LONDON BEIJING HONG TSINGHUA Report and Review in Physics Vol2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China NEW JERSEY \Hp SINGAPORE World Scientific SHANGHAI KONG TAIPEI CHENNAI
Contents Preface vii 1 Basic Principles 1 2 Quantum Fields 3 21 Commutators 3 211 Identical particles principle 3 212 Projection operator 4 213 Creation and annihilation operators 4 214 Symmetrized and anti-symmetrized states 7 215 Commutators between creation and annihilation operators 10 22 The equations of motion 12 221 Field operators 12 222 The generator of time translation 15 223 Transition amplitude 16 224 Causality principle 17 225 Path integral formulas 17 226 Lagrangian and action 19 227 Covariance principle 20 23 Scalar field 21 231 Lagrangian 22 232 Klein-Gordon equation 22 233 Solutions of the Klein-Gordon equation 23 234 The commutators for creation and annihilation operators in p-space 24 ix
x Principles of Physics 235 The homogeneity of spacetime 27 24 The complex scalax field 32 241 Lagrangian of the complex boson field 32 242 Symmetry and conservation law 33 243 Charge conservation 36 25 Spinor fermions 36 251 Lagrangian 36 252 The generator of time translation 38 253 Dirac equation 38 254 Dirac matrices 39 255 Dirac-Pauli representation 40 256 Lorentz transformation for spinors 42 257 Covariance of the spinor fermion Lagrangian 44 258 Spatial reflection 45 259 Energy-momentum tensor and Hamiltonian operator 47 2510 Lorentz invariance 48 2511 Symmetric energy-momentum tensor 49 2512 Charge conservation 51 2513 Solutions of the free Dirac equation 52 2514 Hamiltonian operator in p-space 58 2515 Vacuum state 59 2516 Spin state 59 2517 Helicity 62 2518 Chirality 62 2519 Spin statistics relation 63 2520 Charge of spinor particles and antiparticles 2521 Representation in terms of the Weyl spinors 63 64 26 Vector bosons 65 261 Massive vector bosons 65 262 Massless vector bosons 79 27 Interaction 88 271 Lagrangian with the gauge invariance 89 272 Nonabelian gauge symmetry 90 3 Quantum Fields in the Riemann Spacetime 97 31 Lagrangian in the Riemann spacetime 97 32 Homogeneity of spacetime 99 33 Einstein equations 101
Contents xi 34 The generator of time translation 102 35 The relations of terms in the total action 105 36 Interactions 106 4 Symmetry Breaking 109 41 Scale invariance 109 411 Lagrangian with the scale invariance 109 412 Conserved current for the scale invariance 110 413 Scale invariance for the total Lagrangian 112 42 Ground state energy 113 43 Symmetry breaking 115 431 Spontaneous symmetry breaking 115 432 Continuous symmetry 116 44 The Higgs mechanism 118 45 Mass and interactions of particles 120 5 Perturbative Field Theory 123 51 Invariant commutation relations 123 511 Commutation functions 123 512 Microcausality 126 513 Propagator functions 127 52 n-point Green's function 130 521 Definition of n-point Green's function 130 522 Wick rotation 131 523 Generating functional 133 524 Momentum representation 134 525 Operator representation 134 526 Free scalar fields 135 527 Wick's theorem 136 528 Feynman rules 137 53 Interacting scalar field 139 531 Perturbation expansion 140 532 Perturbation <f>4 theory 142 533 Two-point function 145 534 Four-point function 149 54 Divergency in n-point functions 150 541 Divergency in integrations 150 542 Power counting 152
xii Principles of Physics 55 Dimensional regularization 153 551 Two-point function 153 552 Four-point function 155 56 Renormalization 157 57 Effective potential 160 169 6 From Quantum Field Theory to Quantum Mechanics 169 61 Non-relativistic limit of the Klein-Gordon equation 62 Non-relativistic limit of the Dirac equation 171 63 Spin-orbital coupling 173 64 The operator of time translation in quantum mechanics 175 65 Transformation of basis 177 66 One-body operators 181 67 Schrodinger equation 183 7 Electromagnetic Field 187 71 Current density 187 72 Classical limit 189 73 Maxwell equations 190 74 Gauge invariance 191 75 Radiation of electromagnetic waves 191 76 Poisson equation 193 77 Electrostatic energy of charges 194 78 Many-body operators 195 79 Potentials of charge particles in the classical limit 197 8 Quantum Mechanics 199 81 Equations of motion for operators in quantum mechanics 199 811 Ehrenfest's theorem 200 812 Constants of motion 201 813 Conservation of angular momentum 201 82 Elementary aspects of the Schrodinger equation 203 83 Newton's law 205 84 Lorentz force 207 85 Path integral formalism for quantum mechanics 208 851 Feymann's path integral for one-particle systems 208 852 Lagrangian function in quantum mechanics 212 853 Hamilton's equations 213
