Asymptotic Conditions and Perturbation Theory

The formulation of field theories based on few postulates but not using Lagrangian, Hamiltonian, or field equations has been investigated by many authors. In such a theory the coupled integral equations for Green's functions play the role of a substitute for field equations and serve to determine various physical quantities from the theory. In this paper we give a general prescription for the systematic solution of the coupled integral equations in perturbation theory.We discuss two kinds of Green's functions: (1) the retarded functions, and (2) the time-ordered functions. Especially in the solution of the latter we have to use dispersion relations, and one finds a complete correspondence between the numbers of subtractions in the dispersion relations and the types of interactions in the conventional field theory; furthermore, the so-called renormalized coupling constants can be introduced into our theory through the boundary conditions supplementing the substracted dispersion relations. On the contrary, however, it is not possible even to define the unrenormalized coupling constants in our scheme. In other words, all unobservable divergent quantities are completely eliminated in our formulation, and no divergences occur in the course of the entire calculations.We apply this method to quantum electrodynamics to illustrate the above statement and also to show how one can get convergent unambiguous solutions in agreement with the conventional renormalized quantum electrodynamics. In discussing quantum electrodynamics, it is necessary to discover how to express the requirement of gauge invariance without referring to Lagrangian, Hamiltonian, or field equations; it is found that a set of equations which is a straightforward generalization of the Ward identity meets this requirement.