N-Particle Kinematics and Group-Theoretical Treatment of Phase Space I. Nonrelativistic

Abstract

This paper is a group‐theoretical study of the kinematics of n nonrelativistic particles. A systematic method is given to construct a new complete set of commuting observables. The method is based on the existence of a group (the ``great group'') which acts transitively on the phase‐space manifold and preserves the phase‐space volume element; the observables are then Casimir operators of the great group and of some of its subgroups, including the usual three‐dimensional rotation group. Among these collective observables, in addition to the total angular momentum, the most interesting is the ``togetherness operator'' which describes the simultaneous localization of the n particles. This operator is a generalization to n > 2 of the square of orbital angular momentum; its use allows to generalize to n particles the familiar centrifugal barrier arguments.