Relativistic Combination of Internal and Spin Symmetries inS-Matrix Formulation

Abstract

The relativistic combination of internal and spin symmetries is considered as a restriction on the scattering operator. A physical principle for the statement of symmetries other than space-time is formulated. In the simplest case, the intuitive idea that interactions do not depend on some observables is expressed by the statement that the $S$ operator belongs to a subring of the ring of observables. This principle leads to the usual results for purely internal symmetries, for spin independence of forces, and also for their combinations as in Wigner's supermultiplet theory. The combination of Poincar\'e group, spin independence, and internal symmetry leads to a group which is not a (finite-dimensional) Lie group and not locally compact. Nevertheless, simple homomorphic representations of the little group exist and suffice for physical applications. The nonrelativistic $\mathrm{SU}(6)$ multiplet structure and the selection rules are unchanged in the relativistic case. In addition to the deficiencies of the nonrelativistic model, the relativistic $S$ operator has highly unrealistic properties, although it is not trivial ($S\ensuremath{\ne}1$). A scheme for symmetry breaking is described which may be expected to remedy these deficiencies.