Lectures 1-19 are part of the "Advanced Quantum Theory" MMathPhys course
Lectures 20-25 are part of the "Nonequilibrium Statistical Physics" MMathPhys course
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.
Path Integrals in Quantum
Mechanics (Fabian Essler)
- 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.
- Path integrals and transfer matrices of classical statistical models.
- Transfer matrices for one-dimensional systems in classical statistical mechanics. Transfer matrices in D=2.
Quantum Many-Particle Systems (Fabian Essler)
bosons and fermions, Fock space, single-particle and
- 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.
Phase transitions (Fabian Essler)
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.
Physics, Phase Transitions and
- Stochastic processes: the Langevin and Fokker-Planck equation. Brownian motion of single particle.
Field Theory (Uli Haisch)
theory and Lie algebra primer: basic concepts, SU(N),
- 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, U(1) symmetry, Goldstone’s theorem,
SU(2) × U(1) symmetry, vector fields, Maxwell's theory, scalar electrodynamics.
Canonical Quantisation (Uli Haisch)
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 (Uli Haisch)
of interactions, interaction picture, Feynman
- Applications: tree-level decay and scattering processes of scalar and U(1) gauge fields.
- Path integrals: effective action, Feynman diagrams from path integrals.