Discrete phase space - III: The Divergence-free S-matrix elements

In the arena of the discrete phase space and continuous time, the theory of S-marix is formulated. In the special case of Quantum-Electrodynamics (QED), the Feynman rules are precisely developed. These rules in the fourmomentum turn out to be identical to the usual QED, except for the vertex function. The new vertex function is given by an infinite series which can only be treated in an asymptotic approximation at the present time. Preliminary approximations prove that the second order self-energies of a fermion and a photon in the discrete model have convergent improper integrals. In the final section, a sharper asymptotic analysis is employed. It is proved that in case the number of external photon or fermion lines is at least one, then the S-matrix elements converge in all orders. Moreover, there are no infra-red divergences in this formulation.