PX4307.5 Gauge Theories for Particle Physics
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 timedependent) unitary matrixvalued 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 aims
To develop ideas used in gauge theories and apply these to the field of particle physics. To describe the theory underpinning the Standard Model of Particle Physics should and to highlight the symmetry properties of the theory. To consider Quantum Electrodynamics (QED) in some detail, and to illustrate its success by comparison with experiments.
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.
 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
 Group theory: definition of a group, examples of discrete groups; continuous groups, Lie groups, examples: U(1), SU(2), SU(3)
 Gauge invariance: symmetries and conservation laws; current conservation; Noether's theorem; the gauge principle; examples: Maxwell's equations, quantum electrodynamics
 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; selfinteraction  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, problemsolving, selfstudy
Study time
Type  Required 

Lectures  15 sessions of 1 hour (20%) 
Private study  60 hours (80%) 
Total  75 hours 
Private study description
Working through lecture notes, solving problems, wider reading, discussing with others taking the module, revising for exam, practising on past exam papers
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group B1
Weighting  Study time  

2 hour online examination (Summer)  100%  
Answer 2 questions from 3

Feedback on assessment
Personal tutor, group feedback
Courses
This module is Optional for:

TMAAG1PE Master of Advanced Study in Mathematical Sciences
 Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
 Year 1 of G1PE Master of Advanced Study in Mathematical Sciences
 Year 1 of TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 1 of TMAAG1PC Postgraduate Taught Mathematics (Diploma plus MSc)
 Year 4 of UPXAF304 Undergraduate Physics (BSc MPhys)
 Year 4 of UPXAF303 Undergraduate Physics (MPhys)
This module is Option list A for:
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 3 of UMAAG100 Undergraduate Mathematics (BSc)
 Year 4 of UMAAG101 Undergraduate Mathematics with Intercalated Year
This module is Option list B for:
 Year 4 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 4 of UPXAFG31 Undergraduate Mathematics and Physics (MMathPhys)
This module is Option list C for:

UMAAG105 Undergraduate Master of Mathematics (with Intercalated Year)
 Year 3 of G105 Mathematics (MMath) with Intercalated Year
 Year 5 of G105 Mathematics (MMath) with Intercalated Year

UMAAG103 Undergraduate Mathematics (MMath)
 Year 3 of G103 Mathematics (MMath)
 Year 4 of G103 Mathematics (MMath)

UMAAG106 Undergraduate Mathematics (MMath) with Study in Europe
 Year 3 of G106 Mathematics (MMath) with Study in Europe
 Year 4 of G106 Mathematics (MMath) with Study in Europe