Three-Dimensional Bethe-Salpeter Equation Applied to the Nucleon-Nucleon Interaction

A tractable, relativistic theory for the two-nucleon system is constructed. As a starting point, the Bethe-Salpeter (BS) equation is utilized. In order to reduce the BS equation to three dimensions, the Blankenbecler-Sugar method is generalized to include spin-½ particles. An instantaneous-interaction approximation to the BS equation is also investigated, and results similar to those obtained with the Blankenbecler-Sugar method are derived. Finally, as an application, the generalized potential or interaction kernel is approximated to order $g^2$ and compared to the one-pion-exchange potential (OPEP) used in conjunction with the Schrödinger equation. The singlet states are treated numerically, and results are presented which show that for a lab kinetic energy of 400 MeV, the phase shifts calculated from the relativistic theory differ from those calculated by solving the Lippmann-Schwinger or Schrödinger equation with OPEP by about 20%.