Tentative field-theoretical interpretation of hadronic potentials

This paper is divided into two parts. In the first part we propose a generalization of d'Alembert's field equation which depends on one parameter $\omega$. For each value of $\omega$ the corresponding field equation is Poincaré invariant, admits causal solutions attached to timelike world lines, and is semilinear. In the static limit some of these causal solutions are the functions $r^{\rho }$, where the exponent $\rho$ depends on $\omega$ and where $r$ is the distance between the point where the field is calculated and the straight line defining the motion of the source. The general causal solutions are simple and interesting modifications of the preceding ones. In the second part we consider a system of $N$ pointlike particles interacting via any causal scalar field, and using the methods of relativistic predictive mechanics we calculate the Newtonian and post-Newtonian approximations of the equations of motion. The Newtonian approximation coincides with classical potential theory. The post-Newtonian approximation is unambiguously determined by the Newtonian approximation. The combined results of these two parts suggest some field-theoretical justifications to current ideas about hadronic potentials.