Description
Symmetries lie very much at the heart of fundamental physics. The mathematical language describing symmetries is called Group Theory. Symmetries can be discrete (eg reflections) or continuous (eg rotations) and both are important in physics. This project will likely focus on the continuous symmetries which are described by Lie groups.
There is a beautiful classification of all Simple Lie groups, aided by the use of Dynkin diagrams:
All fundamental particles are in representations of Lie groups. The standard model of particle physics is based on a quantum field theory with an internal symmetry group SU(3)xSU(2)xU(1) (SU(n+1)=An in the above classification). Grand unified theories (GUTs) and string theory attempt to reproduce this gauge group from simpler but larger groups.
In this project we will begin by looking at group theory, Lie groups and representation theory. But from there the project is very broad and
there is the possibility to go in many different directions according to your tastes. The less physics-inclined student could focus on the classification of simple Lie groups and their representations, the Cartan Classification. The more physics inclined might focus instead more on understanding gauge field theory, the standard model, GUTs etc. You may also want to look at finite (permutation) groups and its representations, there are some very nice applications of those in physics one could look at also. Or one could consider the generalisation of Lie groups to super Lie groups and their classification which also have many applications in physics.
Pre-requisites
Co-requisites
If you are unsure about the pre- and co-requisites please feel free to get in touch.
Resources
You will first need a basic familiarity with group theory. The above wikipedia links are a good start. After that
H. Georgi, Lie Algebras in Particle Physics (Perseus Books 1999) (More Physical, you can find the pdf online )
H. Samelson, Notes on Lie Algebras (Springer 1990)(pdf online )
H.F. Jones, Groups, Representations and Physics (CRC 1998)
R. Cahn, Semi Simple Lie Algebras And Their Representations (Frontiers In Physics Series Volume 59 )(pdf online )
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