RelativisticU(6, 6)Theory

Abstract

Infinite-dimensional unitary representations of the noncompact group $U(6, 6)$ are employed to classify elementary particles and, following ideas related to those of Fronsdal, are used to construct relativistic $S$-matrix elements. Like the previously treated relativistic theories where finite-dimensional representations of $U(6, 6)$ were used, a particular $S$-matrix element shows no symmetry higher than that of the appropriate hybrid subgroup. The over-all $U(6, 6)$ symmetry may give new relations between form factors for different processes but will not, in general, give anything beyond the results of the previous formulations for the scattering processes. The unitarity of the $S$ matrix is compatible with the subgroup hierarchy, provided that an infinity of multiplets for elementary particles exists and provided that all such multiplets possess the same mass. The crucial point of our formulation is that if mass differences are introduced, these affect not the relativistic invariance but the unitarity of the $S$ matrix.