9603005v2

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Poisson spaces with a transition probability

N. P. Landsman

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 $p(\rho,\sigma)=\dl_{\rho\sigma}$, 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 $P$.

Medatada:

oai_identifier:

oai:arXiv.org:quant-ph/9603005

categories:

quant-ph

comments:

23 pages, LaTeX, many details added

doi:

10.1142/S0129055X97000038

arxiv_id:

quant-ph/9603005

journal_ref:

Rev.Math.Phys. 9 (1997) 29-58

created:

1996-03-05

updated:

1996-06-10

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