DescriptionThe 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 focus is on classical physics, Quantum Mechanics III would be a useful co-requisite. 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 be 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. PrerequisitesMathematical Physics II or equivalent (required). Co-requisitesQuantum Mechanics III or equivalent (strongly recommended). Electromagnetism III or equivalent may also be useful but is not essential. ResourcesGood places to start are the Wikipedia pages: Also make sure you are familiar with Lagrangian mechanics.
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email: Douglas Smith