Equivalence of the four-point interaction and the Yukawa interaction. I. A Boson model

Abstract

It is shown that the compositeness conditions by themselves do not ensure the complete equivalence of the Yukawa-type interaction with the corresponding four-point interaction. In our study of the corresponding situation in the context of the Lee model versus the separable-potential model, we find in the limit in which the bare Yukawa coupling constant becomes infinite, the finite-energy wave functions calculated from the two theories coincide. It is shown, however, that although the finite-energy spectra of the two theories are the same, the Yukawa theory has additional spectral contributions at infinity. It is further demonstrated that it is only by removing all contributions from the spectrum at infinity that the two theories can be made essentially identical. It is therefore suggested that the proofs of the equivalence of the four-fermion interaction and the Yukawa-type interaction need to be reexamined.