Nonrelativistic Interaction Between Two Nucleons

Abstract

The problem of the derivation of a two-nucleon Schrödinger equation from quantum field theory has been investigated, where only those mesons which are exchanged between the nucleons are taken into account. One expects that the two-particle Schrödinger picture will be useful if the energy (less rest energy) is small ($\stackrel{\sim }{<}\frac{{\mu }^2}{M}$), and if the important matrix elements are those which couple states of small momenta ($\stackrel{\sim }{<}\mu$). The procedure which has been followed has been to go over to Fock space in the manner of Tamm and Dancoff, and then to decouple the two-particle Tamm-Dancoff amplitude from all the others by a series of canonical transformations (the over-all coupling is assumed weak). Unlike the methods developed by Lévy-Klein and Bethe-Salpeter, the characteristic difficulties such as energy-dependent and non-Hermitian potentials are avoided. By way of application, the formalism is used to analyze the nonrelativistic nuclear forces for the neutral scalar and charge-symmetric pseudoscalar theories (with both pseudoscalar and pseudovector coupling). In this approximation, it is shown that there is agreement with the results of Lévy-Klein. In the course of the calculations, it is made evident that the "nonadiabatic velocity-dependent" corrections of Lévy-Klein appear even when the nucleons are taken to be fixed sources. Within the context of the method of canonical transformations, there is no justification for dropping these corrections as has been suggested by Brueckner and Watson. Finally, there is some evidence that the Tamm-Dancoff approximation is not an improvement over weak-coupling perturbation theory when applied to the nuclear-force problem, at least when the coupling constant is small.