Fundamental Symmetries (VS1) (U02594)

? Credit Points : 10 ? SCQF Level : 11 ? Acronym : PHY-5-VFundSym

Symmetry principles are now an essential part of our understanding of the underlying structure of physics. The course sets out the mathematical preliminaries: Lie groups, Lie algebras and their representations, and then goes on to describe the Lorentz group. The course aims to complement the courses on Particle Physics, General Relativity and Relativistic Quantum Field Theory, but might also be of wider interest.

Entry Requirements

? This course is only available to part year visiting students.

? This course is a variant of the following course : U01428

? Pre-requisites : Year 3 Mathematical Physics, including Groups & Symmetries (desirable), or equivalent. Knowledge of linear algebra is essential.

Subject Areas

Home subject area

Undergraduate (School of Physics), (School of Physics, Schedule Q)

Delivery Information

? Normal year taken : 5th year

? Delivery Period : Semester 1 (Blocks 1-2)

? Contact Teaching Time : 3 hour(s) per week for 11 weeks

All of the following classes

Type	Day	Start	End	Area
Lecture	Monday	15:00	15:50	KB
Lecture	Thursday	15:00	15:50	KB

? Additional Class Information : Workshop/tutorial sessions, as arranged.

Summary of Intended Learning Outcomes

After completing this course students should:
1) have a good understanding of Lie Groups and Lie Algebras;
2) understand the relation between a Lie Group and its associated Lie Algebra, including some basic ideas about
the global structure of Lie Groups (such as universal covering groups);
3) have learnt about semi-simple and simple Lie Algebras, the role of the Cartan-Killing metric and of the adjoint representation (of both Groups and Algebras), the canonical representation of semi-simple Lie Algebras in terms of their Cartan subalgebras, and the resulting description in terms of root systems and their corresponding Weyl groups;
4) understand the classification of Lie Algebras using Coxeter/Dynkin diagrams and the roots systems for all the classical and exceptional Lie Algebras;
5) know how to construct the adjoint representation explicitly from the corresponding root system;
6) have learnt about weights and will be able to construct arbitrary irreducible representations using these;
7) understand the basic ideas of representation theory including the Clebsch-Gordan series and 3- and 6-j coefficients, and projection operators;
8) have an understanding of the mathematical structure of the Lie Algebras and Groups most relevant for particle physics, namely SU(3) and the Lorentz group SO(3,1);
9) understand the role of Clifford (Dirac) algebras to spinor representations of orthogonal groups, especially the
Lorentz group, and how fields correspond to non-compact representations of the Poincare group.