ON THE UNIQUENESS OF THE BERRY CONNECTION

Berry phase not only natural, but unique, both in the Abelian and the non-Abelian cases. Invariance is invoked under unitary transformations of the probability amplitudes in quantum systems to force the connection to be invariant under the unitary group. Because the action is not free, the horizontal subspace chosen by the connection has to be invariant under the little group, and that makes it orthogonal to the fiber's direction, yielding the conventional Berry connection. The argument works just as well for the non-Abelian case, where the fibers are orthonormal frames (Stiefel manifolds), and the state space a Grassmanian, with a transitive but not free unitary action.