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.

By the end of the module the student will be able to:

• Convert between the Schrodinger, Heisenberg and interaction pictures of quantum mechanics
• Derive the Euler-Lagrange equations, conjugate fields, Hamiltonian density, and Hamilton equations for a field theory specified by its Lagrangian density
• Canonically quantise a classical field theory by writing down its equal-time commutation relations (ETCRs)
• Obtain creation and annihilation operators for particles by Fourier expansion of field operators
• Use Noether’s theorem to derive conserved quantities such as electrical charge given a continuous internal symmetry of a field theory
• Use Noether’s theorem to derive formulae for the energy, momentum and angular momentum of a field
• Write the energy, momentum and charge operators of a field in terms of creation and annihilation operators, and hence identify the particle content of the field theory
• Quantise the real and complex Klein-Gordon fields (spin-0), Dirac field (spin-1/2), and electromagnetic and Proca fields (spin-1)
• Demonstrate that the Dirac field must be quantised using anticommutation relations
• Write down the Lagrangian density for quantum electrodynamics, and develop the S-matrix expansion in the interaction picture
• Write the Hamiltonian for a system of interacting particles in second-quantised (occupation number representation) form
• Derive the Hamiltonian for the weakly interacting Bose gas, and solve using the Bogoliubov transformation
• Derive the BCS Hamiltonian for superconductivity, and solve using the Bogoliubov-Valatin transformation
• Write down a simple tight-binding Hamiltonian and solve using a Bloch transformation
• Write down the Hubbard model for strongly correlated systems, and explain why it is a Mott insulator at half filling
• Derive the antiferromagnetic Heisenberg model as a limit of the Hubbard model
• Derive and use the Holstein-Primakoff transformation to find the spin wave spectra of ferromagnets and antiferromagnets