Many-Body Problems and Quantum Field Theory

Transcription

1 Philippe A. Martin Francois Rothen Many-Body Problems and Quantum Field Theory An Introduction Translated by Steven Goldfarb, Andrew Jordan and Samuel Leach Second Edition With 102 Figures, 7 Tables and 23 Exercises Springer

2 1. Classical Fields and Their Associated Particles Introduction The Quantum Harmonic Oscillator Review of Properties Coherent States The Forced Oscillator Normal Ordering The Electromagnetic Field and the Photon Maxwell's Equations Gauge Transformations Decomposition of the Field into Longitudinal and Transverse Components Hamiltonian of the Interaction of Radiation with Non-Relativistic Matter Fourier Analysis of the Classical Free Field Photons and Electromagnetic Waves The Elastic Field and the Phonon Elastic Waves and Elastic Energy Elastic Waves and Energy of an Isotropic Solid Fourier Analysis of Elastic Waves An Ensemble of Phonons and Classical Elastic Waves The Classical Linear Chain The Quantum Linear Chain 47 Exercises Fermions and Bosons The Principle of Symmetrization Identical Particles One-Particle States Periodic Boundary Conditions and the Thermodynamic Limit n-particle States Symmetrization 60

3 XII Symmetry of Composite Particles Occupation Number Representation Degenerate Gases The Ground State of n Bosons The Ground State of n Fermions Stability of Matter Nucleo-Electronic Plasma at High Density Fermions and Gravitation 83 Exercises Systems with Variable Particle Number Introduction Formalism of the Second Quantization Fock Space Creation and Annihilation Operators States of the Fock Space Normal Order One-Body Operators Free Evolution and Symmetries Two-Body Operators Reduced Density Matrices and Correlations Correlations in Free Fermi and Bose Gases Quantum Physics and the Concept of a Perfect Gas 121 Exercises Electron Gas The Hartree-Fock Method The Variational Principle The Hartree-Fock Equations Electron Gas in the Hartree-Fock Approximation Electron Gas and Its Hamiltonian The Hartree-Fock Energy The Dielectric Function Screening and the Plasmon Response to an External Charge Evolution of a Charge Fluctuation The RPA Dielectric Function 152 Exercises Fermion Pairing and Superconductivity Does There Exist an Analogue of the Bose Condensation for Fermions? The Phenomenology of Superconductivity Experimental Facts The Phenomenological Approach 163

4 XIII Macroscopic Quantum Fluids Existence of the Energy Gap BCS Theory The Effective Interaction Between Electrons Application of the Variational Method to Superconductivity Sign Ambiguity Variational Class of BCS States How to Calculate with a BCS State Search for a Minimum-Energy State The Energy Gap Spatial Extension of a Cooper Pair Particle Number and Phase in Superconductivity Is it Necessary to Fix the Particle Number or the Phase? Analogy with Statistical Physics High-r c Superconductivity 194 Exercises Nucleon Pairing and the Structure of the Nucleus Introduction A Broad Outline of the Nuclear Structure The Short Range of Nuclear Forces The Liquid Drop Model More About Nuclear Forces The Shell Model of the Nucleus The One-Particle Potential Inside the Nucleus Interpretation of the Magic Numbers Distribution of the Energy Levels in the Shell Model Pairing of the Nucleons Nature of the Residual Interaction Pairing Interaction: Further Experimental Facts The Interaction Responsible for the Pairing Applying the BCS Theory to the Nucleus Which Nuclear Properties are Affected by Nucleon Pairing? Superconductivity, Superfluidity and Nuclei An Excited Nucleus in a Rotational State Moment of Inertia of a Deformed Nucleus 222 Exercises 224

5 The Superfluidity of Liquid Helium Experimental Facts Phase Diagram of He Properties of the Superfluid Phase of He II Quantum Liquid and the Two-Fluid Model The Superfluid Phase and Quantum Liquid Dissipation in a Superfluid Second Sound The Energy Spectrum of He II Excitations of He II Non-Viscous Flow Through a Capillary Imperfect Bose Gas Bogoliubov's Approximation and Transformation Bose Gas or Liquid? Superfluidity of the Light Isotope 3 He A Fermi Liquid Superfluidity of 3 He 254 Exercises 255 Quantum Fields Introduction The Quantum-Electromagnetic Field The Free Field Canonical Variables Invariant Commutation Function and Microcausality Emission of Photons by a Classical Source Coherent States of Photons Emission and Absorption of Photons by an Atom Spontaneous Emission Photons and Matter in Equilibrium Photon Statistics Massive Scalar Field Neutral Scalar Field The Yukawa Potential Charged Scalar Field Spin and Statistics The Lagrangian Formalism The Gauge Invariance Principle and Field Interactions Mass Generation Electrons and Phonons Non-Relativistic Fermi Field The Quantum-Elastic Field Electron-Phonon Interactions 317 Exercises 321

6 XV 9. Perturbative Methods in Field Theory Introduction The Green Functions Definition The Free-Particle Green Function Particle in an External Field Simplified Example: The Cooper Pair Perturbative Expansion of the Scattering Operator Time-Dependent Perturbation Theory The Scattering Operator Fermions and Bosons in Interaction The Wick Theorem for Time-Ordered Products Time-Ordered Contractions and Propagators Feynman Diagrams Applications Physical Interpretation of the Diagrams Electromagnetic Interactions: Compton Scattering Quantum Electrodynamics: Radiative Corrections Electron-Phonon Interactions Diagram Summation 384 Exercises Perturbative Methods in Many-Body Problems General Properties The One-Body Green Function Perturbative Calculation of the Green Function Particle in an External Field and the Connected-Graph Theorem Interacting Particles Approximation Schemes for the Electron Gas Hartree-Fock Approximation RPA Approximation 414 Exercises 420 Bibliography 423 Index 431