Frame functions in finite-dimensional Quantum Mechanics and its Hamiltonian formulation on complex projective spaces

This work concerns some issues about the interplay of standard and geometric (Hamiltonian) approaches to finite-dimensional quantum mechanics, formulated in the projective space. Our analysis relies upon the notion and the properties of so-called frame functions, introduced by A.M. Gleason to prove his celebrated theorem. In particular, the problem of associating quantum state with positive Liouville densities is tackled from an axiomatic point of view, proving a theorem classifying all possible correspondences. A similar result is established for classical observables representing quantum ones. These correspondences turn out to be encoded in a one-parameter class and, in both cases, the classical objects representing quantum ones result to be frame functions. The requirements of $U(n)$ covariance and (convex) linearity play a central r\^ole in the proof of those theorems. A new characterization of classical observables describing quantum observables is presented, together with a geometric description of the $C^*$-algebra structure of the set of quantum observables in terms of classical ones.