Some Aspects of the Covariant Two-Body Problem. I. The Bound-State Problem

A study has been made of the bound states of the Bethe-Salpeter equation for the nucleon-antinucleon system, including the ladder and pair-annihilation diagrams. For simplicity, nucleons and mesons were taken to be scalar, the latter having zero rest mass. Pair effects enter only in $S$-states with the bound states corresponding to the poles of the meson propagator ${D_F}^{\prime }$. The Bethe-Salpeter equation is closely related to the integral equation for the generalized vertex operator $\Gamma$; this has been solved by using an integral-transform method similar to that of Wick and Cutkosky, under the assumption that the nucleon mass is large compared to the binding energy. After performing a self-energy subtraction, the energy eigenvalues are found as a function of the coupling constant. These have the form given by the usual Bohr formula plus corrections. Finally, some comments are made with respect to the extension of the formalism to mesons with nonzero mass and spinor nucleons.