In quark-model calculations of the meson spectrum, fully covariant two-body Dirac equations dictated by Dirac's relativistic constraint mechanics gave a good fit to the entire meson mass spectrum (excluding flavor mixing) with constituent world scalar and vector potentials depending on just one or two parameters. In this paper, we investigate the properties of these equations that made them work so well by solving them numerically for quantum electrodynamics (QED) and related field theories. The constraint formalism generates a relativistic quantum mechanics defined by two coupled Dirac equations on a 16-component wave function which contain Lorentz-covariant constituent potentials that are initially undetermined. An exact Pauli reduction leads to a second-order relativistic Schrödinger-like equation for a reduced eight-component wave function determined by an effective interaction—the quasipotential. We first determine perturbatively to lowest order the relativistic quasipotential for the Schrödinger-like equation by comparing that form with one derived from the Bethe-Salpeter equation. Insertion of this perturbative information into the minimal interaction structures of the two-body Dirac equations then completely determines their interaction structures. Then we give a procedure for constructing the full 16-component solution to our coupled first-order Dirac equations from a solution of the second-order equation for the reduced wave function. Next, we show that a perturbative treatment of these equations yields the standard spectral results for QED and related interactions. The relativistic potentials in our exact Schrödinger-like equations incorporate detailed minimal interaction and dynamical recoil effects characteristic of field theory yet, unlike the approximate Fermi-Breit forms, do not lead to singular wave functions for any angular momentum states. Hence, we are able to solve them numerically and compare the resultant nonperturbative energy eigenvalues to their perturbative counterparts and hence to standard field-theoretic results. We find that nonperturbative solution of our equation produces energy levels that agree with the perturbative spectrum through order ${\alpha }^4$. Surprisingly, this agreement depends crucially on inclusion of coupling between upper-upper and lower-lower components of our 16-component Dirac wave functions and on the short-distance behavior of the relativistic quasipotential in the associated Schrödinger-like equation. To examine speculations that the effective potentials (including the angular momentum barrier) for some states in the $e^{+}{e}^{-}$ system may become attractive for small separations, we study whether our equations predict pure QED resonances in the $e^{+}{e}^{-}$ system which might correspond to the anomalous positron peaks in the yield of $e^{+}{e}^{-}$ pairs seen in heavy-ion collisions. For the ${}_{}{}^3{}_{}{}^{}P_0^{}{}_{}{}^{}$ state we find that, even though the quasipotential becomes attractive at separations near 10 fm and overwhelms the centrifugal barrier, the attraction is not strong enough to hold a resonance. This result contradicts recent predictions of such states by other authors based on numerical solutions of three-dimensional truncations of the Bethe-Salpeter equation for which the QED bound-state wave equation has been treated successfully only by perturbation theory.

• Received 26 December 1991

DOI:https://doi.org/10.1103/PhysRevD.46.5117