  The conventional formulation of the non-adiabatic (Aharonov-Anandan) phase is
based on the equivalence class $\{e^{i\alpha(t)}\psi(t,\vec{x})\}$ which is not
a symmetry of the Schr\"{o}dinger 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 non-adiabatic phase. This re-formulation 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. The non-adiabatic phase may well be regarded as a special case of the
adiabatic phase in this re-formulation, contrary to the customary understanding
of the adiabatic phase as a special case of the non-adiabatic phase. Some
explicit examples of geometric phases are discussed to illustrate this
re-formulation.
