Divergence-Free Iterative Expansion of theSMatrix in a Field Theory

A new method is proposed for evaluating the $S$-matrix as a series expansion in powers of the coupling constant. The method is applicable to field theories which in the usual formulation have ultraviolet divergences in self-energy and vertex parts and require self-energy, coupling constant, and wave-function renormalization. The procedure cannot be applied in its present form to theories which allow boson self-energy terms. In this new procedure the usual form of the Hamiltonian for the coupled system is retained. The theory results in an iterated solution in powers of the physical coupling constant and yields a series, each term of which is finite without subtractions or renormalization. It agrees up to all orders examined with the finite $S$-matrix elements obtained by renormalizing the old formulation of the scattering problem.