Separable solutions for directly interacting particle systems

Abstract

The problem of constructing a representation of the Poincaré group corresponding to a directly interacting system of a finite number of particles and satisfying the condition that the interaction be separable is considered by expansion of the group generators in powers of $\frac{1}{c^2}$. It is established that the problem has a solution to order $\frac{1}{c^2}$, but, except in special cases, the solution requires that the interaction contain three-body terms to order $\frac{1}{c^2}$ if it is the sum of two-body terms only to nonrelativistic order. Furthermore, there is considerable arbitariness in the $\frac{1}{c^2}$-order interaction term, and we discuss the possibility and significance of removing this arbitrariness by a unitary transformation. Finally, we discuss higher-order terms in $\frac{1}{c^2}$, where we present arguments to show that an $N$-particle system will eventually have some $N$-body interaction terms at some order in $\frac{1}{c^2}$ even though it contains only two-body terms nonrelativistically, and we then present some applications.