matrix structure for an electromagnetic-like interaction would correspond to the Feynman gauge. If one inserts Eq.() into () and brings the free Dirac operator () to the right of the matrix hyperbolic functions and uses standard gamma matrix commutators and anticommutators and one arrives at in which The (covariant) structure of these equations are analogous to those of a Dirac equation for each of the two particles, with and playing the roles that and do in the single particle Dirac equation Over and above the usual kinetic part and time-like vector and scalar potential portions, the spin-dependent modifications involving and the last set of derivative terms are two-body recoil effects absent for the one-body Dirac equation but essential for the compatibility (consistency) of the two-body equations. The connections between what are designated as the vertex invariants and the mass and energy potentials are Comparing Eq.() with the first equation of this article one finds that the spin-dependent vector interactions are Note that the first portion of the vector potentials is timelike (parallel to while the next portion is spacelike (perpendicular to . The spin-dependent scalar potentials are The parametrization for and takes advantage of the Todorov effective external potential forms (as seen in the above section on the two-body Klein Gordon equations) and at the same time displays the correct static limit form for the Pauli reduction to Schrödinger-like form. The choice for these parameterizations (as with the two-body Klein Gordon equations) is closely tied to classical or quantum field theories for separate scalar and vector interactions. This amounts to working in the Feynman gauge with the simplest relation between space- and timelike parts of the vector interaction,. The mass and energy potentials are respectively so that Applications and limitations The TBDE can be readily applied to two body systems such as positronium, muonium, hydrogen-like atoms, quarkonium, and the two-nucleon system. These applications involve two particles only and do not involve creation or annihilation of particles beyond the two. They involve only elastic processes. Because of the connection between the potentials used in the TBDE and the corresponding quantum field theory, any radiative correction to the lowest order interaction can be incorporated into those potentials. To see how this comes about, consider by contrast how one computes scattering amplitudes without quantum field theory. With no quantum field theory one must come upon potentials by classical arguments or phenomenological considerations. Once