The Tamm-Dancoff Formalism and the Symmetric Pseudoscalar Theory of Nuclear Forces

Abstract

The general method of deducing the Tamm-Dancoff equaltimes formalism, as generalized by L\'evy, from the relativistic two-body equation of Bethe-Salpeter and Schwinger is given. Only processes which are finite ab initio are considered. The essence of the procedure is the relation between a set of conventional matrix elements of the Tamm-Dancoff formalism and the Feynman diagram which summarizes them; this relationship provides a convenient guide for enumerating all matrix elements of a specified type and precludes the possibility of omission of any members of the set. Rules are also given for writing down any matrix element. The method is then applied to the derivation of the fourth-, sixth-, and eighth-order adiabatic potentials on the symmetrical pseudoscalar-pseudoscalar theory. Some discrepancies with the results of L\'evy are noted: In connection with the fourth-order potential these are first, that a more careful treatment of the energy denominators of the leading two-pair terms brings to light contributions that cancel with all other two-pair matrix elements that are of relative order $\frac{\ensuremath{\mu}}{M}$ compared to the leading ones; second, that the one-pair terms do not vanish but yield a repulsive interaction which substantially alters the qualitative picture of the fourth-order potential; third, that for the no-pair terms the result should agree with the previously calculated fourth-order potential for the pseudoscalar-pseudovector theory. The sixth- and eighth-order results are also in disagreement with L\'evy. Finally, an analysis of the problem of many-particle forces is given and explicit results obtained for the leading terms of the three- and four-particle forces as well as for certain smaller contributions to the three-particle interaction.