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• W3099330539 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 $p:Ptimes P-> [0,1]$, 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 $p(rho,sigma)=dl_{rhosigma}$, 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 $P$." @default.
• W3099330539 created "2020-11-23" @default.
• W3099330539 creator A5060266645 @default.
• W3099330539 date "1997-01-01" @default.
• W3099330539 modified "2023-09-24" @default.
• W3099330539 title "Poisson Spaces with a Transition Probability" @default.
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• W3099330539 doi "https://doi.org/10.1142/s0129055x97000038" @default.
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