Discrete Phase Space-Continuous Time Relativistic Klein-Gordon and Dirac Equations, and a New Non-Singular Yukawa Potential

This paper deals with the second quantization of interacting relativistic Fermionic and Bosonic fields in the arena of discrete phase space and continuous time. The mathematical formulation involves partial difference equations. The corresponding Feynman diagrams and a new $S^{\#}$-matrix theory is developed. In the special case of proton-proton Moller scattering via an exchange of a neutral meson, the explicit second order element $\langle f | S^{\#}_{(2)} |i \rangle$ is deduced. In the approximation of very low external three-momenta, a new Yukawa potential is explicitly derived from $\langle f | S^{\#}_{(2)} |i \rangle$. Moreover, it is rigorously proved that this new Yukawa potential is divergence-free. The mass parameter of the exchanged meson may be set to zero to obtain a type of scalar Boson exchange between hypothetical Fermions. This provides a limiting case of a new Coulomb type potential directly from the new singularity free Yukawa potential. A divergence-free Coulomb potential between two Fermions at two discrete points is shown to be proportional to the Euler beta function. Within this relativistic discrete phase space continuous time, a single quanta is shown to occupy the hyper-tori $S^{1}_{n^1} \times S^{1}_{n^3} \times S^{1}_{n^3}$ where $S^{1}_{n}$ is a circle of radius $\sqrt{2n+1}$.