ラダー近似で解ける相対論的束縛模型とそれがラダー近似について意味すること

We make a study to assess the accuracy of the ladder approximation, by employing a solvable twobodyb positron-theoretical composite model which is defined in terms of the Bethe-Salpeter equation in the ladder approxmation with various Fermi-type direct instantaneous interactions in 1 + 1 dimensional world. In this model, we obtain two sets of even- and odd-parity bound-state solutions. One set is obtained with a kind of renormalization of the coupling constant, and the other is gotten without any renormalization. We examine the wave functions of all the obtained bound-state solutions in detail, as they allow the probabilistic interpretation. Judging from some considerations, we have the following consequences: (A) the wave functton of the odd-parity bound-state solution obtained with a renormalization includes a remarkable and undesired feature (against a natural requirement) in the cases where the binding energy is large. This remarkable and undesired feature is owing to the inadeqeacy of the ladder approximation. (B) As for the remaining three bound-state solutions, qualitative features of their wave functions are acceptable. We suppose and emphasize that the ladder approximation (in the real world) may yield even undesired features against the physical requirements concerned with the wave functions (including the 3 + 1-dimensional generalization of a natural requirement stated above), at least in the case where the coupling constants of short-range interactions are renormalized.