Field-Theory Restrictions on the Unification of Space-Time and Internal Symmetries

The problem of combining the Poincaré group of space-time symmetries and an internal symmetry group $S$ in an over-all group $G$ is considered. It is argued that such an over-all group must have representations that are defined on the space-time manifold. As a consequence it is shown that the over-all symmetry group can at most contain the internal symmetry group as a proper subgroup and that the factor group $\frac{G}{S}$ must be isomorphic to a covering group of $P$. However, if we also take account of the fact that the existence of a symmetry group leads to conservation laws, then $G$ must be the direct product of $S$ and a covering group of $P$. As an additional result we show that the energy-momentum tensor does not contain an internal part, in contrast to the angular-momentum tensor.