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Collectivity and geometry. I. General approach

### 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 3*n* degrees of freedom are characterized by a definite irrep of the group Sp(6*n*) 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(6*n*). 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 *e* *t* *a* *l*. 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.

© 1984 American Institute of Physics

Received 03 March 1983
Accepted 22 July 1983

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1984-05-01

2016-09-29

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