Relativistic wave equations in momentum space

Relativistic equal-time wave equations obtained from field theory which describe bound states of $N$ Dirac particles inevitably involve Casimir-type positive-energy projection operators ${\Lambda }_{+}\mathrm{}\left(i\right)\mathrm{}$. For $N>\mathrm{}2\mathrm{}$, these operators are vital if the equations are to admit normalizable solutions. Such equations, which are of integro-differential form, have been used in the past to obtain relativistic corrections to, e.g., level shifts for a variety of simple atomic systems, and to provide a theoretical basis for the Dirac-Hartree-Fock type of equations for many-electron atoms. Here we initiate a study of such equations without making an expansion in powers of $\frac{v}{c}$. We work in momentum space, where the free-particle projection operators are simple functions of $\overrightarrow{\mathrm{p}}$ and the wave equation is essentially no more complicated than in the nonrelativistic case. In the present paper we describe techniques for finding the eigenvalues of $h_{+}\mathrm{}(1,2)=h_D\mathrm{}\left(1\right)+h_D\mathrm{}\left(2\right)+{\Lambda }_{++}V{\Lambda }_{++}$, where $h_D\mathrm{}\left(i\right)\mathrm{}$ is the free-particle Dirac Hamiltonian and $V$ is a local potential with a ${|{\overrightarrow{\mathrm{r}}}_1-{\overrightarrow{\mathrm{r}}}_2|}^{-1\mathrm{}}$ singularity. Numerical results are presented for the case of a pure Coulomb potential and a Coulomb-plus-Breit potential, for a wide range of mass ratios $\frac{m_1}{m_2}$ and coupling strength $\frac{e_1{e}_2}{4\pi }$. In the $m_2=\infty$ limit, comparison is made with the Dirac equation. The results are used to discuss the magnitude of level shifts associated with virtual-pair production in such two-body systems.