Groups, Lie Algebras, Symmetries in Physics

Abstract

The first problems in this chapter deal with basic properties of groups and of group representations. Fundamental results following from Schur lemma are introduced since the beginning in the case of finite groups, with simple applications of character theory, in the study of vibrational levels of symmetric systems. Other problems concern the notion and properties of Lie groups and Lie algebras, mainly oriented to physical examples: rotation groups ${ SO}_2$, $SO_3$, $SU_2$, translations, Euclidean group, Lorentz transformations, dilations, Heisenberg group, $SU_3$, with their physically relevant representations. The last section starts with some examples and applications of symmetry properties of differential equations, provides a group-theoretical interpretation of the Zeeman and Stark effects, and finally is devoted to obtaining the symmetry properties of the hydrogen atom (the group ${ SO}_4$) and of the three-dimensional harmonic oscillator (the group $U_3$) in quantum mechanics.