Singularity-Free Breit Equation from Constraint Two-Body Dirac Equations

Abstract

We examine the relation between two approaches to the quantum relativistic two-body problem: (1) the Breit equation, and (2) the two-body Dirac equations derived from constraint dynamics. The Breit equation is known to be pathological when singularities appear at finite separations $r$ in the reduced set of coupled equations for attractive potentials even when the potentials themselves are not singular there. They then give rise to unphysical bound states and resonances. In contrast, the two-body Dirac equations of constraint dynamics do not have these pathologies in many nonperturbative treatments. To understand these marked differences, we first express these contraint equations in a hyperbolic form. These coupled equations are then re-cast into two equivalent equations: (1) a covariant Breit-like equation with potentials that are exponential functions of certain ``generator'' functions, and (2) a covariant orthogonality constraint on the relative momentum. This reduction enables us to show in a transparent way that finite-$r$ singularities do not appear as long as the the exponential structure is not tampered with and the exponential generators of the interaction are themselves nonsingular for finite $r$. These Dirac or Breit equations, free of the structural singularities which plague the usual Breit equation, can then be used safely under all circumstances, encompassing numerous applications in the fields of particle, nuclear, and atomic physics which involve highly relativistic and strong binding configurations.