| # | Date | Read | Topic |
| 1. | Jan. 28, Monday | FW 1 | Course info. Illustration of perturbation theory and "Feynman diagrams". Second quantization. Hamiltonian and Hilbert space. Periodic boundary conditions. Particle in the box. Many distinguishable particles in the box. Basis of the Hilbert space and matrix elements of the Hamiltonian. |
| 2. | Jan 30, Wednesday | FW 1-2 | Identical particles in the box. Fermions and bosons in first quantization. Occupation numbers. Hilbert space for identical particles. Vacuum, Fock states, and Fock space. Creation and annihilation operators. |
| 3. | Feb 1, Friday | FW 1-2 | HW1 given. Hamiltonian in the second quantized form. Occupation numbers. Fermi and Bose statistics. Fields. Particle density operator. |
| 4. | Feb 4, Monday | FW 1-2, part 3 | Particle density operator. Fourrier transform. Tight-binding model. Ideal Fermi gas. Fermi sea. Fermi energy and Fermi momentum. Density of states. |
| 5. | Feb 6, Wednesday | FW 4-5 | Ideal Fermi gas approximation: interaction versus kinetic energy. r_s parameter. Wigner crystal. Particle-hole pairs and quasiparticles. |
| 6. | Feb 8, Friday | FW 5-6 | Specific heat of a degenerate Fermi gas. Interaction representation. Schroedinger, Heisenberg, and interaction pictures. |
| 7. | Feb 11, Monday | FW 6 | Interaction picture. Perturbation theory. Chronological ordering. S-matrix. Adiabatic switching on. |
| 8. | Feb 13, Wednesday | FW 7 | Gell-Mann and Low theorem. Ordinary perturbation theory from S-matrix. Green's functions. Definitions. Time ordering. |
| 9. | Feb 15, Friday | FW 7 | HW2 given. Relation to observables. Expectation value of a single-particle operator. The average potential energy of interactions. The exact ground state energy. |
| 10. | Feb 18, Monday | FW 7 | Green's function of free Fermi gas. The number of particles from the Green's function. |
| 11. | Feb 20, Wednesday | FW 7 | Green's function of free Fermi gas from the equation of motion. Analytic structure. Plemelj formulas. Lehmann representation. |
| 12. | Feb 22, Friday | FW 7 | Translational invariance. Lehmann representation. Spectral functions A and B. Advanced and retarded Green's functions. |
| 13. | Feb 25, Monday | FW 7 | Analytic structure of Green's functions. Asymptotics at omega->infinity. Particle and hole propagators. Poles of Green's functions. Physical meaning of poles. |
| 14. | Feb 27, Wednesday | FW 7, 8 | Energy and lifetime of quasiparticles. Wick's theorem. Perturbation theory for Green's functions. Green's functions in interaction representation. |
| 15. | Feb 29, Friday | FW 8 | HW3 given. Expectation values over the ground state. Time ordering. Normal ordering. Contraction. |
| 16. | Mar 3, Monday | FW 8, 9 | Contraction. Contraction withing normal ordering. Wick's theorem. Examples. Feynman diagrams. First order diagrams in coordinate representation. |
| 17. | Mar 5, Wednesday | FW 9 | First order diagrams in coordinate representation. Bubble diagram. Cancellation of disconnected diagrams. Topologically distinct connected diagrams. Feynman rules in coordinate space. Examples: first order. |
| 18. | Mar 7, Friday | FW 9 | Examples: second order. Feynman diagrams in momentum space. Feynman rules in momentum space. Example: First order diagrams. |
| 19. | Mar 10, Monday | FW 9 | HW4 given. First order diagrams. Dyson's equations. Self-energy. Proper self-energy. Integral equations and summation of geometric series. Proper energy in the first order. Corrections to the dispersion of particles. |
| Mar 12, Wednesday | FW 9 | No class. March meeting. | |
| Mar 14, Friday | FW 9 | No class. March meeting. | |
| 20. | Mar 24, Monday | FW 9, 46 | Effective interaction. Polarization insertions. Proper polarization insertion. Vertex parts. Proper vertex parts. Skeleton diagrams. Dyson's equations. |
| 21. | Mar 26, Wednesday | FW 10 | Hartree-Fock approximation. Fock's approach following Landau-Lifshitz III. Variational approach. Slater determinants. Hartree-Fock equations. The energy of the ground state. |
| 22. | Mar 28, Friday | FW 10 | Hartree and Fock diagrams. Hartree-Fock equations on G. Conventional Hartree-Fock equations from diagrams. HF for translationally invariant systems. |
| 23. | Mar 31, Monday | FW 11 | Imperfect Fermi gas. Singular potentials and gas parameter. Scattering theory. Differential cross section. Scattering amplitude. Integral equation for the wave function in momentum space. |
| 24. | Apr 2, Wednesday | FW 11 | HW5 given. Integral equation for the scattering amplitude. Scattering of hard spheres. Born approximation. Ladder diagrams. |
