Theoretische Physik SS 09 Exercises on Theoretical Astroparticle Physics

Lie groups and Lie algebras are of highest importance for the understanding of the standard model (SM), grand unified theories (GUT’s) and particle physics in general. The symmetries given by them imply that the particles are organized into multiplets which are built out of the representations of the groups. This way many particles have been predicted before their experimental detection. Therefore the first exercise sheets will deal with simple lie algebras, their representations and their classification. If you want to learn more about group theory, we recommend the books given in the bibliography at the end. 1. A Lie group G is roughly a group which at the same time is a differentiable manifold. The group action · : G × G → G and the inversion ()−1: G → G are also required to be differentiable maps. One can first divide them into Abelian and non-Abelian groups, but the Abelian ones are very easy to handle so our focus lies on the non-Abelian Lie groups. The simplest of them is SU(2), the group of Special Unitary 2 × 2 matrices. SU(2):