Geometry of collective motions

A general projection method for decomposing the kinetic energy of an N-particle system into collective and intrinsic parts defined respectively on the orbits and the orbit space of a Lie transformation group is given. Specific targets of the application of the method are the kinematical group GL+(3,R) and the quotient set GL+(3,R)/SO(3) for their importance in microscopic formulation of nuclear collective motions. For these two cases the orbit spaces in the particle configuration space are shown to be identifiable with the Grassman and Stiefel manifolds of 3-planes and 3-frames respectively. Finally the corresponding decomposition of the N-particle Hilbert space is considered. It is proposed that an appropriate basis function for the GL+(3,R) collective model is provided by an irreducible representation of the boson SU(6) group. (author)