The singularities of asymptotic quantum field theory

Abstract

It has been shown recently that the L.S.Z. formulation of quantum field theory can produce finite solutions for theS-matrix elements to all orders in perturbation theory. In this paper these solutions for the scattering of an arbitrary number of scalar bosons are examined in momentum space and their singularity structure is found. It is shown that the extension off the mass shell, as carried out byPugh (1), of the fundamental L.S.Z. equations, yields solutions which are the continuation of the right-hand side of the unitarity relation,R+R †=R † R to unphysical off-mass-shell regions. The analytic mass-shell amplitudes are constructed using the unitarity condition and this continuation. By induction it is shown that the perturbation solutions of asymptotic quantum field theory possess in any order only the singularities which are required by unitarity. These have been shown byStapp (3) to correspond to the Landau singularities of ordinary quantum field theory.