Equivalence of the four-point interaction and the Yukawa interaction. II. A fermion model

Abstract

It is explicitly demonstrated that, just as in the case of the four-point bosonic interaction, the compositeness conditions by themselves do not ensure a complete equivalence of the Yukawa interaction with the corresponding four-fermion interaction. We study the corresponding situation in the context of the Lee model versus the separable-potential model, with the N and θ fields quantized as fermions and V as a boson. We find that the spectrum of the Lee model consists of that of the separable potential together with some other contributions. It is shown that in the limit in which the bare coupling constant of the Yukawa theory becomes infinite, these additional contributions move to infinity. It is further shown that the finite-energy wave functions calculated from the two theories coincide in the strong-coupling limit (this limit ensures the compositeness condition). It is demonstrated, in the same manner as for the bosonic case, that in order to make the two theories essentially identical, in addition to taking the strong-coupling limit, all the spectral contributions at infinity in the Yukawa theory have to be removed. It is, thus, suggested that the proofs of equivalence of the four-fermion interaction and the Yukawatype interaction may need to be reexamined.