Collectivity and geometry. I. General approach

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Abstract
In the last decade an extensive literature appeared in which the microscopic collective behavior of nuclei was associated with definite irreducible representations (irreps) of either the O(n) or Sp(6) groups, where n=A−1 and A is the number of nucleons. It became clear that the two approaches are equivalent, as problems with 3n degrees of freedom are characterized by a definite irrep of the group Sp(6n) and for its subgroup Sp(6)×O(n) the irrep of O(n) determines that of Sp(6) and vice versa. Thus one can consider that collective effects appear when one introduces the constraint that in the many-body system the states are restricted to a definite irrep of O(n) [and thus also of Sp(6)] and the Hamiltonians are in the enveloping algebra of Sp(6) rather than in that of Sp(6n). Once Sp(6) becomes the paramount group of collective motions, the problem is to determine the matrix elements of the generators of Sp(6) in a basis characterized by irreps of its subgroups. What subgroups to choose? Rowe and Rosensteel have taken Sp(6)⊇U(3) and Sp(6)⊇CM(3), where the latter has also been considered by Biedenharn et al. In the present series of papers we analyze the problem in the chain Sp(6)⊇Sp(2)×𝒪(3), as we show that in the boson limit, i.e., when n≫1, the Casimir operator of Sp(2) goes into the Casimir operator of U(5), i.e., the corresponding chain is U(6)⊇U(5)⊇𝒪(3). In the case Sp(6)⊇U(3)⊇𝒪(3), the boson limit is U(6)⊇U(3)⊇𝒪(3). Thus in this series of papers we look at the microscopic collective model from what could be called the vibrational rather than the rotational point of view.