A criterion for internal symmetry in an $S$-matrix framework is formulated and a method to determine permitted internal symmetries by a rigorous application of unitarity and crossing is developed. Applying this method to particular cases, one finds amplitude relations that are usually obtained by assuming invariance under certain Lie groups. For example, applications are made to isotopic-spin invariance in the coupled pion-kaon system and to $\mathrm{SU}\left(3\right)\mathrm{}$ invariance in the scattering of the 8 pseudoscalar mesons. Other higher symmetries are discussed. By clarifying the essential assumptions which go into dynamical derivations of symmetries, our method permits more general derivations from weaker assumptions. It is shown that the usual dynamical schemes used to derive internal symmetries essentially assume most of their results and in addition contain many inessential assumptions. For example, the bootstrap approach requires the existence of poles and the bootstrap conditions, whereas our approach requires neither.

• Received 1 September 1966

DOI:https://doi.org/10.1103/PhysRev.156.1624