On the algebraic formulation of collective models. II. Collective and intrinsic submanifolds

Abstract A general procedure is given for decomposing N -particle configuration space into orbits of a kinematical collective group and a smooth transversal. Evaluation of the Laplace-Beltrami operator, in terms of vector fields tangent to the orbit and to the transversal, gives the decomposition of the kinetic energy into collective and intrinsic parts. By such geometric means, the essence of the various transformations of coordinates is revealed in a simple coordinate free way. Specifically considered are the kinematical collective groups SO (3) and GL + (3, R ). It is also shown that, with center-of-mass motion removed, N -particle configuration space can be considered as an orbit of the group GL + (3, R ) × SO ( N − 1). This decomposition of the configuration space is observed to induce a corresponding decomposition of the N -particle Hilbert space into irreducible subspaces with respect to both interesting spectrum generating algebras for collective motion and of the symmetric group. Thus the connection is established with the algebraic approach to the microscopic realization of collective states, with full respect for the Pauli principle.

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