Finite Subgroups of Continuous Groups

Abstract

AbstractWe study non-Abelian discrete groups and their representations as finite subgroups of continuous groups. There exist continuous symmetries in nature such as Lorentz and gauge symmetries. In particle physics, since there are three families in quarks and leptons, continuous flavor symmetries such as the SU(3) group have long been considered. Although the non-Abelian discrete flavor symmetries has become one of the most attractive models to understand the origin of the observed lepton mixing matrix, there exists an important question how such a non-Abelian discrete flavor symmetry could arise. One desired solution is that the discrete flavor symmetry is a remnant symmetry of the continuous (gauge) symmetry. In the framework of the quantum field theory this could occur due to the spontaneous symmetry breaking of a non-zero VEV of a scalar field. This leads to a reduction of the original group to a subgroup of unbroken symmetry in the low-energy physics. In general, a different VEV alignment can lead to a different breaking pattern (different low-energy physics), so that a classification of the symmetry breaking patterns is important.