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

Fairbairn, W.M.; Fulton, T.; Klink, W.H.

doi:10.1063/1.1704204

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

1964

Published in:

J.Math.Phys. 5 (1964) 8, 1038

DOI:

10.1063/1.1704204

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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 SU 3. 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 SU 2 to isotopic spin are extended to the subgroups of SU 3. The structure of SU 3 is utilized to suggest how charge and hypercharge operators are to be assigned in the subgroups. The results obtained are similar to those of SU 2 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 SU 3 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 SU 3/C is included in the appendix.

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

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It may appear puzzling that generators for all the Σ groups given in are 3‐dimensional and therefore seem to provide a counterexample to the result we have just obtained. These matrices generate 3‐dimensional representations of subgroups of SU3. In those cases, where the subgroups of SU3 differ from those of SU3/C, we must factor out the center to obtain the groups corresponding to φ = 1. The 3‐dimensional representations would then disappear. These considerations do not affect our character tables

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There is also no longer any reason why σ groups other than Σ(216) should not be analyzed. For convenience we will continue the analysis with Σ(216). The results we obtain will be typical of the other Σ groups

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