  We develop the widest possible generalisation of the well-known connection
between quantum mechanical Bargmann invariants and geometric phases. The key
notion is that of null phase curves in quantum mechanical ray and Hilbert
spaces. Examples of such curves are developed. Our generalisation is shown to
be essential to properly understand geometric phase results in the cases of
coherent states and of Gaussian states. Differential geometric aspects of null
phase curves are also briefly explored.
