First, we review the second quantization formalism, and apply it to set up the many-body problem for a solid in a general form, including electron-electron interactions and interactions between electrons and various lattice excitations. Then, we apply both perturbational methods and variational methods to calculate material properties, such as effective masses of electrons, bulk moduli, etc.
To overcome the limitations of perturbation theory, we need to systematize the way in which perturbation series are set up. For this purpose, we introduce Green's functions, and see how to interpret them, in particular how to extract material properties from Green's functions. To calculate the Green's function, we introduce the Gell-Man & Low theorem, Wick's theorem, vacuum polarization. We find that Feynman diagrams allow to calculate the Green's function in a systematic manner, and that they can be resummed using Dyson series. Diagrammatics allows to calculate the amplitudes corresponding to particular Feynman diagrams.
Then, we apply the diagrammatic techniques to calculate dielectric functions, the optical response, plasmon and polariton dispersion relations, and the conductivity of materials. This illustrates the use of the Green's functions and Feynman diagrams in a concrete many-body problem.
The optical response brings us to consider in the next part of the course Kubo response theory, for more general cases such as magnetic response.
Finally, we venture into a regime where perturbation theory around the Fermi sphere fails, and a variational approach shows the way forward: the Bardeen-Cooper-Schrieffer theory for superconductivity. We review the basic properties and experiments that led to the development of that theory, set up the theory, and calculate basis properties such as the critical temperature and critical magnetic field from it.