Tamm-Dancoff Methods and Nuclear Forces

It is shown that by the use of 3-dimensional Feynman diagrams an equal-time two-particle wave equation can be written down. The procedure is the same one as Bethe and Salpeter used in deriving the 4-dimensional two-body equation. In this way all the terms of the new Tamm-Dancoff method of Dyson, in the form of the L\'evy-Klein expansion of the interaction function, can easily be recorded.The connection between the Bethe-Salpeter and Dyson equations is discussed. The spurious energy denominators of the fourth order interaction, in the adiabatic limit, can be eliminated by including all the contributions to the fourth order term arising from the 4- and 5-particle amplitudes that are coupled back to the two-particle amplitudes. The potential derived in the adiabatic limit to the fourth order term of the interaction operator is the same as the L\'evy-Klein potential derived from the old Tamm-Dancoff method.The Appendix contains a discussion of the Bethe-Salpeter equation and a simple method of derivation of equal-time equations from the 4-dimensional theories.