The Construction of Potentials in Quantum Field Theory

An inductive method, based on the formal solution of the Schrödinger equation in the form given by Lippmann and Schwinger, is used to construct scattering potentials in quantum field theories of various types. The method is applied to a linear theory without nucleon pair production, to a general nonlinear theory without pair production, and finally to a general nonlinear theory with pair production. The $n\mathrm{th}$ order potential is obtained by the solution of a Schrödinger type equation involving the potential of lower order; the series of potentials so obtained is, however, not a power series in the coupling constant. Rough estimates indicate that some of the problems of convergence associated with the usual perturbation expansion are obviated by this method. Simple illustrative examples are given for the linear and nonlinear theories.