| 25. | Apr 4, Friday | FW 11 | First order in (k_F a) from Born approximation. Replacing Born scattering length by an exact one. Effective two-particle interaction in the medium. Bethe-Salpiter equation. Galitskii's integral equations. |
| 26. | Apr 7, Monday | FW 11, 3 | Solution of Galitskii's integral equations through the order (k_F a)^2. Proper self-energy to this order. Physical quantities: spectrum renormalization, effective mass, lifetime of quasiparticles, chemical potential, ground state energy, sound velocity. Degenerate electron gas. Jellium model. Kinetic and potential energy and length scales. |
| 27. | Apr 9 , Wednesday | FW 3, 12 | Expansion in r_s. Ground state energy calculation in the first order in e^2. Fermi surface integrals. Divergencies in the second and higher orders. Specifics of a long range interaction. Reordering of perturbation series. Ring diagrams. |
| 28. | Apr 10 , Thursday, 5:30pm | FW 12 | Ring diagrams. Divergence of the effective mass in HF approximation. Density-density correlation function and the total polarization insertion. Proper polarization in RPA approximation. Evaluation of \Pi_0(q). Real part. |
| 29. | Apr 11 , Friday | FW 12 | Evaluation of \Pi_0 (q). Imaginary part. Creation of particle-hole pair and the structure of Im\Pi_0(\omega,q) in a frequency-momentum plane. Useful limits of \Pi_0. Correlation energy in terms of \Pi(\omega,q). |
| 30. | Apr 14, Monday | FW 12 | Correlation energy in RPA approxiamation. Effective interaction in RPA. Static limit. Thomas-Fermi screening. Friedel oscillations. Linear response and collective modes. Introduction. |
| 31. | Apr 16 , Wednesday | FW 13 | General theory of linear response. External perturbation. The retarded density correlation function and generalized susceptibility. Retarded and chronological correlation functions. |
| 32. | Apr 18 , Friday | FW 14 | Screening in electron gas. Yukawa potential. Friedel oscillations at large distances. 2k_F singularities. Plasma oscillations. Poisson, Euler, and continuity equations for motion of the classical plasma. Linearization of equations. |
| Apr 21 , Monday | FW | No classes, Passover. | |
| 33. | Apr 23 , Wednesday | FW 15 | HW6 given. Solution of linearized equations. Classical plasma frequency. Plasma oscillations in electron gas. Collective modes: damping and dispersion. Plasmon dispersion in the limit of long wavelengths. Sound waves in "ideal" Fermi gas (first sound). |
| 34. | Apr 24, Thursday, 5:30pm | FW 16 | Zero sound in an imperfect Fermi gas. The zero sound speed. Comparison with ordinary sound. Physical picture of zero sound. Linear response for conductivity: correction to Kubo formula. |
| 35. | Apr 25, Friday | AGD 39.2 | Conductivity of an ideal Fermi gas. Damping in Green's function. Naive bubble with damped Green's functions. Drude formula. Wrong scattering time. Gauge invariance and ladder diagrams for polarization operator. Transport scattering time and correct Drude formula. Field theory at T>0. Grand canonical ensemble. Real time Green's functions. Matsubara time and Matsubara Green's functions. |
| 36. | Apr 28, Monday | FW 23 | Observables: single-particle operators. Energy and theromodynamic potential. Ideal gas. Perturbation theory. |
| 37. | Apr 30, Wednesday | FW 24 | Perturbation theory. Connected diagrams. (Anti)periodicity in imaginary time. Wick's theorem. Feynman rules. Noninteracting Green's function in momentum representation. |
| 38. | May 1, Thursday, 5:30pm | FW | Diagrams and Dyson's equations. Frequency sums. Lehmann representation for Matsubara Green's functions. Spectral function. |
| 39. | May 2, Friday | FW | HW7 given. Real time temperature Green's functions. Analytical properties. Matsubara susceptibility. |
| 40. | May 5, Monday | AGD | The theorem about analytical continuation for Matsubara susceptibility. Instabilities in particle-hole and particle-particle channels. Logarithmic ivergence of 2k_F susceptibility in particle-hole channel for nested Fermi surfaces. CDW, SDW, and SC instabilities. Parquet diagrams in 1D. Bosonization. Right and left densities. |
| 41. | May 7, Wednesday | Commutator of right densities. Schwinger anomalous term. Bosonization of free fermions: Hamiltonian. Interactions: g-ology. Renormalization of velocity. | |
| May 9, Friday | No class. | ||
| 42. | May 12, Monday | Free boson field theory. Vertex operators and bosonization of fermion fields. Spin-charge separation. |
