PRINCIPLES OF PHYSICS. \Hp. Ni Jun TSINGHUA. Physics. From Quantum Field Theory. to Classical Mechanics. World Scientific. Vol.2. Report and Review in

Transcription

1 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

2 Contents Preface vii 1 Basic Principles 1 2 Quantum Fields 3 21 Commutators Identical particles principle Projection operator Creation and annihilation operators Symmetrized and anti-symmetrized states Commutators between creation and annihilation operators The equations of motion Field operators The generator of time translation Transition amplitude Causality principle Path integral formulas Lagrangian and action Covariance principle Scalar field Lagrangian Klein-Gordon equation Solutions of the Klein-Gordon equation The commutators for creation and annihilation operators in p-space 24 ix

3 x Principles of Physics 235 The homogeneity of spacetime The complex scalax field Lagrangian of the complex boson field Symmetry and conservation law Charge conservation Spinor fermions Lagrangian The generator of time translation Dirac equation Dirac matrices Dirac-Pauli representation Lorentz transformation for spinors Covariance of the spinor fermion Lagrangian Spatial reflection Energy-momentum tensor and Hamiltonian operator Lorentz invariance Symmetric energy-momentum tensor Charge conservation Solutions of the free Dirac equation Hamiltonian operator in p-space Vacuum state Spin state Helicity Chirality Spin statistics relation Charge of spinor particles and antiparticles 2521 Representation in terms of the Weyl spinors Vector bosons Massive vector bosons Massless vector bosons Interaction Lagrangian with the gauge invariance Nonabelian gauge symmetry 90 3 Quantum Fields in the Riemann Spacetime Lagrangian in the Riemann spacetime Homogeneity of spacetime Einstein equations 101

4 Contents xi 34 The generator of time translation The relations of terms in the total action Interactions Symmetry Breaking Scale invariance Lagrangian with the scale invariance Conserved current for the scale invariance Scale invariance for the total Lagrangian Ground state energy Symmetry breaking Spontaneous symmetry breaking Continuous symmetry The Higgs mechanism Mass and interactions of particles Perturbative Field Theory Invariant commutation relations Commutation functions Microcausality Propagator functions n-point Green's function Definition of n-point Green's function Wick rotation Generating functional Momentum representation Operator representation Free scalar fields Wick's theorem Feynman rules Interacting scalar field Perturbation expansion Perturbation <f>4 theory Two-point function Four-point function Divergency in n-point functions Divergency in integrations Power counting 152

5 xii Principles of Physics 55 Dimensional regularization Two-point function Four-point function Renormalization Effective potential From Quantum Field Theory to Quantum Mechanics Non-relativistic limit of the Klein-Gordon equation 62 Non-relativistic limit of the Dirac equation Spin-orbital coupling The operator of time translation in quantum mechanics Transformation of basis One-body operators Schrodinger equation Electromagnetic Field Current density Classical limit Maxwell equations Gauge invariance Radiation of electromagnetic waves Poisson equation Electrostatic energy of charges Many-body operators Potentials of charge particles in the classical limit Quantum Mechanics Equations of motion for operators in quantum mechanics Ehrenfest's theorem Constants of motion Conservation of angular momentum Elementary aspects of the Schrodinger equation Newton's law Lorentz force Path integral formalism for quantum mechanics Feymann's path integral for one-particle systems Lagrangian function in quantum mechanics Hamilton's equations 213

6 Contents xiii 854 Path integral formalism for multi-particle systems Three representations Schrodinger representation Heisenberg representation Interaction representation S Matrix de Broglie waves Statistical interpretation of wave functions Heisenberg uncertainty principle Stationary states Applications of Quantum Mechanics Harmonic oscillator Classical solution Hamiltonian operator in terms of at and a Eigenvalues and eigenstates Wave functions Schrodinger equation for a central potential Schrodinger equation in the spherical coordinates Separation of variables Angular momentum operators Eigenvalues of J2 and Jz Matrix elements of angular momentum operators Spherical harmonics Radial equation Hydrogen atom Statistical Mechanics Equi-probability principle and statistical distributions Average of an observable A Statistical average Average using canonical distribution Average using grand canonical distribution Functional integral representation of partition function 104 First law of thermodynamics Second law of thermodynamics Entropy increase principle 259

7 xiv Principles of Physics 1052 Extensiveness of In Z Thermodynamic quantities in terms of partition function Kelvin formulation of the second law of thermodynamics Carnot theorem Clausius inequality Characteristic functions Maxwell relations Gibbs-Duhem relation Isothermal processes Derivatives of thermodynamic quantities Third law of thermodynamics Thermodynamic quantities expressed in terms of grand partition function Relation between grand partition function and partition function Systems with particle number changeable Thermodynamic relations for open systems Equilibrium conditions of two systems Phase equilibrium conditions Equilibrium distributions of nearly independent particle systems Derivations of the distribution functions of single particle from the macro-canonical distribution Partition function of independent particle systems About summations in calculations of independent particle system Fluctuations Absolute and relative fluctuations Fluctuations in systems of canonical ensemble Fluctuations in systems of grand canonical ensemble Classic statistical mechanics and quantum corrections Classic limit of statistical distribution functions Quantum corrections Equipartition theorem 298

8 Contents xv 11 Applications of Statistical Mechanics Ideal gas Partition function for mass center motion Ideal gas of single-atom molecules Internal degrees of freedom Weakly degenerate quantum gas Bosegas Bose-Einstein condensation Thermodynamic properties of BEC Photon gas Fermi gas General Relativity Classical energy-momentum tensor Equation of motion in the Riemann spacetime Weak field limit Static weak field limit-newtonian gravitation Equation of motion in Newtonian approximation Harmonic coordinate Weak field approximation in the harmonic gauge Spherical solutions for stars Spherically symmetric spacetime Einstein equations for static fluid The metric outside a star Interior structure of a star Structure of a Newtonian star Simple model for the interior structure of stars Pressure of relativistic Fermi gas White dwarfs Neutron Stars Normal solutions 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

9 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