Contents xiii 854 Path integral formalism for multi-particle systems 214 86 Three representations 216 861 Schrodinger representation 216 862 Heisenberg representation 217 863 Interaction representation 218 87 S Matrix 220 88 de Broglie waves 221 89 Statistical interpretation of wave functions 223 810 Heisenberg uncertainty principle 224 811 Stationary states 228 9 Applications of Quantum Mechanics 231 91 Harmonic oscillator 231 911 Classical solution 231 912 Hamiltonian operator in terms of at and a 232 913 Eigenvalues and eigenstates 233 914 Wave functions 235 92 Schrodinger equation for a central potential 236 921 Schrodinger equation in the spherical coordinates 236 922 Separation of variables 236 923 Angular momentum operators 237 924 Eigenvalues of J2 and Jz 238 925 Matrix elements of angular momentum operators 241 926 Spherical harmonics 241 927 Radial equation 244 928 Hydrogen atom 244 10 Statistical Mechanics 251 101 Equi-probability principle and statistical distributions 251 256 102 Average of an observable A 254 1021 Statistical average 254 1022 Average using canonical distribution 254 1023 Average using grand canonical distribution 255 103 Functional integral representation of partition function 104 First law of thermodynamics 257 105 Second law of thermodynamics 259 1051 Entropy increase principle 259
xiv Principles of Physics 1052 Extensiveness of In Z 262 1053 Thermodynamic quantities in terms of partition function 263 1054 Kelvin formulation of the second law of thermodynamics 265 1055 Carnot theorem 266 1056 Clausius inequality 267 1057 Characteristic functions 268 1058 Maxwell relations 269 1059 Gibbs-Duhem relation 269 10510 Isothermal processes 270 10511 Derivatives of thermodynamic quantities 271 106 Third law of thermodynamics 272 107 Thermodynamic quantities expressed in terms of grand partition function 273 108 Relation between grand partition function and partition function 275 109 Systems with particle number changeable 276 1091 Thermodynamic relations for open systems 276 279 289 1092 Equilibrium conditions of two systems 277 1093 Phase equilibrium conditions 278 1010 Equilibrium distributions of nearly independent particle systems 279 10101 Derivations of the distribution functions of single particle from the macro-canonical distribution 10102 Partition function of independent particle systems 285 10103 About summations in calculations of independent particle system 287 1011 Fluctuations 288 10111 Absolute and relative fluctuations 288 10112 Fluctuations in systems of canonical ensemble 10113 Fluctuations in systems of grand canonical ensemble 290 1012 Classic statistical mechanics and quantum corrections 291 10121 Classic limit of statistical distribution functions 291 10122 Quantum corrections 296 10123 Equipartition theorem 298
Contents xv 11 Applications of Statistical Mechanics 301 111 Ideal gas 301 1111 Partition function for mass center motion 303 1112 Ideal gas of single-atom molecules 304 1113 Internal degrees of freedom 305 112 Weakly degenerate quantum gas 311 113 Bosegas 314 1131 Bose-Einstein condensation 314 1132 Thermodynamic properties of BEC 318 114 Photon gas 319 115 Fermi gas 322 12 General Relativity 329 121 Classical energy-momentum tensor 329 122 Equation of motion in the Riemann spacetime 332 123 Weak field limit 334 1231 Static weak field limit-newtonian gravitation 334 1232 Equation of motion in Newtonian approximation 337 1233 Harmonic coordinate 338 1234 Weak field approximation in the harmonic gauge 339 124 Spherical solutions for stars 343 1241 Spherically symmetric spacetime 343 1242 Einstein equations for static fluid 346 1243 The metric outside a star 348 1244 Interior structure of a star 348 1245 Structure of a Newtonian star 350 1246 Simple model for the interior structure of stars 351 1247 Pressure of relativistic Fermi gas 353 125 White dwarfs 356 126 Neutron Stars 359 1261 Normal solutions 359 1262 Solutions with void 361 Appendix A Tensors 365 Al Vectors 365 A 2 Higher rank tensors 366 A3 Metric tensor 368 A4 Flat spacetime 368
xvi Principles of Physics A5 Lorentz transformation 369 A51 Infinitesimal Lorentz transformation 369 A52 Finite Lorentz transformation 371 A6 Christoffel symbols 375 A7 Riemann spacetime 377 A8 Volume 379 A9 Riemann curvature tensor 381 A 10 Bianchi identities 382 A 11 Ricci tensor 383 A 12 Einstein tensor 383 Appendix B Functional Formula 385 Appendix C Gaussian Integrals 387 Cl Gaussian integrals 387 C2 functions 388 C3 Gaussian integrations with source 389 Appendix D Grassmann Algebra 391 Appendix E Euclidean Representation 397 Appendix F Some Useful Formulas 399 Appendix G Jacobian 403 Appendix H Geodesic Equation 405 Bibliography 409 Index 413