Two-body Dirac equations

Abstract

P. A. M. Dirac took the matrix square root of the Klein-Gordon equation to obtain his relativistic wave equation for a single spin-one-half particle. In this paper, we use Dirac's constraint mechanics and supersymmetry to perform the same operation on the relativistic description of two spinless particles to obtain consistent descriptions of two interacting particles, either or both of which may have spin one-half. The resulting coupled quantum wave equations correctly incorporate relativistic kinematics as well as heavy-particle limits to one-body Dirac or Klein-Gordon equations. The 16-component wave equations for the system of two spin one-half particles separate exactly into four decoupled four-component equations for the analogs of “upper” and “lower” components of the Dirac equation. Perturbative treatment of our equations through 0( α 4) automatically reproduces the appropriate fine structure. Furthermore, like the decoupled forms of Dirac's equation, the two-body versions have spin-dependent pieces that make non-perturbative quantum-mechanical sense. This feature eliminates the need for extra smoothing parameters in the potential or finite particle size in phenomenological applications.