Many-body forces and the cluster decomposition

Direct-interaction dynamics is considered in the relativistic Hamiltonian constraint formalism. It is proven that the Todorov-Komar equations for an $N$-particle system ($N>\mathrm{}2\mathrm{}$) of mutually interacting particles have no solutions that permit interaction if only two-body forces are admitted. The inclusion of many-body forces leads to a system of equations that determines allowed classes of such forces recursively. Starting with given two-body forces that are allowed for the two-body problem, three-body, four-body, etc. forces can be obtained in successive steps from the solutions of equations which we specify. When the interactions are separable, i.e., when they vanish sufficiently fast with increasing distance, the cluster decomposition holds: for large enough separation the dynamics of each cluster becomes independent of the dynamics of all other clusters while maintaining the internal dynamics.