An introduction into Lie Group, Lie Algebra, Representations and Spin

Abstract

In this thesis we will go through what a Lie group and a Lie algebra is. How they are linked to one another and we will have a look at their representations. The main focus will be towards matrix Lie groups and their associated Lie algebras. In particularly we will be looking into the matrix Lie groups SO(3) and SU(2) with their associated Lie algebras so(3) and su(2), resp. This is to describe the spin of a particle. Since the representations of SO(3) can only describe particles with an integer spin, i.e. bosons, we will look at the matrix Lie group SU(2) and show that this Lie group can describe particles of half-integer spins through representations, fermions. The main motivations was to understand the spin of an electron. The electron is a fermion and has spin-1/2. Therefore we will be focussing on theorems which will assist us to achive our quest to understand the spin.