  An operator generalisation of the notion of geometric phase has been recently
proposed purely based on physical grounds. Here we provide a mathematical
foundation for its existence, while uncovering new geometrical structures in
quantum systems. While probing the average of any observable it is found that a
quantum system exhibits different ray spaces and associated fibre bundle
structures. The generalised geometric phase is understood as (an)holonomy of a
connection over these fibre bundles. The underlying ray spaces in general are
found to be pseudo-Kahler manifolds, and its symplectic structure gets
manifests as the generalised geometric phase.
