INTRODUCTION TO QUANTUM FIELDS IN CLASSICAL BACKGROUNDS

By X. FILIU

CONTENTS I CANONICAL QUANTIZATION OVERVIEW A TASTE OF QUANTUM ?ELDS THE HARMONIC OSCILLATOR AND ITS VACUUM STATE FREE QUANTUM ?ELDS AND VACUUM ZERO-POINT ENERGY QUANTUM ?UCTUATIONS IN THE VACUUM STATE AMPLITUDE OF ?UCTUATIONS OBSERVABLE EFFECTS OF VACUUM ?UCTUATIONS PARTICLE INTERPRETATION OF QUANTUM ?ELDS QUANTUM ?ELD THEORY IN CLASSICAL BACKGROUNDS EXAMPLES OF PARTICLE CREATION TIME-DEPENDENT OSCILLATOR THE SCHWINGER EFFECT PRODUCTION OF PARTICLES BY GRAVITY THE UNRUH EFFECT REMINDER: CLASSICAL AND QUANTUM MECHANICS LAGRANGIAN FORMALISM THE ACTION PRINCIPLE EQUATIONS OF MOTION FUNCTIONAL DERIVATIVES HAMILTONIAN FORMALISM THE HAMILTON EQUATIONS OF MOTION QUANTIZATION OF HAMILTONIAN SYSTEMS DIRAC NOTATION AND HILBERT SPACES EVOLUTION IN QUANTUM THEORY QUANTIZING A DRIVEN HARMONIC OSCILLATOR CLASSICAL OSCILLATOR UNDER FORCE QUANTIZATION THE “IN” AND “OUT” REGIONS EXCITED STATES RELATIONSHIP BETWEEN “IN” AND “OUT” STATES CALCULATIONS OF MATRIX ELEMENTS FROM HARMONIC OSCILLATORS TO ?ELDS QUANTIZATION OF FREE ?ELDS FROM OSCILLATORS TO ?ELDS QUANTIZING ?ELDS IN ?AT SPACETIME A ?RST LOOK AT MODE EXPANSIONS THE SCHRÖDINGER EQUATION FOR A QUANTUM ?ELD OVERVIEW OF CLASSICAL ?ELD THEORY CHOOSING THE ACTION FUNCTIONAL REQUIREMENTS FOR THE ACTION FUNCTIONAL EQUATIONS OF MOTION FOR ?ELDS REAL SCALAR ?ELD GAUGE SYMMETRY AND GAUGE ?ELDS THE U(1) GAUGE SYMMETRY ACTION FOR GAUGE ?ELDS ENERGY-MOMENTUM TENSOR FOR ?ELDS CONSERVATION OF THE EMT QUANTUM ?ELDS IN EXPANDING UNIVERSE SCALAR ?ELD IN FRW UNIVERSE MODE FUNCTIONS MODE EXPANSIONS QUANTIZATION OF SCALAR ?ELD THE VACUUM STATE AND PARTICLE STATES BOGOLYUBOV TRANSFORMATIONS MEAN PARTICLE NUMBER CHOICE OF THE VACUUM STATE THE INSTANTANEOUS LOWEST-ENERGY STATE THE MEANING OF VACUUM VACUUM AT SHORT DISTANCES ADIABATIC VACUUM A QUANTUM-MECHANICAL ANALOGY QUANTUM ?ELDS IN DE SITTER SPACETIME AMPLITUDE OF QUANTUM ?UCTUATIONS CORRELATION FUNCTIONS FLUCTUATIONS OF AVERAGED ?ELDS FLUCTUATIONS IN VACUUM AND NONVACUUM STATES A WORKED-OUT EXAMPLE FIELD QUANTIZATION IN DE SITTER SPACETIME GEOMETRY OF DE SITTER SPACETIME QUANTIZATION OF SCALAR ?ELDS THE BUNCH-DAVIES VACUUM EVOLUTION OF ?UCTUATIONS RINDLER SPACETIME UNIFORMLY ACCELERATED MOTION COORDINATES IN THE PROPER FRAME METRIC OF THE RINDLER SPACETIME QUANTUM ?ELDS IN THE RINDLER SPACETIME LIGHTCONE MODE EXPANSIONS THE BOGOLYUBOV TRANSFORMATIONS DENSITY OF PARTICLES THE UNRUH TEMPERATURE THE HAWKING EFFECT THERMODYNAMICS OF BLACK HOLES THE HAWKING RADIATION SCALAR ?ELD IN A BH SPACETIME THE KRUSKAL COORDINATES FIELD QUANTIZATION CHOICE OF VACUUM THE HAWKING TEMPERATURE THE HAWKING EFFECT IN 3+1 DIMENSIONS REMARKS ON OTHER DERIVATIONS THERMODYNAMICS OF BLACK HOLES EVAPORATION OF BLACK HOLES LAWS OF BH THERMODYNAMICS BLACK HOLES IN HEAT RESERVOIRS 10 THE CASIMIR EFFECT 101 VACUUM ENERGY BETWEEN PLATES 102 REGULARIZATION AND RENORMALIZATION 103 RENORMALIZATION USING RIEMANN’S ZETA FUNCTION II PATH INTEGRAL METHODS 11 PATH INTEGRAL QUANTIZATION 111 EVOLUTION OPERATORS PROPAGATORS 112 PROPAGATOR AS A PATH INTEGRAL 113 LAGRANGIAN PATH INTEGRAL 12 EFFECTIVE ACTION 121 GREEN’S FUNCTIONS OF A HARMONIC OSCILLATOR 1211 GREEN’S FUNCTIONS 1212 WICK ROTATION EUCLIDEAN OSCILLATOR 122 INTRODUCING EFFECTIVE ACTION 1221 EUCLIDEAN PATH INTEGRALS 1222 DE?NITION OF EFFECTIVE ACTION 1223 THE EFFECTIVE ACTION “RECIPE” 123 BACKREACTION 1231 GAUGE COUPLING 1232 COUPLING TO GRAVITY 1233 POLARIZATION OF VACUUM AND SEMICLASSICAL GRAVITY 13 FUNCTIONAL DETERMINANTS AND HEAT KERNELS 131 EUCLIDEAN ACTION FOR ?ELDS 1311 TRANSITION TO EUCLIDEAN METRIC 1312 EUCLIDEAN ACTION FOR GRAVITY 132 EFFECTIVE ACTION AS A FUNCTIONAL DETERMINANT 133 ZETA FUNCTIONS AND HEAT KERNELS 1331 RENORMALIZATION USING ZETA FUNCTIONS 1332 HEAT KERNELS 1333 THE ZETA FUNCTION “RECIPE” 14 CALCULATION OF HEAT KERNEL 141 PERTURBATIVE ANSATZ FOR THE HEAT KERNEL 142 TRACE OF THE HEAT KERNEL 143 THE SEELEY-DEWITT EXPANSION 15 RESULTS FROM EFFECTIVE ACTION 151 RENORMALIZATION OF EFFECTIVE ACTION 1511 LEADING DIVERGENCES 1512 RENORMALIZATION OF CONSTANTS 152 FINITE TERMS IN THE EFFECTIVE ACTION 1521 NONLOCAL TERMS 1522 EMT FROM THE POLYAKOV ACTION 153 CONFORMAL ANOMALY