Microscopic study of fermionic collective motion. (I). Functional representation of collective motion

Based on the shell structure of the finite nuclear many fermion system (FMFS), the coherent states related to the Spin(2r) group are defined. The global and local functional representations of the FMFS state-vectors and operators, defined on the coset space Spin(2r)/U(r), are constructed. The nonuniqueness of the coherent state functional representations is overcome by the imposition of a consistency condition on the wave functions. The influence of the boundary of the coset space Spin(2r)/U(r) on the local functional representation is physically removed only for the bound states of FMFS. The reason for the non-hermitian behavior of the local functional representation is exposed. Finally, using Bargmann's theory, the boson representation of FMFS are directly calculated from the local functional representation of FMFS. Thus, in this paper, we have demonstrated that the kinematics of the collective behavior of FMFS can be described in three non-equivalent representations: the fermion representation, the global functional representation and the local functional representation.