Gauge-independent formalism for parallel transport, geodesics and geometric phase

For a general quantal system, physical properties of the finite difference between a pair of density operators are derived and a complete set of gener ators for the associated 2-subspace is obtained. Each infinitesimal step in any general ray-space evolution takes place in the local 2-subspace and ca n be equated to a 'spin' rotation for the equivalent spin-1/2 particle. Hen ce a completely general Hamiltonian implementing a given ray space evolutio n comprises Pauli operators in the local 2-subspaces, constructed using the given density operator and its differential. Dynamical phase identifies wi th the phase in a rotating reference frame in which the 'spin' remains stat ionary. The transformation from this rotating frame to the laboratory frame effects a parallel transport evolution, producing geometric phase. A densi ty operator equation is derived for geodesics. Geometric phase arises from the parallel transport of the local 'spin' and equals minus half the integr al of 2-subspace solid angles.