Contents 1 Introduction 5 2 Scales and Complexity 11 2.1 Time scales . 12 2.2 L: Length scales . 12 2.3 N: particle number . 13 2.4 C: Complexity and diversity. 13 3 Quantum Fields: Overview 17 4 Collective Quantum Fields 27 4.1 Harmonic oscillator: a zero-dimensional field theory . 27 4.2 Collective modes: phonons . 34 4.3 The Thermodynamic Limit . 39 4.4 Continuum Limit . 42 4.5 Exercises for chapter 4 . 47 5 Conserved Particles 53 5.1 Commutation and Anticommutation Algebras . 54 5.1.1 Heuristic Derivation for Bosons . 55 5.2 What about Fermions? . 57 5.3 Field operators in different bases . 57 5.4 Fields as particle creation and annihilation operators. 59 5.5 The vacuum and the many body wavefunction . 62 5.6 Interactions . 63 5.7 Identical Conserved Particles in Thermal Equilibrium . 68 5.7.1 Generalities . 68 5.7.2 Identification of the Free energy: Key Thermodynamic Properties . 71 5.7.3 Independent Particles . 73 5.8 Exercises for chapter 5 . 74 1 Chapter 0. c P. Coleman 04 6 Simple Examples of Second-quantization 79 6.1 Jordan Wigner Transformation . 79 6.2 The Hubbard Model . 86 6.3 Gas of charged particles . 88 6.3.1 Link with first quantization . 88 6.4 Non-interacting particles in thermal equilibrium . 90 6.4.1 Fluid of non-interacting Fermions . 91 6.4.2 Fluid of Bosons: Bose Einstein Condensation . 94 6.5 Exercises for chapter 6 . 100 7 Greens Functions 105 7.1 Interaction representation . 107 7.1.1 Driven Harmonic Oscillator . 111 7.2 Greens Functions . 116 7.2.1 Green's function for free Fermions . 117 7.2.2 Green's function for free Bosons . 120 7.3 Adiabatic concept . 121 7.3.1 Gell-Man Low Theorem . 123 7.3.2 Generating Function for Free fermions . 125 7.3.3 The Spectral Representation . 127 7.4 Many particle Green's functions . 130 7.5 Landau's Fermi Liquid Theory . 132 7.6 Exercises for chapter 7 . 141 8 Feynman Diagrams: T=0 145 8.1 Heuristic Derivation . 146 8.2 Developing the Feynman Diagram Expansion . 152 8.2.1 Symmetry factors . 159 8.2.2 Linked Cluster Theorem . 161 8.3 Feynman rules in momentum space . 163 8.3.1 Relationship between energy, and the S-matrix . 164 8.4 Examples . 166 8.4.1 Hartree Fock Energy . 166 8.4.2 Response functions . 170 8.4.3 Magnetic susceptibility of non-interacting electron gas . 172 8.4.4 Electron in a scattering potential . 177 8.5 The self-energy . 180 8.5.1 Hartree-Fock Self-energy . 181 8.6 Large-N electron gas . 182 8.7 Exercises for chapter 8 . 188 2 c 2004 P. Coleman Chapter 0. 9 Finite Temperature Many Body Physics 193 9.1 Imaginary time . 196 9.1.1 Representations . 197 9.2 Imaginary Time Green Functions . 199 9.2.1 Periodicity and Antiperiodicity . 200 9.2.2 Matsubara Representation . 201 9.3 The contour integral method . 204 9.4 Generating Function and Wick's theorem . 208 9.5 Feynman diagram expansion . 211 9.5.1 Feynman rules from Functional Derivative . 212 9.5.2 Feynman rules in frequency/momentum space . 216 9.5.3 Linked Cluster Theorem . 218 9.6 Examples of the application of the Matsubara Technique . 219 9.6.1 Hartree Fock at a finite temperature. 220 9.6.2 Electron in a disordered potential . 221 9.7 Interacting electrons and phonons . 229 9.7.1 α2F : the electron-phonon coupling function . 237 9.7.2 Mass Renormalization by the electron phonon interaction . 240 9.7.3 Migdal's theorem. 243 9.8 Appendix A . 245 9.9 Exercises for chapter 9 . 247 10 Fluctuation Dissipation Theorem and Linear Response Theory 253 10.1 Introduction . 253 10.2 Fluctuation dissipation theorem for a classical harmonic oscillator . 255 10.3 Quantum Mechanical Response Functions. 257 10.4 Fluctuations and Dissipation in a quantum world . 259 10.4.1 Spectral decomposition I: the correlation function S(t t ) . 259 − 0 10.4.2 Spectral decomposition II: the response function χ(t t ) . 260 − 0 10.4.3 Quantum Fluctuation dissipation Theorem . 260 10.4.4 Spectral decomposition III: fluctuations in imaginary time . 261 10.5 Calculation of response functions . 261 10.6 Spectroscopy: linking measurement and correlation . 265 10.7 Electron Spectroscopy . 269 10.7.1 Formal properties of the electron Green function . 269 10.7.2 Tunneling spectroscopy . 270 10.7.3 ARPES, AIPES and inverse PES . 273 10.8 Spin Spectroscopy . 275 10.8.1 D.C. magnetic susceptibility . 275 10.8.2 Neutron scattering . 275 10.8.3 NMR . 279 10.9 Electron Transport spectroscopy . 281 3 Chapter 0. c P. Coleman 04 10.9.1 Resistivity and the transport relaxation rate . 281 10.9.2 Optical conductivity . 284 10.9.3 The f-sum rule. 285 10.10Exercises for chapter 10 . 288 11 Electron transport Theory 293 11.1 Introduction . 293 11.2 The Kubo Formula . 297 11.3 Drude conductivity: diagramatic derivation . 300 11.4 Electron Diffusion . 306 11.5 Weak Localization . 311 11.6 Exercises for chapter 11 . 318 12 Path Integrals and Phase transitions 323 12.1 Introduction: Broken symmetry, coherent states and path integrals. 323 12.2 Coherent states and Grassman mathematics . 328 12.2.1 Completeness and matrix elements . 330 12.3 Path integral for the partition function . 332 12.4 General evaluation of Path Integral for non-interacting Fermions . 337 12.5 Hubbard Stratonovich transformation . 339 12.6 Superconductivity and BCS theory . 342 12.6.1 Introduction: Superconductivity pre-history . 342 12.6.2 The BCS Hamiltonian . 344 12.6.3 Computing Tc . 348 12.6.4 The structure of the Boguilubov quasiparticle and the BCS wavefunction350 12.6.5 The Nambu Greens function . 356 12.6.6 Twisting the phase: the Anderson Higg's mechanism . 360 A Appendix: Grassman Calculus . 364 A.1 Differentiation and Integration . 364 A.2 Change of variable . 365 A.3 Gaussian Integrals . ..