Three-Particle Nonrelativistic Kinematics and Phase Space

Abstract

The kinematics of a nonrelativistic three-particle system is studied with the help of the general method devised by Lévy-Leblond and Lurçat. Basis states are constructed which are eigenstates, in addition to the total momentum-energy, angular momentum, etc., of new observables; among these, the ``togetherness tensor'' describes the simultaneous localization of the three particles and therefore is of great physical interest. All of these observables arise as Casimir operators of a ``great group'' acting on the three-particle phase-space manifold in a transitive way, and of some of its subgroups. In the present case, by trying to keep all the particles on the same footing (``democracy'' arguments), we are led to choose the SU3 group as a particularly convenient ``great group''. We thus recover completely the Dragt classification of non-relativistic three-particle states. The explicit calculation of the basis functions is done in a new way, by analytical methods, solving partial derivative equations. This enables us to establish the most general form of these basis functions.