Unitary declension of dynamical symmetries for the time-dependent harmonic oscillator

Abstract

Elaboration is given to a previous discussion concerning dynamical symmetries of the Lewis–Riesenfeld time‐dependent harmonic oscillator. A boson operator formalism is used to define a generator of impicitly time‐dependent unitary transformations. Bilinear combinations of the transformed boson operators are shown to span the symplectic algebra Sp(2n,IR) regarded as a larger noninvariance group for the Lewis–Riesenfeld Hamiltonian. The multiplet structure of the embeddings, Sp(2n) ↓SU(n), and SU (n+1) ↓SU(n), is determined for the case n=3, using established branching rules. Boson operator realizations are presented for the multiplet structures in each SU(n) declension of these dynamical groups.