Finite and Disconnected Subgroups of SU3 and their Application to the Elementary-Particle Spectrum

Abstract

An attempt is made to fit the symmetries of the currently observed elementary‐particle spectrum into the structure of finite or disconnected subgroups of SU₃. Surprisingly, the detailed properties of these subgroups have not been elucidated previously. As a first step, therefore, character tables and other relevant properties are derived for these groups. Next, the classification of elementary particles is made on the basis of the representations of the groups discussed. The techniques previously employed by Case, Karplus, and Yang for the application of finite subgroups of SU₂ to isotopic spin are extended to the subgroups of SU₃. The structure of SU₃ is utilized to suggest how charge and hypercharge operators are to be assigned in the subgroups. The results obtained are similar to those of SU₂ and isotopic spin. There is an upper limit, for any given group, to the dimension of the irreducible representation. For some of the groups considered, these upper limits are eight and even ten. There exist finite groups which can accommodate the eight baryons in one of the irreducible representations. However, when one looks at scattering problems, use of the finite groups, as expected, gives charge or hypercharge conservation only modulo an integer determined by the group. Charge independence is also lost. In a representative group analyzed in detail, the imposition of exact charge conservation leads automatically to the exact conservation of hypercharge and to the full SU₃ symmetry. Exact charge and hypercharge conservation can be maintained for the disconnected groups, but the maximum dimension of the irreducible representations is six, and only charge symmetry, not charge independence, is satisfied. A short discussion of the representations of the group SU₃/C is included in the appendix.