Relativistic bound-state equations in three dimensions.

Abstract

First, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasipotential approaches'' this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasipotential equations: They may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Second, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Third, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasipotential equations do. \textcopyright{} 1996 The American Physical Society.