Partial Dynamical Symmetry as an Intermediate Symmetry Structure

Symmetries play an important role in dynamical systems. They provide quantum numbers for the classification of states, determine selection rules and facilitate the calculation of matrix elements. An exact symmetry occurs when the Hamiltonian of the system commutes with all the generators (gi) of the symmetry-group, [H , gi ] = 0. In this case, all states have good symmetry and are labeled by the irreducible representations (irreps) of the group. The Hamiltonian admits a block structure so that inequivalent irreps do not mix and all eigenstates in the same irrep are degenerate. In a dynamical symmetry the block structure of the Hamiltonian is retained, the states preserve the good symmetry but in general are no longer degenerate (splitting but no mixing). When the symmetry is completely broken [H , gi ] 6= 0, and none of the states have good symmetry. In-between these limiting cases there may exist intermediate symmetry structures, called partial (dynamical) symmetries for which the symmetry is neither exact nor completely broken. Models based on spectrum generating algebras, such as those developed 1,2 by F. Iachello and his colleagues, form a convenient framework for discussing these different types of symmetries. In such models the Hamiltonian is written in terms of the generators of a Lie algebra, called the spectrum generating algebra. A dynamical symmetry occurs if the Hamiltonian can be written in terms of the Casimir operators (ĈGi) of a chain of nested algebras G1 ⊃ G2 ⊃ . . . ⊃ Gn [α1] [α2] . . . [αn] (1)