The nonmodular topological phase and phase singularities

Generalizing an earlier definition of the noncyclic geometric phase (R.Bhandari, Phys.Lett.A, 157, 221 (1991)), a nonmodular topological phase is defined with reference to a generic time-dependent two-slit interference experiment involving particles with N internal states in which the internal state of both the beams undergoes unitary evolution. A simple proof of the shorter geodesic rule for closure of the open path is presented and several useful new insights into the behaviour of the dynamical and geometrical components of the phase shift presented. An effective hamiltonian interpretation of the observable phase shifts is also presented.