Infinite-Component Field Theories, Fubini Sum Rules, Completeness, and Current Algebra. I. Discrete Spectra

Abstract

It is shown that the Born-approximation scattering amplitudes in a class of infinite-component field theories satisfy Fubini sum rules. The contributions to the sum rules are analyzed, and completeness relations are obtained. These are found to differ radically from the naive expectations. Singularities associated with the vertices give rise to cuts in the scattering amplitudes; the discontinuities contribute to the sum rule and hence to the completeness relations. Such contributions are incompatible with current algebra and with locality of the second-quantized form of the theory. Spacelike solutions, on the other hand, seem to be less relevant than has been feared.