Analysis of Berry's phase by the evolution operator method

Berry (1984) discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase. Aharonov and Anandan (1987), on the other hand, showed that an eigenstate transporting round a closed circuit in the projective Hilbert space suffices to generate the topological phase. The authors employ the evolution operator method to study the propagation of an eigenstate. They show that Berry's phase can be represented as a closed path integral in the Hilbert space and is independent of the choice for the base ( mod n(t))). The treatments of Berry, and Aharonov and Anandan, are shown to correspond to two different choices of the base. Therefore their two approaches are unified; they have acquired a more general viewpoint on the origin of Berry's phase.