Exact three-dimensional reduction of the Bethe-Salpeter equation

Abstract

The partial-wave Bethe-Salpeter equation in the ladder approximation is converted into a set of two one-dimensional equations. The resulting reduced equations determine two off-energy-shell, on-mass-shell amplitudes, one for the scattering of two positive-energy and the other for two negative-energy particles. The connection between the potentials of the reduced equations and the many-particle intermediate states is displayed by their series expansion. Two versions of the equations which are convenient for numerical computations are also obtained, and one of them is a one-channel equation which involves only the physical amplitude for the scattering of two positive-energy particles. The potentials are determined by auxiliary equations which are two dimensional but nonsingular and exactly solvable by standard numerical methods. One of the results of the derivation of the reduced equations is a set of reduction relations which express the off-mass-shell amplitude as a functional of a restricted shell amplitude.