Part A Quantum Theory, Part A Classical Mechanics, B7.1 Classical Mechanics, B7.2 Electromagnetism.
This course is intended to give an introduction to some aspects of many-particle systems, field theory and related ideas. These form the basis of our current theoretical understanding of particle physics, condensed matter and statistical physics. An aim is to present some core ideas and important applications in a unified way. These applications include the classical mechanics of continuum systems, the quantum mechanics and statistical mechanics of many-particle systems, and some basic aspects of relativistic quantum field theory.
Note: This double unit is offered by the Physics Department.
24 lectures in MT and 16 lecture in HT
Path Integrals in Quantum Mechanics
- Mathematical tools for describing systems with an infinite number of degrees of freedom: functionals, functional differentiation; Multi-dimensional Gaussian integrals.
- Quantum mechanical propagator as a path integral. Semiclassical limit. Free particle.
- Quantum statistical mechanics in terms of path integrals. Harmonic oscillator.
- Perturbation theory for non-Gaussian functional integrals. Anharmonic oscillator. Feynman diagrams.
Quantum Many-Particle Systems
- Second Quantization: bosons and fermions, Fock space, single-particle and two-particle operators.
- Applications to the Fermi gas, weakly interacting Bose condensates, magnons in (anti)ferromagnets, and to superconductivity.
- quantum field theory as a low-energy description of quantum many-particle systems.
Classical Field Theory
- Group theory and Lie algebra primer: basic concepts, \(\mathrm{SU}(N)\), Lorentz group.
- Elements of classical field theory: fields, Lagrangians, Hamiltonians, principle of least action, equations of motion, Noether's theorem, space-time symmetries.
- Applications: scalar fields, spontaneous symmetry breaking, \(\mathrm{U}(1)\) symmetry, Goldstone's theorem, \(\mathrm{SU}(2) × \mathrm{U}(1)\) symmetry, vector fields, Maxwell's theory, scalar electrodynamics.
Canonical Quantisation
- Free real and complex scalar fields: Klein-Gordon field as harmonic oscillators, Heisenberg picture.
- Propagators and Wick's theorem: correlators, causality, Green's functions.
- Free vector fields: gauge fixing, Feynman propagator.
Interacting Quantum Fields
- Perturbation theory: classification of interactions, interaction picture, Feynman diagrams.
- Applications: tree-level decay and scattering processes of scalar and U(1) gauge fields.
- Path integrals: effective action, Feynman diagrams from path integrals.
Statistical Physics, Phase Transitions and Stochastic Processes
- Transfer matrices: one-dimensional systems in classical statistical mechanics.
- Transfer matrices in \(D=2\) and their relation to path integrals.
- Phase transition in the 2D Ising model: Peierls argument.
- Landau Theory of phase transitions: phase diagrams, first-order and continuous phase transitions.
- Landau-Ginzburg-Wilson free energy functionals. Examples including liquid crystals.
- Critical phenomena and scaling theory.
- Stochastic processes: the Langevin and Fokker-Planck equation. Brownian motion of single particle.