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 16component wave function which contain Lorentz-covariant constituent potentials that are initially undetermined. An exact Pauli reduction leads to a second-order relativistic Schrodinger-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 Schrodinger-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 Schrodinger-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 a4. Surprisingly, this agreement depends crucially on inclusion of coupling between upper-upper and lower-lower components of our 16component Dirac wave functions and on the short-distance behavior of the relativistic quasipotential in the associated Schrodinger-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 'P, 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...

A large number of treatments of the meson spectrum have been tried that consider mesons as quark -anti quark bound states. Recently, we used relativistic quantum "constraint" mechanics to introduce a fully covariant treatment defined by two coupled Dirac equations. For field-theoretic interactions, this procedure functions as a "quantum mechanical transform of Bethe-Salpeter equation". Here, we test its spectral fits against those provided by an assortment of models: Wisconsin model, Iowa State model, Brayshaw model, and the popular semi-relativistic treatment of Godfrey and Isgur. We find that the fit provided by the two-body Dirac model for the entire meson spectrum competes with the best fits to partial spectra provided by the others and does so with the smallest number of interaction functions without additional cutoff parameters necessary to make other approaches numerically tractable. We discuss the distinguishing features of our model that may account for the relative overall success of its fits. Note especially that in our approach for QCD, the resulting pion mass and associated Goldstone behavior depend sensitively on the preservation of relativistic couplings that are crucial for its success when solved nonperturbatively for the analogous two-body bound-states of QED. * hcrater@utsi.edu

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Recently we used Dirac's constraint mechanics and supersymmetries to derive two coupled compatible 16-component Dirac equations that govern two relativistic spinning particles interacting through world scalar and vector potentials. They reduce exactly to four decoupled four-component local Schrodinger-like equations with energy-dependent quasipotentials @, . Their nonperturbative covariant structure [leading to perturbative and 0 ( 1 /c2) expansions that agree with field-theoretic approaches] suit these equations ideally for phenomenological applications in which the potentials have some links with relativistic field theories. (These equations are exactly solvable for singlet positronium producing a spectrum correct through order a4.) Here we use our equations to extend the validity of various one-or two-parameter models for the heavy-quark static potential to the relativistic light-quark regime. These models include the leading-log model (for all length scales) of Adler and Piran and Richardson's potential modified by flavor-dependent vacuum corrections. They significantly improve the good results that we obtained using Richardson's potential alone. Both nonperturbative and perturbative properties of the constraint approach are responsible for the spin-dependent consequences of the potential that result in a good overall fit to the meson masses. The nonperturbative structure dictated by the compatibility of our two Dirac equations enforces an approximate chiral symmetry that may account for the goodness of our pion fit. Perturbatively, for weak potentials, the upper-upper components of our equations reduce to the appropriate Todorov equation and then for low velocities to the Breit Hamiltonian. Thus, our approach reproduces the semirelativistic spin-dependent consequences of a quantum field theory. We strengthen this connection by deriving the Todorov inhomogeneous quasipotential equation for @, from the Bethe-Salpeter equation using an operator generalization of Sazdjian's quantum-mechanical transform of the Bethe-Salpeter equation. Consequently our covariant compatible coupled Dirac equations provide a nonperturbative framework for extrapolating O ( 1 /c2) field-theoretic results into the highly relativistic regime of bound light quarks.