Poisson Spaces with a Transition Probability

The common structure of the space of pure states P of a classical or a quantum mechanical system is that of a Poisson space with a transition probability. This is a topological 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 (in a specific formulation intrinsic to the given context). In classical mechanics, where p(ρ,σ) = δρσ , unitarity 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). Motivated by E. M. Alfsen, H. Hanche-Olsen and F. W. Shultz (Acta Math. 144 (1980) 267–305) and F.W. Shultz (Commun. Math. Phys. 82 (1982) 497–509), we give axioms guaranteeing that P is the space of pure states of a unital C∗-algebra. We give an explicit construction of this algebra from P.