1 Second Quantisation1．1 Identical Particles1．2 The \"Continuous\" Fock Representation1．3 The \"Discrete\" Fock Representation1．4 Exercises1．5 Self-Examination Questions2 Many-Body Model Systems2．1 Crystal Electrons2．1．1 Non-interacting Bloch Electrons2．1．2 The Jellium Model2．1．3 The Hubbard Model2．1．4 Exercises2．2 Lattice Vibrations2．2．1 The Harmonic Approximation2．2．2 The Phonon Gas2．2．3 Exercises2．3 The Electron-Phonon Interaction2．3．1 The Hamiltonian2．3．2 The Effective Electron-Electron Interaction2．3．3 Exercises2．4 Spin Waves2．4．1 Classification of Magnetic Solids2．4．2 Model Concepts2．4．3 Magnons2．4．4 The Spin-Wave Approximation2．4．5 Exercises2．5 Self-Examination Questions3 Green's Funetions3．1 Preliminary Considerations3．1．1 Representations3．1．2 Linear-Response Theory3．1．3 The Magnetic Susceptibility3．1．4 The Electrical Conductivity3．1．5 The Dielectric Function3．1．6 Spectroscopies， Spectral Density3．1．7 Exercises3．2 Double-Time Green's Functions3．2．1 Equations of Motion3．2．2 Spectral Representations3．2．3 The Spectral Theorem3．2．4 Exact Expressions3．2．5 The Kramers-Kronig Relations3．2．6 Exercises3．3 First Applications3．3．1 Non-Interacting Bloch Electrons3．3．2 Free Spin Waves3．3．3 The Two-Spin Problem3．3．4 Exercises3．4 The Quasi-Particle Concept3．4．1 One-Electron Green's Functions3．4．2 The Electronic Self-Energy3．4．3 Quasi-Particles3．4．4 Quasi-Particle Density of States3．4．5 Internal Energy3．4．6 Exercises3．5 Self-Examination Questions4 Systems of Interacting Particles4．1 Electrons in Solids4．1．1 The Limiting Case of an Infinitely Narrow Band4．1．2 The Hartree-Fock Approximation4．1．3 Electronic Correlations4．1．4 The Interpolation Method4．1．5 The Method of Moments4．1．6 The Exactly Half-filled Band4．1．7 Exercises4．2 Collective Electronic Excitations4．2．1 Charge Screening （Thomas-Fermi Approximation）4．2．2 Charge Density Waves， Plasmons4．2．3 Spin Density Waves， Magnons4．2．4 Exercises4．3 Elementary Excitations in Disordered Alloys4．3．1 Formulation of the Problem4．3．2 The Effective-Medium Method4．3．3 The Coherent Potential Approximation4．3．4 Diagrammatic Methods4．3．5 Applications4．4 Spin Systems4．4．1 The Tyablikow Approximation4．4．2 \"Renormalised\" Spin Waves4．4．3 Exercises4．5 The Electron-Magnon Interaction4．5．1 Magnetic 4f Systems （s-f-Model）4．5．2 The Infinitely Narrow Band4．5．3 The Alloy Analogy4．5．4 The Magnetic Polaron4．5．5 Exercises4．6 Self-Examination Questions5 Perturbation Theory （T = 0）5．1 Causal Green's Functions5．I．1 \"Conventional\" Time-dependent Perturbation Theory5．1．2 \"Switching on\" the Interaction Adiabatically5．1．3 Causal Green's Functions5．1．4 Exercises5．2 Wick's Theorem5．2．1 The Normal Product5．2．2 Wick's Theorem5．2．3 Exercises5．3 Feynman Diagrams5．3．1 Perturbation Expansion for the Vacuum Amplitude5．3．2 The Linked-Cluster Theorem5．3．3 The Principal Theorem of Connected Diagrams5．3．4 Exercises5．4 Single-Particle Green's Functions5．4．1 Diagrammatic Perturbation Expansions5．4．2 The Dyson Equation5．4．3 Exercises5．5 The Ground-State Energy of the Electron Gas （Jellium Model）5．5．1 First-Order Perturbation Theory5．5．2 Second-Order Perturbation Theory5．5．3 The Correlation Energy5．6 Diagrammatic Partial Sums5．6．1 The Polarisation Propagator5．6．2 Effective Interactions5．6．3 Vertex Function5．6．4 Exercises5．7 Self-Examination Questions6 Perturbation Theory at Finite Temperatures6．1 The Matsubara Method6．1．1 Matsubara Functions6．1．2 The Grand Canonical Partition Function6．1．3 The Single-Particle Matsubara Function6．2 Diagrammatic Perturbation Theory6．2．1 Wick's Theorem6．2．2 Diagram Analysis of the Grand-Canonical Partition Function6．2．3 Ring Diagrams6．2．4 The Single-Particle Matsubara Function6．3 Self-Examination QuestionsSolutions of the ExercisesIndex