Introductory description

The electromagnetic field is a gauge field. Gauge changes to the vector potential (Aμ → Aμ - ∂μ Φ with Φ an arbitrary function of position and time), combined with multiplication of the wavefunction of particles with charge q by the phase factor, exp( iqΦ) , leave all physical properties unchanged. This is called a gauge symmetry. In particle physics, this idea is generalized to (space- and time-dependent) unitary matrix-valued fields multiplying spinor wavefunctions and fields. This generalization of the theory of an electron in an electromagnetic field is the basis for current theories of elementary particles.

The module starts with the theory of the electron in the electromagnetic field making the gauge symmetry explicit. It then discusses the gauge symmetries appropriate for the various theories and approximate theories used to describe other elementary particles and their interactions with their corresponding gauge fields.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1. Introduction and revision: relativistic quantum mechanics and notation; the Klein Gordon equation; the Dirac equation and interpretation of negative energy solutions; quantum numbers and spin; revision of matrices, Hermitan, unitary, determinants
2. Group theory: definition of a group, examples of discrete groups; continuous groups, Lie groups, examples: U(1), SU(2), SU(3)
3. Gauge invariance: symmetries and conservation laws; current conservation; Noether's theorem; the gauge principle; examples: Maxwell's equations, quantum electrodynamics
4. Quantum field theories: brief outline of the deeper theory; Feynman rules and diagrams
   Non Abelian gauge theories: SU(2) and the electroweak interaction; SU(3) and QCD; local nonAbelian gauge theory; gauge fields; self-interaction
5. Quantum electrodynamics: perturbation theory; scattering and cross sections

Learning outcomes

By the end of the module, students should be able to:

• Explain the theoretical framework of the Standard Model
• Explain the symmetry properties associated with gauge invariance
• Calculate amplitudes for simple QED processes
• Discuss qualitatively properties of the strong and weak interactions

Indicative reading list

Gauge Theories in Particle Physics, I.J.R.Aitchison and A.J.G.Hey, IOP Publishing

View reading list on Talis Aspire

Subject specific skills

Knowledge of mathematics and physics. Skills in modelling, reasoning, thinking

Transferable skills

Analytical, communication, problem-solving, self-study