Canonical realizations of the Poincaré group. I. General theory

The canonical realizations of the full Poincaré group in classical mechanics are studied by means of a general formalism introduced in preceding papers. The resulting classification displays significant analogies with the quantum one. The irreducible realizations, corresponding to positive, zero, and imaginary mass particles with or without spin, are discussed. Also the irreducible realizations of the homogeneous Lorentz group are classified. Particular attention is given to the nonirreducible realization describing a system of two free particles and to a discussion of the physical meaning of the ’’center-of-mass’’ and ’’internal’’ variables. It is seen that this formalism provides a most natural framework for the introduction of a direct interaction between the particles according to the well-known prescription given by Bakamjian and Thomas. Finally, some simple models of relativistic ’’rigid’’ systems are discussed.