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. Some problems related to expressing the kinetic energy in terms of vector fields on these manifolds are resolved. In particular, non-integrable coordinates previously used by one of the present authors is shown to arise from the imposition of unacceptable conditions. Finally the authors consider the corresponding decomposition of the N-particle Hilbert space. 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.