Poisson spaces with a transition probability

N. P. Landsman

doi:10.1142/S0129055X97000038

paper

Poisson spaces with a transition probability

arXiv:quant-ph/9603005 · doi:10.1142/S0129055X97000038

Abstract

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 $p : P \times P - > [0, 1]$ , 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 $p(ρ,σ)=\dl_{ρσ}$, 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 ({\em Acta Math.} {\bf 144} (1980) 267-305) and F.W. Shultz ({\em Commun.\ Math.\ Phys.} {\bf 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 $P$ .

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Classical behaviour in quantum mechanics: a transition probability approach

Poisson spaces with a transition probability

Abstract

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