A group theoretic treatment of the geometric phase

We define a new unitary operator in the Hilbert space of a quantum system which parallel transports the state of the system along an arbitrary curve in the projective Hilbert space. This operator is geometrical even for an open curve in the sense that it depends uniquely only on the curve and is independent of the Hamiltonian. Using this, when the curve is closed, the geometric phases discovered by Pancharatnam, Berry and Aharanov-Anandan are obtained