2 1 PARTITION FUNCTION 14 2 2 CASIMIR SCALING OF POLYAKOV LOOPS 14 3 INVARIANT MEASURE AND IRREDUCIBLE REPRESENTATIONS 14 3 1 THE PETER–WEYL THEOREM 14 4 PROBLEMS FERMIONS ON A LATTICE 15 1 DIRAC EQUATION 15 1 1 COUPLING TO GAUGE FIELDS 15 2 GRASSMANN VARIABLES 15 2 1 GAUSSIAN INTEGRALS 15 2 2 PATH INTEGRAL FOR DIRAC THEORY 15 3 FERMION FIELDS ON A LATTICE 15 3 1 LATTICE DERIVATIVE 15 3 2 NAIVE FERMIONS ON THE LATTICE 15 3 3 WILSON FERMIONS 15 3 4 STAGGERED FERMIONS 15 4 NIELSEN–NINOMIYA THEOREM 15 5 GINSPARG–WILSON RELATION AND OVERLAP FERMIONS 15 5 1 OVERLAP FERMIONS 15 5 2 LOCALITY 15 6 YUKAWA MODELS ON THE LATTICE 15 6 1 HIGGS SECTOR OF STANDARD MODEL 15 6 2 SUPERSYMMETRIC YUKAWA MODELS 15 7 COUPLING TO LATTICE GAUGE FIELDS 15 7 1 FINITE TEMPERATURE AND DENSITY 15 8 PROGRAMS FOR CHAP 15 15 9 PROBLEMS