Group Theory and Physics
Detailed Project Description
Symmetries lie very much at the heart of fundamental physics. This project explores the mathematical language of symmetry—Group Theory—with a particular focus on Continuous Groups (Lie Groups) and their representations. Understanding how these groups act on vector spaces is essential for fundamental physics, as every fundamental particle in the universe is interpreted as a state within a representation of a specific Lie group. It is also beautiful mathematics!
The project will begin with introducing Lie groups, Lie algebras and their representations following from the GMP module. A potential final objective of this project is to investigate the classification of Simple Lie groups and possibly also their finite representations. The classification of simple Lie groups is one of the most elegant results in modern mathematics. It reveals that the vast array of possible continuous symmetries can be distilled into just four infinite families and five unique, exceptional cases, uncovering a deep and surprising order at the foundation of algebraic structure and physical laws. The classification can be interpreted into classifying Root Systems— sets of vectors in Euclidean space that completely encode the structure of the algebra. The geometry of these roots (lengths and angles) is beautifully summarised by Dynkin diagrams.
Figure 1: The Cartan Classification, showing the four infinite families \(A_n, B_n, C_n, D_n\) and the five exceptional groups.
Other potential directions could be more physics oriented looking at Lie groups in the standard model, GUTs, string theory etc.
Mode of Operation and Evidence of Learning
The project will revolve around learning through reading with focus on the underlying theory, mathematical rigour, and the development of deep conceptual understanding. Students will demonstrate their understanding by solving relevant problems, exploring examples and theoretical applications of the material, and clearly communicating it in both written and oral formats.
Prerequisites
Recommended Reading
At this stage the wikipedia links above are the best place to explore. During the project we will look at some or all of the following (and other) books:
- H. Georgi, Lie Algebras in Particle Physics (Perseus): The classic text for physicists. It prioritises the application of \(SU(3)\) in the Quark Model and GUTs. [ PDF Online]
- Nick Dorey, Symmetries,Fields and Particles Part III notes [ PDF Online]
- Hans Samelson, Notes on Lie Algebras: A more mathematical, rigorous text deriving the Cartan classification. [ PDF Online ]
- K. Erdmann & M. Wildon, Introduction to Lie Algebras (Springer): Again more mathematical, but possibly clearer than Samelson with the Cartan classification.[ PDF Online ]
- Brian Hall, Lie Groups, Lie Algebras, and Representations: An elementary introduction to Lie groups, Lie algebras, and their representations. Designed to be accessible to students of mathematics or physics. Classificatin fo representations of SL(3) but not the full Cartan classifcation.[ PDF Online ]
- Robert Cahn, Semi-Simple Lie Algebras and Their Representations: A focused guide on the Dynkin diagram method for calculating weights and dimensions. Very practical! [ PDF Online ]