Alternative Formulations of Scattering in Model Field Theories

A comparison is made between the expression for the boson-fermion scattering amplitude, in a model theory, obtained by renormalizing Feynman diagrams, and the one that originates from the dressed-particle picture. A proof is given of a surmise, previously verified to sixth order, that the two expressions for the transition amplitude are identical, although the former stems from a theory that leads to paradoxes from which the latter is exempt. The proof of the identity of the two representations is extended to transition amplitudes in which a weak interaction, considered to first order, is modified by a strong one considered to all orders. It is shown how the operations in the dressed-particle picture directly lead to iterations in terms of the physical mass and coupling constant, whereas the expressions obtained from the Feynman diagrams require renormalization before they have this form.