(Course description last updated for academic year 2014-15).
Prerequisites

This course is only suitable for students whose mathematics is VERY strong. Physics students taking this course may need to do some supplementary reading on Lie group theory, for which the following are recommended:

G. 't Hooft, Lectures on Lie Groups in Physics (given at the University of Utrecht, 2007), available at http://www.phys.uu.nl/~thooft/lectures/lieg07.pdf

For more advanced topics: Symmetries, Lie Algebras and Representations,  Fuchs J and Schweigert C, (C.U.P 1997), available in the Rayleigh Library.

Synopsis

Quantum field theory (QFT) is the basic theoretical framework for describing elementary particles and their interactions (excluding gravity) and is essential in the understanding of string theory. It is also used in many other areas of physics including condensed matter physics, astrophysics, nuclear physics and cosmology.  The Standard Model, which describes the basic interactions of particle physics, is a particular type of QFT known as a gauge theory.  Gauge theories are invariant under symmetry transformations defined at each point in spacetime which form Lie Group under composition.  To quantise a gauge theory, it is necessary to eliminate non-physical degrees of freedom and this requires additional theoretical tools beyond those developed in the introductory quantum field theory course.

A variety of new concepts and methods are first introduced in the simpler context of scalar field theory.  The functional integral approach provides a formal non-perturbative definition of any QFT which also reproduces the usual Feynman rules. The course discusses in a systematic fashion the treatment of the divergences which arise in perturbative calculations. The need for regularisation in QFT is explained, and the utility of dimensional regularisation in particular is emphasised.  It is shown how renormalisation introduces an arbitrary mass scale and renormalisation group equations which reflect this arbitrariness are derived.  Various physical implications are then discussed.

The rest of the course is concerned specifically with gauge theories. The peculiar difficulties of quantising gauge fields are considered, before showing how these can be overcome using the functional integral approach in conjunction with ghost fields and BRST symmetry.  A renormalisation group analysis reveals that the coupling constant of a quantum gauge theory can become effectively small at high energies.  This is the phenomenon of asymptotic freedom, which is crucial for the understanding of QCD: the gauge theory of the strong interactions. It is then possible to perform perturbative calculations which may be compared with experiment.  Further properties of gauge theories are  discussed, including the possibility that a classical symmetry may be broken by quantum effects, and how these can be analysed in perturbation theory.   Such anomalies have important implications for the way in which gauge particles and fermions interact in the Standard Model.

BOOKS

An Introduction to Quantum Field Theory, Peskin M E and Schroeder D V (Addison-Wesley 1996) Quantum Theory of Fields, Vols. 1 & 2, Weinberg S (CUP 1996)

Course section:

Other Information

Staff
Dr Ronald Reid-EdwardsLecturer