On the Geometric Structure of State Space and Operators in Quantum Mechanics

This paper examines the geometric structure underlying quantum state space and operator representations. Building on the interpretation of quantum phase in terms of holonomy and parallel transport, Hilbert space is analyzed as a representation of transport structure in complex line and vector bundles. Quantum states are interpreted as equivalence classes under phase transformations, and operators arise as generators of infinitesimal transformations associated with the connection. The construction provides a geometric perspective on state space, inner products, and operator structure within standard quantum mechanics.