For a general evolution of a quantal system, the geometric phase measured with reference to a given initial state is derived as an integral of a function of the pure state density operator by invoking the Pancharatnam connection continuously.

In mid-1950s, Pancharatnam [1] encountered the geometric phase associated with the evolution along a geodesic triangle on the Poincaré sphere. We generalize his 3-vertex phase and employ it as the fundamental building block, to geometrically construct a general ray-space expression for geometric phase. In terms of a reference ray used to specify geometric phase, we delineate clear geometric meanings for gauge transformations and gauge freedom, which are generally regarded as mere mathematical abstractions.

We present an overview of polarized neutron experiments observing SU(2) phases. The first experimental separation of geometric and dynamical phases, the explicit verification of Pauli anticommutation and the first observation of interference amplitudes and phases in noncyclic evolutions are described. These experiments elucidate the physics of phases and phase jumps propounded by the Pancharatnam connection.

A general gauge-invariant formalism for parallel transport, geodesics and geometric phase based on the pure state density operator is propounded. A single-query quantum search algorithm is proposed.