Normalization Condition and Normal and Abnormal Solutions of the Bethe-Salpeter Equation

Abstract

The normalization constants for normal and abnormal solutions of the Bethe-Salpeter equation are explicitly calculated in the Wick-Cutkosky model. In the case of the vanishing total four-momentum, it is shown that the abnormal solutions with odd $\kappa$ have a negative norm, where $\kappa$ is the Wick-Cutkosky quantum number, and that the corresponding scattering Green's function contains all the normal and abnormal solutions as the residues of poles. It is demonstrated explicitly for $\kappa =0,1,2,3\mathrm{}$ that the first conclusion remains true also in the case of an infinitesimally positive mass. As for the case of massless bound states, its special character is emphasized, and solid harmonics are constructed corresponding to the "little group" for a massless particle. Non-Cutkosky integral equations are obtained for the weight functions of the integral representation for the Bethe-Salpeter amplitude.