Dynamical symplectic symmetry in the quasiparticle-phonon nuclear model

It is shown that under some restrictions the quasiparticle-phonon nuclear model (QPM) has the dynamical symplectic symmetric limit. The conditions for the phonon amplitudes under which the generators of the $Sp(2d, R)$ algebra and of its central inhomogeneous extension – the $WSp(2d,R)$ algebra – can be constructed from the phonon and biphonon QPM operators are derived. The contribution of the pure $Sp(2d,R)$ symmetry into the general $WSp(2d,R)$ symmetry of the QPM is discussed. The exact Dyson boson realizations of the phonon and biphonon operators in the spaces of the Perelomov partially coherent states and Barut–Girardello coherent states are obtained.