Phase change during a cyclic quantum evolution

A new geometric phase factor is defined for any cyclic evolution of a quantum system. This is independent of the phase factor relating the initial and final state vectors and the Hamiltonian, for a given projection of the evolution on the projective space of rays of the Hilbert space. Some applications, including the Aharonov-Bohm effect, are considered. For the special case of adiabatic evolution, this phase factor is a gauge-invariant generalization of the one found by Berry.