We start with a general introduction and reminder on classical field theory. This is studied with a number of simple examples in mind, such as scalar bosons and massive vector bosons. We also look at gauge fields, symmetries, conserved currents, coordinate transforms and component transforms in the framework of classical field theory. The electromagnetism of charged scalar bosons is investigated as an example of a gauge theory.
Next, we make the transition from classical to quantum field theory. For this purpose we focus on the path integral (Lagrangian) formalism - this is complementary to the Hamiltonian formalism investigated in the course of solid state physics 2. We derive the propagator and interpret it (as greens function, and as a description of particles as quanta of excitations of a field). We investigate how classical and quantum fields differ. This difference is particularly striking for fermionic fields where we have to introduce Grassmann variables. We first study quantum field theory for exactly solvable models.
Then we switch to interacting fields .These are no longer exactly solvable, and we introduce the perturbative technique of Feynman diagrams. Problems of convergence (and solutions with regularization and renormalization) are discussed. Next, we turn to quantum statistical field theory, necessary to study fields at finite temperature. The free energy is calculated as a function of temperature, using the technique of Matsubara summations. In this part we use the phi^4 theory as textbook example.
Finally, we apply the techniques to quantum electrodynamics. For this theory we focus on how to calculate scattering amplitudes and the S-matrix.