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)
- 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.
Phase transitions (Fabian Essler)
- 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.
Statistical
Physics, Phase Transitions and
Stochastic Processes
(Ramin Golestanian)
-
Stochastic
processes: the Langevin and
Fokker-Planck equation. Brownian
motion of single particle.
Classical
Field Theory (Uli Haisch)
- Group
theory and Lie algebra primer: basic concepts, 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, U(1) symmetry,
Goldstone’s theorem,
SU(2) × U(1)
symmetry, vector fields, Maxwell's theory, scalar
electrodynamics.
Canonical Quantisation (Uli Haisch)
- 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 (Uli Haisch)
- 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.