Many-Body Problems and Quantum Field Theory : An Introduction

by Martin, Philippe A.; Rothen, Francois; Goldfarb, Steven

• ISBN13: 9783540411536
• ISBN10: 3540411534
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: Springer Verlag

Summary

The book gives an introduction to the concepts and methods of many-body problems and quantum fields for graduate students and researchers. The formalism is developed in close conjunction with the description of a number of physical systems: cohesion and dielectric properties of the electron gas, superconductivity, superfluidity, nuclear matter and nucleon pairing, matter and radiation, interaction of fields by particle exchange and mass generation. Emphasis is put on analogies between the various systems rather than on advanced or specialized aspects, with the purpose of illustrating common ideas in different domains of physics. The exposition is self-contained and displays in a coherent way all details of derivations starting from a basic knowledge of quantum mechanics and classical electromagnetism.

Table of Contents

Classical Fields and Their Associated Particles	p. 3
Introduction	p. 3
The Quantum Harmonic Oscillator	p. 6
Review of Properties	p. 6
Coherent States	p. 10
The Forced Oscillator	p. 15
Normal Order	p. 17
The Electromagnetic Field and the Photon	p. 19
Maxwell's Equations	p. 19
Gauge Transformations	p. 20
Decomposition of the Field into Longitudinal and Transverse Components	p. 22
Hamiltonian of the Interaction of Radiation with Non-Relativistic Matter	p. 24
Fourier Analysis of the Classical Free Field	p. 26
Photons and Electromagnetic Waves	p. 30
The Elastic Field and the Phonon	p. 33
Elastic Waves and Elastic Energy	p. 33
Elastic Waves and Energy of an Isotropic Solid	p. 35
Fourier Analysis of Elastic Waves	p. 39
An Ensemble of Phonons and Classical Elastic Waves	p. 41
The Classical Linear Chain	p. 42
The Linear Quantum Chain	p. 46
Exercises	p. 49
Fermions and Bosons	p. 51
The Principle of Symmetrization	p. 51
Identical Particles	p. 51
One-Particle States	p. 52
Periodic Boundary Conditions and the Thermodynamic Limit	p. 54
n-Particle States	p. 58
Symmetrization	p. 60
Symmetry of Composite Particles	p. 64
Occupation Number Representation	p. 65
Degenerate Gases	p. 68
The Ground State of n Bosons	p. 68
The Ground State of n Fermions	p. 75
Stability of Matter	p. 79
Nucleo-Electronic Plasma at High Density	p. 82
Fermions and Gravitation	p. 84
Exercises	p. 86
Systems with Variable Particle Number	p. 89
Introduction	p. 89
Formalism of the Second Quantization	p. 90
Fock Space	p. 90
Creation and Annihilation Operators	p. 92
States of Fock Space	p. 96
Normal Order	p. 99
One-Body Operators	p. 101
Free Evolution and Symmetries	p. 104
Two-Body Operators	p. 108
Reduced Density Matrices and Correlations	p. 113
Correlations in Free Fermi and Bose Gases	p. 118
Quantum Physics and the Concept of a Perfect Gas	p. 121
Exercises	p. 123
Electron Gas	p. 127
The Hartree-Fock Method	p. 127
The Variational Principle	p. 127
The Hartree-Fock Equations	p. 129
Electron Gas in the Hartree-Fock Approximation	p. 133
Electron Gas and Its Hamiltonian	p. 133
The Hartree-Fock Energy	p. 137
The Dielectric Function	p. 142
Screening and the Plasmon	p. 142
Response to an External Charge	p. 145
Evolution of a Charge Fluctuation	p. 148
The RPA Dielectric Function	p. 152
Exercises	p. 157
Fermion Pairing and Superconductivity	p. 161
Does There Exist an Analogue of the Bose Condensation for Fermions?	p. 161
The Phenomenology of Superconductivity	p. 163
Experimental Facts	p. 163
The Phenomenological Approach	p. 165
Macroscopic Quantum Fluids	p. 169
Existence of the Energy Gap	p. 173
BCS Theory	p. 174
The Effective Interaction Between Electrons	p. 174
Application of the Variational Method to Superconductivity	p. 176
Sign Ambiguity	p. 179
Variational Class of BCS States	p. 180
How to Calculate with a BCS State	p. 182
