Non-Abelian geometric phase, Floquet theory and periodic dynamical invariants

For a periodic Hamiltonian, periodic dynamical invariants may be used to ob tain non-degenerate cyclic states. This observation is generalized to the d egenerate cyclic states, and the relation between the periodic dynamical in variants and the Floquet decompositions of the time-evolution operator is e lucidated. In particular, a necessary condition for the occurrence of cycli c non-adiabatic non-Abelian geometrical phase is derived. Degenerate cyclic states are obtained for a magnetic dipole interacting with a precessing ma gnetic field.