Seminar VALTER MORETTI anticipated

[Emanuele Tuillier Illingworth]

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                              JOINT SESSION
                     NON-COMMUTATIVE GEOMETRY SEMINAR
                                   AND
                 ANALYSIS, MATH-PHYS, AND QUANTUM SEMINAR

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                     Thursday 12 February 2015, 9:15
                 SISSA main building, lecture room A-134


                         Valter Moretti (Trento)
     Frame functions in finite-dimensional Quantum Mechanics and its
           Hamiltonian formulation on complex projective spaces


Summary: I will discuss 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 role 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.