Geometric Hamiltonian formulation of Quantum Mechanics on complex projective spaces

By Davide Pastorello (U. Trento)

The topic of the talk is the geometric Hamiltonian formulation of finite-dimensional Quantum Mechanics and its interplay with the standard formulation. In the Quantum-Hamiltonian picture, the phase space is chosen to be the complex projective space P(H) constructed out of the Hilbert space H of the considered quantum theory. I focus on the problem of associating quantum states (density matrices on H) to Liouville densities and quantum observables (self-adjoint operators on H) to real scalar functions on phase space P(H) in order to obtain a classical-like theory. Within an axiomatic approach I construct a general prescription for associating a real scalar function f_A on P(H) to every quantum observable A such that the dynamics given by the Hamiltonian vector field on P(H) defined by f_H is equivalent to the dynamics given by the Schrödinger equation given by the Hamiltonian operator H. For this purpose, the Kähler structure of the complex projective space is exploited, the so-called frame functions are defined as crucial tools, the Liouville measure on P(H) (quantum expectation values are computed as integrals w.r.t. this measure in a classical-like fashion) is introduced. I describe the C*-algebra of quantum observables in terms of classical-like observables and I discuss how this framework can be applied to study composite quantum systems and quantum esntanglement. Finally, some ideas to extend the considered formulation to infinite-dimensional case are discussed.

References

[1] V. Moretti and D. Pastorello. Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem Ann. Henri Poincaré 14,1435-1443 (2013).

[2] V. Moretti and D. Pastorello. Frame functions in finite-dimensional Quantum Mechanics and its Hamiltonian formulation on complex projective spaces, Submitted paper arXiv:1311.1720.