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This course is intended to give an introduction to some aspects
of field theory and related ideas. These are important in particular
for treating systems with an infinite number of degrees of freedom.
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.
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## Course Contents

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The mathematical description of systems with an infinite
number of degrees of freedom: functionals, functional
differentiation, and functional integrals. Multi-dimensional
Gaussian integrals. Random fields: properties of a Gaussian field.
Perturbation theory for non-Gaussian functional integrals. Path
integrals and quantum mechanics. Treatment of free particle and of
harmonic oscillator.
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Stochastic processes and path integrals: the Langevin and
Fokker-Planck equation. Brownian motion of single particle.
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The link between quantum mechanics and the statistical
mechanics of one-dimensional systems via Wick rotation. Transfer
matrices for one-dimensional systems in statistical mechanics.
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Classical field theory: fields, Lagrangians and Hamiltonians.
The least action principle and field equations. Space-time and
internal symmetries: U(1) example, Noether current. The idea of an
irreducible representation of a group. Irreducible representations
of SU(2) and application to global internal symmetry. Simple
representations of the Lorentz group via SU(2)xSU(2) without proof.
U(1) gauge symmetry, action of scalar QED and derivation of
Maxwell's eqns in covariant form.
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Canonical quantisation and connection to many body theory:
quantised elastic waves; quantisation of free scalar field theory;
many-particle quantum systems.
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Path integrals and quantum field theory: generating functional
and free particle propagator for scalar and U(1) gauge fields (in
Lorentz gauge).
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Perturbation theory at tree level for decay and scattering
processes. Examples from pure scalar theories and scalar QED.
Goldstone theorem.
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Canonical transformations in quantum field theory: Bogoliubov
transformations applied to bose condensates, magnons in
antiferromagnets, and to BCS theory.
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Landau theory and phase transitions: phase diagrams,
first-order and continuous phase transitions.
Landau-Ginsburg-Wilson free energy functionals. Examples including
liquid crystals. Critical phenomena and scaling theory.
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