Quantum kinematic approach to the geometric phase. I. General formalism

Abstract

A new approach to the theory of the geometric phase in quantum mechanics, based entirely on kinematic ideas, is developed. It is shown that a gauge and reparametrization invariant phase can be naturally associated with any smooth open curve of unit vectors in Hilbert space. Expression of this phase in terms of total and dynamical parts, rewriting it exclusively in terms of the density matrix, the special roles of geodesics and the Bargmann invariants all emerge very naturally. Application of the formalism to standard quantum evolution, adiabatic or otherwise, gives back familiar results and also answers questions relating to effects of unitary transformations and symmetries. An application to the case of light beams propagating along an arbitrary fibre, in the geometrical optic limit, is worked out in detail. Connections to differential geometric approaches, and extension to nonunitary evolution, are also worked out.