Axiomatic Perturbation Theory for Retarded Functions

Abstract

A construction of the retarded n-point functions of perturbation theory is given within the Lehmann, Symanzik, and Zimmermann framework and without the specification of an interaction Lagrangian. An intermediate-state expansion of retarded functionals is employed to define a systematic set of equations representing approximations to the (integral) unitarity conditions; the requirement of symmetry of the n − 1 retarded coordinates of an n-point retarded function enters in an essential way. The class of solutions to these equations contains the renormalized perturbation theory retarded functions corresponding to local renormalizable Lagrangian interactions, as well as more singular functions corresponding to nonrenormalizable interactions; if the latter are excluded all the n-point functions may be successively determined to all orders in the renormalized coupling constants. The construction is explicitly performed for the first radiative corrections to the 2- and 3-point functions of a self-interacting neutral scalar boson field, yielding the finite renormalized results of perturbation theory. Similar but slightly singular results are quoted for the π-π scattering amplitude.