In the nonrelativistic case of the Schrodinger equation, composite particles correspond to Regge poles in scattering amplitudes {poles in the complex plane of angular momentum). It has been suggested that the same may be true in relativistic theory. In that case, the scattering amplitude in which such a particle is exchanged behaves at high energies like s~~'& Lsinvo (t) 7 ', where s is the energy variable and t the momentum transfer variable. When t=t~, the mass squared of the particle, then n

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... equals an integer n related to the spin of the particle. In contrast, we may consider the case of a field theory in which the exchanged particle is treated as elementary and we examine each order of perturbation theory. When N & 1, we can usually not renormalize successfully; when e & 1 and the theory is renormalizable, then the high-energy behavior is typically s"(ttn) 'd (t). Thus an experimental distinction is possible between the two situations. That is particularly interest-ing in view of the conjecture of Blankenbecler and Goldberger that the nucleon may be composite and that of Chew and Frautschi that all strongly interacting particles may be composite dynamical combinations of one another. We suggest a set of rules for finding the high-energy behavior of scattering cross sections according to the Regge pole hypothesis and apply them to m--7I, m-Ã, and N-37 scattering. We show how these cross sections differ from those expected when there are "elementary" nucleons and mesons treated in renormalized perturbation theory. For the case of 37-37 scattering, we analyze some preliminary experimental data and find indications that an "elementary" neutral vector meson is probably not present. Various reactions are proposed to test the "elementary" or "composite" nature of other baryons and mesons. Higher energies may be needed than are available at present.