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 SU<sub>3</sub> 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.