| Relativistic equal-time wave equations obtained from field theory which describe bound states of N Dirac particles inevitably involve free or external-field positive-energy projection operators (LAMDA)(,+)(i). For N > 2 these operators are vital if the equations are to admit normaliz- able solutions. Such equations have been used in the past to obtain relativistic corrections to simple atomic systems, and to provide a theoretical basis for the Dirac-Hartree-Fock type of equations for many-electron atoms. These equations also find applications in ele- mentary particle physics in describing bound states of quarks. Here we initiate a numerical study of such equations, avoiding an expan- sion in powers of v/c. We work in momentum space, where the free projection operators are simple functions of (')p. We describe tech- niques for finding the eigenvalues and eigenfunctions of H(,+)(1,2) = h(,D)(1) + h(,D)(2) + (LAMDA)(,++)V(LAMDA)(,++) where h(,D)(i) is the free Dirac Hamiltonian and V is a local potential with either a (VBAR)(')r(,1)-(')r(,2)(VBAR)('-1) singularity in the case of atomic systems, or a (VBAR)(')r(,1)-(')r(,2)(VBAR) behavior plus a Coulomb-like singular- ity in the case of bound quarks. Results are presented for both pure Coulomb and a Coulomb plus Breit potential for the atomic case, and for a pure Lorentz scalar in the linear potential case. In the atomic case a wide range of m(,1)/m(,2) and coupling strength (gamma) is studied and the m(,2) = (INFIN) limit is compared with the Dirac equation. The magni- tude of level shifts associated with virtual pair production in such two-body systems is discussed. For intermediate values of (gamma) a com- parison is made between the numerical results and those of pertur- bation theory. We find that there can often be large corrections to perturbative results even for not terribly large values of v/c. We also study the strong coupling limit and find the value (gamma)(,max) for which the lowest-lying bound state disappears from the spectrum. For the linear potential, the relativistic eigenfunctions are used to study, in the absence of pair effects, the dipole amplitude of radiative transitions between bound states of charmonium. We again make a comparison with a v/c expansion and find that in our model of the (c,c) system perturbation theory may not be yielding reliable results. |
