GEOMETRIC PHASES, GAUGE SYMMETRIES AND RAY REPRESENTATION

Abstract

The conventional formulation of the nonadiabatic (Aharonov–Anandan) phase is based on the equivalence class {eiα(t)ψ(t,x)} which is not a symmetry of the Schrödinger equation. 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 nonadiabatic phase. This reformulation reproduces all the successful aspects of the nonadiabatic 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 nonadiabatic phases in the absence of gauge fields, and it allows a unified treatment of all the geometric phases. The nonadiabatic phase may well be regarded as a special case of the adiabatic phase in this reformulation, contrary to the customary understanding of the adiabatic phase as a special case of the nonadiabatic phase. Some explicit examples of geometric phases are discussed to illustrate this reformulation.