Equivalence of a New Wave Equation to the Breit and Salpeter Equations* John

It is proven that a recently derived simple configuration-space wave equation for two spin-l/2 particles is equivalent to the Breit equation and to the Salpeter equation in first order perturbation theory. The wave equation is based on a simple quasipotential approximation. The potential in the equation is given by the Blankenbecler-Sugar correction series. The proof holds for an arbitrary combination of scalar and vector interactions. Submitted to Physical Review D, Brief Reports * Work supported in part by the Department of Energy, Contract No. DE-AC03-76SF00515 and Associated Western Universities-DOE Summer Faculty Fellowship. ** Permanent address. Interest in S-dimensional two-body bound state equations for spin-l/2 particles has revived in the last 15 years, due to the success of the qij model of mesons. But two important questions have not been answered yet [1,2]. (1) What equation is best? (2) What interaction is best? This note is about equations. Two have dominated: the Breit equation [3] and Salpeter’s reduction [4] of the Bethe-Salpeter equation [5]. As we shall explain below, neither of these “big two” equations can be solved numerically exactly in configuration space. They are only solved in first-order perturbation theory, which is trusted most for non-relativistic systems. Here we will prove the equivalence in first order perturbation theory of these two equations to a recently derived third equation [6] which h as f avorable configuration space behaviour and may be susceptible to an exact numerical solution. Light mesons, where relativistic effects are high, are still treated by first-order perturbation theory [7]. A practical configuration-space equation which treats relativistic effects in some way beyond first-order perturbation theory is badly needed. A one-particle example of what we mean is the Dirac equation with a static source. For weak potentials the level splitting can be calculated satisfactorily either from the exact solution of the equation or from first-order perturbation theory. But for stronger potentials it is clearly better to solve the Dirac equation exactlynumerically if need be. Because of the simple form of the Dirac equation in configuration space, such a numerical solution is easy to carry out. For this reason the Dirac equation is accepted as a relativistic wave equation. We would like to have a solvable relativistic wave equation for two spin-l/2 particles. We first briefly review the two dominant equations. In the CM system the Breit equation is [3] [(7’m + 7’7. p) + (lT”A4 Fora p) E] $(r) = -y”lToW(r)$(r) . (1) Here y”, 7, m refer to one particle and I”, I’, M refer to the other. The total energy