A No-Interaction Theorem in Classical Relativistic Hamiltonian Particle Dynamics

Abstract

It is shown that a relativistically invariant classical mechanical Hamiltonian description of a system of three (spinless) particles admits no interaction between the particles. If a set of ten functions of the canonical variables of the three‐particle system satisfies the Poisson bracket relations characteristic of the ten generators of the inhomogeneous Lorentz group, and‐with the canonical position variables of the particles‐satisfies the Poisson bracket equations which express the familiar transformation properties of the (time‐dependent) particle positions under space translation, space rotation, and Lorentz transformation, then this set of ten functions can only describe a system of three free particles. A significant part of the proof is valid for a system containing any fixed number of particles. In this general case, a simplified form is established for the Hamiltonian and generators of Lorentz transformations, and it is shown that the generators of space translations and space rotations can be put in the standard form characteristic of free‐particle theories. The proof of the latter involves a generalization from one to many three‐vector variables of the angular momentum Helmholtz theorem of Lomont and Moses.