The possibility of covariantly describing a system of a fixed number of particles interacting directly is explored by attempting a direct "integration" of the commutation relations for the inhomogeneous Lorentz group under restrictions appropriate to the term "system of a fixed number of particles." By direct interaction is meant the fact that interaction between the particles is expressed directly in terms of coordinates, momenta, and spins for the particles rather than through the agency of a mediating field. The integration is carried out in considerable generality with the assumption that the infinitesimal generators of the group have expansions in inverse powers of the square of the velocity of light. The result coincides with that obtained earlier by Bakamjian and Thomas, but the method employed yields greater insight into the generality of the result, as well as into how further conditions beyond covariance, such as the property which is here called "separability of the interaction," can be incorporated in the result. The relationship of the result to the complete reducibility of a representation of the inhomogeneous Lorentz group is pointed out. Possible generalizations and applications of the procedures here employed are discussed.

• Received 17 November 1960

DOI:https://doi.org/10.1103/PhysRev.122.275