A model field theory is invented in the following way: Dispersion relations in the energy are assumed to hold for all amplitudes. Unitarity gives the absorptive parts in the "physical" regions. If it is assumed that the absorptive parts are otherwise zero (in violation of crossing symmetry and the Mandelstam representation), then the dispersion relations and unitarity form an infinite set of coupled integral equations for all amplitudes. An exact solution (at least for the simplest amplitudes) to this set of equations can be found, in which all self-masses, etc., are finite. The solution is equivalent to summing a certain class of Feynman graphs, computed in the usual way. For a wide range of coupling constants, there are no "ghost" difficulties.