Equivalent Representations of Collective Hamiltonian and Implication on Generalized Density Matrix Method

We discuss equivalent representations of the collective/bosonic Hamiltonian in the form of Taylor expansion over collective coordinate and momentum. Different expansions are equivalent if they are related by a transformation of collective variables. The independent parameters in the collective Hamiltonian are identified, which are much less in number than it appears. In this sense, the microscopic generalized density matrix method fixes the collective Hamiltonian completely, which seems to solve the old problem of microscopic calculation of the collective Hamiltonian.