ua nt - p h / 06 05 08 1 v 2 1 6 M ay 2 00 6 Geometric phases , gauge symmetries and ray representation

The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is based on the equivalence class {eiα(t)ψ(t, ~x)}. This equivalence class when understood as defining generalized rays in the Hilbert space is not generally consistent with the superposition principle in interference and polarization phenomena. The hidden local gauge symmetry, which arises from the arbitrariness of the choice of coordinates in the functional space, is then proposed as a basic gauge symmetry in the non-adiabatic phase. This reformulation reproduces all the successful aspects of the non-adiabatic phase in a manner manifestly consistent with the conventional notion of rays and the superposition principle. The hidden local symmetry is thus identified as the natural origin of the gauge symmetry in both of the adiabatic and non-adiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. Some explicit examples of geometric phases are discussed to illustrate this re-formulation.