3 ECTS Credits — Semester 2 — Elective
Advanced mathematics for physicists involves the application of mathematical concepts and techniques to solve complex problems in physics. It includes in particular group theory, a branch of mathematics that studies the properties of groups, which are algebraic structures used to describe symmetries and transformations in various mathematical and physical systems. In physics, group theory plays a crucial role in understanding the symmetries of physical laws and predicting the behavior of particles and fields.
Syllabus
This course enables students to acquire a familiarity and working knowledge of the mathematics of symmetry groups, a cross-cutting and structuring concept in modern physics, from condensed matter to particle physics. Special emphasis will be put on the Lie groups relevant to High Energy Physics such as the Poincaré and Lorentz groups; SU(2) and SU(3) in relation with particle physics.
- Chapter 1: General Group theory (definitions and main theorems)
- Chapter 2: Finite and discrete groups. Examples include reflection groups and lattices in relation with crystallography, molecular physics etc. This chapter also introduces the notion of root systems as relevant to general Lie theory.
- Chapter 3: Introduction to Lie groups and Lie algebras
- Chapter 4: Introduction to the representation theory of Lie algebras
Suggested bibliography
- Fulton, W. and Harris, J. Representation Theory. Springer.
- Gilmore, R. Lie groups, Lie algebras and some of their applications. Dover.
- Georgi, H. Lie algebras in particle physics. CRC Press.
- Hamermesh, M. Group theory and its applications to physical problems. Dover
- Kosmann-Schwarzbach, Y. Groups and symmetries. Springer.
- Sternberg, S. Group theory and physics. CUP.
Prerequisites
- Elementary quantum mechanics (Hilbert spaces, operators);
- Elementary linear algebra (vector spaces, matrices etc).