All Possible Symmetries of theSMatrix

We prove a new theorem on the impossibility of combining space-time and internal symmetries in any but a trivial way. The theorem is an improvement on known results in that it is applicable to infinite-parameter groups, instead of just to Lie groups. This improvement is gained by using information about the $S$ matrix; previous investigations used only information about the single-particle spectrum. We define a symmetry group of the $S$ matrix as a group of unitary operators which turn one-particle states into one-particle states, transform many-particle states as if they were tensor products, and commute with the $S$ matrix. Let $G$ be a connected symmetry group of the $S$ matrix, and let the following five conditions hold: (1) $G$ contains a subgroup locally isomorphic to the Poincaré group. (2) For any $M>\mathrm{}0\mathrm{}$, there are only a finite number of one-particle states with mass less than $M$. (3) Elastic scattering amplitudes are analytic functions of $s$ and $t$, in some neighborhood of the physical region. (4) The $S$ matrix is nontrivial in the sense that any two one-particle momentum eigenstates scatter (into something), except perhaps at isolated values of $s$. (5) The generators of $G$, written as integral operators in momentum space, have distributions for their kernels. Then, we show that $G$ is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group.