Proof of the Mandelstam Representation for Every Order in Perturbation Theory

It is proved that every term in the perturbation series for a scattering amplitude satisfies the Mandelstam representation when there are no anomalous thresholds. The absence of anomalous thresholds can be investigated from a few low-order diagrams, or reduced diagrams. Under certain conditions it is shown that their absence in fourth order ensures their absence in every order.