Search for a Minimum-Energy State	p. 186
The Energy Gap	p. 190
Spatial Extension of a Cooper Pair	p. 192
Particle Number and Phase in Superconductivity	p. 194
Is it Necessary to Fix the Particle Number or the Phase?	p. 194
Analogy with Statistical Physics	p. 195
High-Tc Superconductivity	p. 197
Exercises	p. 199
Nucleon Pairing and the Structure of the Nucleus	p. 201
Introduction	p. 201
A Broad Outline of the Nuclear Structure	p. 202
Nuclear Forces	p. 202
The Liquid Drop Model	p. 203
Pairing Energy	p. 206
Shell Energy	p. 207
The Shell Model	p. 209
The Average Potential of the Shell Model	p. 209
Magic Numbers	p. 210
Pairing of the Nucleons	p. 213
Necessity of the Model	p. 213
The Interaction Responsible for the Pairing	p. 215
The Variational Method and the State of Paired Nucleons	p. 217
Determination of the Ground State of a Paired Nucleus	p. 220
Ground and Excited States of a Spherical Paired Nucleus	p. 221
The Spectra of Paired Nuclei	p. 224
Total Angular Momentum of Spherical Even-Even Nuclei	p. 226
Relation Between Pairing and Superfluidity	p. 228
How Can a Nucleus Be a Superfluid?	p. 228
An Excited Nucleus in a Rotational State	p. 229
Moment of Inertia of a Deformed Nucleus	p. 230
Exercises	p. 232
The Superfluidity of Liquid Helium	p. 233
Experimental Facts	p. 233
Phase Diagram of 42He	p. 233
Properties of the Superfluid Phase of He II	p. 235
Quantum Liquid and the Two-Fluid Model	p. 238
The Superfluid Phase and Quantum Liquid	p. 238
Dissipation in a Superfluid	p. 239
Second Sound	p. 244
The Energy Spectrum of He II	p. 248
Excitations of He II	p. 248
Non-Viscous Flow Through a Capillary	p. 252
Imperfect Bose Gas	p. 254
Bogolioubov's Approximation and Transformation	p. 254
Bose Gas or Liquid?	p. 258
Superfluidity of the Light Isotope 3He	p. 260
A Fermi Liquid	p. 260
Superfluidity of 3He	p. 261
Exercises	p. 262
Quantum Fields	p. 265
Introduction	p. 265
The Quantum-Electromagnetic Field	p. 267
The Free Field	p. 267
Canonical Variables	p. 271
Invariant Commutation Function and Microcausality	p. 272
Emission of Photons by a Classical Source	p. 275
Coherent States of Photons	p. 278
Emission and Absorption of Photons by an Atom	p. 280
Spontaneous Emission	p. 283
Equilibrium of Photons and Matter	p. 285
Photon Statistics	p. 286
Massive Scalar Field	p. 288
Neutral Scalar Field	p. 288
The Yukawa Potential	p. 291
Charged Scalar Field	p. 297
Spin and Statistics	p. 303
The Lagrangian Formalism	p. 304
The Gauge Invariance Principle and Field Interactions	p. 308
Mass Generation	p. 313
Electrons and Phonons	p. 319
Non-Relativistic Fermi Field	p. 319
The Quantum-Elastic Field	p. 323
Electron-Phonon Interactions	p. 325
Exercises	p. 329
Perturbative Methods in Field Theory	p. 333
Introduction	p. 333
The Green Functions	p. 335
Definition	p. 335
The Free-Particle Green Function	p. 339
Particle in an External Field	p. 342
Simplified Example: The Cooper Pair	p. 347
Perturbative Expansion of the Scattering Operator	p. 350
Time-Dependent Perturbation Theory	p. 350
The Scattering Operator	p. 354
Fermions and Bosons in Interaction	p. 358
The Wick Theorem for Chronological Products	p. 361
Chronological Contractions and Propagators	p. 364
Feynman Diagrams	p. 368
Applications	p. 374
Physical Interpretation of the Diagrams	p. 374
Electromagnetic Interactions: Compton Scattering	p. 379
Quantum Electrodynamics: Radiative Corrections	p. 386
Electron-Phonon Interactions	p. 390
Diagram Summation	p. 393
Exercises	p. 399
Perturbative Methods in Many-Body Problems	p. 401
General Properties	p. 401
The One-Body Green Function	p. 401
Perturbative Calculation of the Green Function	p. 405
Particle in an External Field and the Connected-Graph Theorem	p. 410
Particles in Interaction	p. 415
Approximation Schemes for the Electron Gas	p. 420
Hartree-Fock Approximation	p. 420
RPA Approximation	p. 422
Exercises	p. 428
Bibliography	p. 431
Index	p. 437
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