Quantum Field Theory is the most complete description we have at present of the physical world. One starts with a classical field theory written in Lagrangian form, and then quantises the classical fields to obtain field operators. If one then expands these field operators as a sum over momentum states, the Fourier coefficients are the creation and annihilation operators for corresponding particles. We carry out this canonical quantization for non-interacting spin-0 (Klein-Gordon), spin-1/2 (Dirac), and spin-1 (electromagnetic and Proca) fields. The problem of negative energy states arising in one-particle relativistic wave equations is solved in QFT; they correspond to the absence of positive energy antiparticles. Finally we write down the Lagrangian for quantum electrodynamics (QED) and start the development of perturbation theory via the S-matrix expansion. Many Particle Theory considers the properties of systems with many interacting particles, such as the electron liquid in a metal. The quantum mechanical properties of such systems can in principle be obtained by solving a large Schrodinger equation, but this is not mathematically tractable. The trick is to rewrite the problem in "second quantised" form using the creation and annihilation operators familiar from QFT. We apply these ideas to various condensed matter systems: superfluids, superconductors, correlated systems, ferromagnets and antiferromagnets. |