Bargmann invariants, null phase curves, and a theory of the geometric phase

Abstract

We present a theory of the geometric phase based logically on the Bargmann invariant of quantum mechanics, and null phase curves in ray space, as the fundamental ingredients. Null phase curves are themselves defined entirely in terms of the (third order) Bargmann invariant, and it is shown that these are the curves natural to geometric phase theory, rather than geodesics used in earlier treatments. The natural symplectic structure in ray space is seen to play a crucial role in the definition of the geometric phase. Logical consistency of the formulation is explicitly shown, and the principal properties of geometric phases are deduced as systematic consequences.