We use the fact that some linear Hamiltonian systems can be considered as "finite level" quantum systems, and the description of quantum mechanics in terms of probabilities, to associate probability distributions with this particular class of linear Hamiltonian systems.
Hamilton-Randers dynamical systems, a particular class of Hamiltonian dynamical systems, are considered as mathematical models for deterministic emergent quantum mechanics. We show that in such framework, local diffeomorphism invariance, reversibility of the effective quantum dynamics and an universal maximal bound for proper acceleration emerge in that class of classical, deterministic and local models. Starting from the elements of Hamilton-Randers spaces, a phenomenological Hilbert space is constructed and associated with the space of wave functions of quantum mechanics. A geometric description for a spontaneous reduction of the quantum states, based on the concentration of measure phenomena as it appears in asymptotic Banach theory and probability theory, is described in general terms. It is also shown how the same concentration of measure phenomena plays a remarkable role in showing conditions for the existence of stable vacua states for the matter Hamiltonian. Furthermore, it is discussed the emergence of a weak equivalence principle from very fundamental principles of mathematical analysis and the basic assumptions of Hamilton-Randers theory. This fact, together with the existence in the theory of a maximal speed and the property of diffeomorphism invariance of the interaction driving the reduction of the quantum state, suggest that the reduction of the quantum state is driven by a gravitational type interaction. Moreover, since such interaction appears only in the dynamical domain when localization happens, it must be associated with a classical interaction. We make the hypothesis that such identification is universal and that indeed gravity is a domain of the dynamics of Hamilton-Randers systems. Hence Hamilton-Randers theory provides an unification scheme where quantum mechanics and classical gravity are both emergent.
In finite dimension (at least), Quantum Mechanics can be formulated as a proper Hamiltonian theory where a notion of phase space is given by the projective space P(H) constructed on the Hilbert space H of the considered quantum theory. It is well-known P(H) can be equipped with a structure of Kähler manifold, in particular we have a symplectic form and a Poisson structure; Quantum dynamics can be described in terms of a Hamiltonian vector field on P(H). In this paper, exploiting the notion and properties of so-called frame functions, I describe a general prescription for associating quantum observables to real functions on P(H), classical-like observables, and quantum states to probability densities on P(H), Liouville densities, in order to obtain a complete and meaningful Hamiltonian formulation of a finite-dimensional quantum theory.
In a recent paper it was shown that all the Hilbert space formulas for quantum probabilities can be realized as functions of geometric properties of the associated projective space, but those functions were expressed using the structures of the associated Hilbert space. In this paper a direct description of all these probabilities is given as formulas involving only the geometric properties of the projective space itself without referring to the associated Hilbert space theory. In large part this depends on a projection theorem for complex projective space which is analogous to the projection theorem for Hilbert spaces. The importance of this is that this exhibits quantum probability in terms of the geometry of a Riemannian metric in a non-linear Kähler manifold without any reference to a linear Hilbert space. As such this is a part of a larger program of the geometrization of physics. This opens the possibility of generalizations of quantum theory in other similar geometric settings. The theory presented includes projective spaces of both finite and infinite dimension. Some comments explain how quantum theory based on a von Neumann algebra is compatible with this approach.