Two-body Dirac equations for particles interacting through world scalar and vector potentials

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Abstract
In a recent Letter, we used ‘‘two-body Dirac equations’’ to make the naive quark model fully relativistic. In this paper, we apply Dirac’s constraint mechanics and supersymmetry not to a string but instead to a system of two spinning particles to derive the two coupled Dirac equations that govern the quantum mechanics of two spin-1/2 particles interacting through world scalar and vector potentials. Along the way, we demonstrate that our equations are compatible, that the scalar and vector interaction structures contribute separately to compatibility when both are present, and that compatibility persists even in the presence of relative-momentum-dependent interactions. We show that extrapolation of supersymmetries associated with the ordinary single-particle Dirac equation to the interacting two-body system eliminates spin complications and reduces the compatibility problem to that of the corresponding spinless system. In addition, supersymmetry introduces physically important nonperturbative recoil effects that contribute to the correct perturbative spectrum while rendering nonsingular certain singular terms of standard semirelativistic approximations. This eliminates the need for singularity-softening parameters or finite particle sizes in phenomenological applications. In order to make clear the underlying structure of our equations, we also derive the corresponding coupled Dirac or Klein-Gordon equations that govern a system of two particles, one or neither of which has spin.