Boson representations of the real symplectic group and their applications to the nuclear collective model

Both non-Hermitian Dyson and Hermitian Holstein-Primakoff representations of the Sp(2d,R) algebra are obtained when the latter is restricted to a positive discrete series irreducible representation 〈λd + n/2,λ1 + n/2〉. For such purposes, some results for boson representations, recently deduced from a study of the Sp(2d,R) partially coherent states, are combined with some standard techniques of boson expansion theories. The introduction of Usui operators enables the establishment of useful relations between the various boson representations. Two Dyson representations of the Sp(2d,R) algebra are obtained in compact form in terms of v = d (d + 1)/2 pairs of boson creation and annihilation operators, and of an extra U(d) spin, characterized by the irreducible representation [λ 1⋯λd]. In contrast to what happens when λ1 = ⋯ = λd = λ, it is shown that the Holstein-Primakoff representation of the Sp(2d,R) algebra cannot be written in such a compact form for a generic irreducible representation. Explicit expansions are, however, obtained by extending the Marumori, Yamamura, and Tokunaga method of boson expansion theories. The Holstein-Primakoff representation is then used to prove that, when restricted to the Sp(2d,R) irreducible representation 〈λd + n/2, λ1 + n/2〉, the dn-dimensional harmonic oscillator Hamiltonian has a U(v)×SU(d) symmetry group. Finally, the results are applied to the Sp(6,R) nuclear collective model to demonstrate the existence of a hidden U(6)×SU(3) symmetry in this model. © 1985 American Institute of Physics.SCOPUS: ar.j