Separable solutions for directly interacting particle systems

The problem of constructing a representation of the Poincar\'e 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 1c². It is established that the problem has a solution to order 1c², but, except in special cases, the solution requires that the interaction contain three-body terms to order 1c² if it is the sum of two-body terms only to nonrelativistic order. Furthermore, there is considerable arbitariness in the 1c²-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 1c², where we present arguments to show that an N-particle system will eventually have some N-body interaction terms at some order in 1c² even though it contains only two-body terms nonrelativistically, and we then present some applications.