Relativistic few-body problem. I. Two-body equations

Abstract

This paper begins with an explanation of the implications of the requirement that a two-body relativistic equation should approach a one-body equation when one of the masses becomes very large. It is found that the Bethe-Salpeter equation does not satisfy this requirement. An infinite family of three-dimensional equations depending on a parameter $\ensuremath{-}1\ensuremath{\le}\ensuremath{\nu}\ensuremath{\le}1$ is constructed, all of which do satisfy this limit. When $|\ensuremath{\nu}|=1$ one of the particles is on its mass shell; when $\ensuremath{\nu}=0$ both particles are equally off mass shell. The fourth order irreducible kernel for this family is studied in the expanded static limit for all $\ensuremath{\nu}$. It is found, both for scalar theories and for a realistic chiral theory of spin \textonehalf{} nucleons interacting with isovector pions, that the leading order terms in the static limit cancel for any $\ensuremath{\nu}$, and that the nonleading terms are independent of energy only for the $|\ensuremath{\nu}|=1$ equation. Other criteria for the selection of a relativistic two-body equation and implications for the form of the two-pion exchange potential are briefly discussed.