Symmetry Groups in Physics

This article is a résumé, intended for experimentalists and other nonspecialists who have a need to know of that part of group representation theory which is employed in the spectroscopic classification of energy levels. A summary of the general properties of semisimple Lie groups, sufficient to make the treatment self-contained, is included. Attention is then focused on Racah's important invariant operators (such as $J^2={J_1}^2+{J_2}^2+{J_3}^2$ for the rotation group), whose eigenvalues provide the essential connection between group theory and physics by serving as the "good quantum numbers" which label the multiplets and energy levels of the system. The rotation and isospin groups are used as examples throughout the discussion. The final section describes the way in which, in entirely analogous fashion, S${\mathrm{U}}_3$ and larger groups are being employed at present in the classification of strong-interaction multiplets.