ar X iv : q ua nt - p h / 96 03 00 5 v 1 5 M ar 1 99 6 Poisson spaces with a transition probability

The common structure of the space of pure states P of a classical and a quantum mechanical system is that of a Poisson space with a transition probability. This is a uniform space equipped with a Poisson structure, as well as with a function p : P × P → [0, 1], with certain properties. The Poisson structure is connected with the transition probabilities through unitarity: each point ρ ∈ P defines a function pρ : σ → p(ρ, σ), whose Hamiltonian flow must leave the transition probabilities invariant. In classical mechanics, where p(ρ, σ) = δρσ, this condition poses no restriction on the Poisson structure. Quantum mechanics is characterized by a specific (complex Hilbert space) form of p, and by the property that the irreducible components of P as a transition probability space coincide with the symplectic leaves of P as a Poisson space. In conjunction, these stipulations determine the Poisson structure of quantum mechanics up to a multiplicative constant (identified with Planck's constant)., we give axioms guaranteeing that P is the space of pure states of a C *-algebra. We give an explicit construction of this algebra from P.