Description

The aim of this project is to explore the role and consequences of symmetry in mathematical physics. You will investigate how the principle of symmetry can be used to define a model (by specifying the Lagrangian), and also examples of how symmetry can be used to find special solutions of the Euler-Lagrange equations. This project will build on topics covered in Mathematical Physics II, and although the initial focus is on classical physics, there will also be aspects which require knowledge of quantum mechanics so Quantum Mechanics III or equivalent is a recommended corequisite.

There are a variety of different types of symmetries: either discrete (e.g. reflections) or continuous (e.g. rotations by an arbitrary angle), and either global or local (e.g. if an angle of rotation could be chosen to vary arbitrarily with position, that would be a local symmetry.) In the Lagrangian formulation, a model has a symmetry if the symmetry transformations change the fields but in such a way that the action is left unchanged. There are many specific symmetries you can investigate. The principle of symmetry underlies General Relativity and Electromagnetism. Generalisations of the "gauge symmetry" of electromagnetism are fundamental in defining gauge theories such as the Standard Model of particle physics. An important consequence of a continuous symmetry is the existence of a conserved charge via Noether's theorem.

The full set of symmetries form a group (essentially since two consecutive symmetry transformations leave the action unchanged, so must be a symmetry transformation) and for continuous symmetries this is described mathematically by Lie groups.

Supersymmetry is a very interesting symmetry. It is used in many particle physics models, and the LHC at CERN may find evidence for supersymmetry. In such models it requires a symmetry between "bosons" and "fermions", particles which have very different properties in quantum theory. This leads to special properties of such theories, and indeed it appears that supersymmetry is an essential ingredient in string theory. However, even in classical physics, supersymmetry can be used to find special "BPS" solutions which are given by solutions of first-order differential equations. The remarkable feature is that these first-order BPS equations automatically imply the second-order Euler-Lagrange equations.

Finally, some models have an interesting type of symmetry known as duality symmetry. E.g. there is a symmetry in electromagnetism if we swap electric and magnetic fields. Generalisations of this occur in some models and this can have important consequences.

Prerequisites

Mathematical Physics II or equivalent Physics module(s) covering Lagrangian and Hamiltonian dynamics, and some quantum mechanics. Special Relativity and Electromagnetism II or equivalent is also very useful but is not essential.

Corequisites

Quantum Mechanics III or equivalent Physics module(s) recommended.

Resources

Good places to start are the Wikipedia pages:

• Symmetry in physics
• Introduction to gauge theory
• Gauge theory
• Lie groups
• Supersymmetry