dorsal/arxiv
View SchemaPoisson spaces with a transition probability
| Authors | N. P. Landsman |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9603005 |
| URL | https://arxiv.org/abs/quant-ph/9603005 |
| DOI | 10.1142/S0129055X97000038 |
| Journal | Rev.Math.Phys. 9 (1997) 29-58 |
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 $P$.
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"abstract": "The common structure of the space of pure states $P$ of a classical or a\nquantum mechanical system is that of a Poisson space with a transition\nprobability. This is a topological space equipped with a Poisson structure, as\nwell as with a function $p:P\\times P-\u003e [0,1]$, with certain properties. The\nPoisson structure is connected with the transition probabilities through\nunitarity (in a specific formulation intrinsic to the given context).\n In classical mechanics, where $p(\\rho,\\sigma)=\\dl_{\\rho\\sigma}$, unitarity\n poses no restriction on the Poisson structure. Quantum mechanics is\ncharacterized by a specific (complex Hilbert space) form of $p$, and by the\nproperty that the irreducible components of $P$ as a transition probability\nspace coincide with the symplectic leaves of $P$ as a Poisson space. In\nconjunction, these stipulations determine the Poisson structure of quantum\nmechanics up to a multiplicative constant (identified with Planck\u0027s constant).\n Motivated by E.M. Alfsen, H. Hanche-Olsen and F.W. Shultz ({\\em Acta Math.}\n{\\bf 144} (1980) 267-305) and F.W. Shultz ({\\em Commun.\\ Math.\\ Phys.} {\\bf 82}\n(1982) 497-509), we give axioms guaranteeing that $P$ is the space of pure\nstates of a unital $C^*$-algebra. We give an explicit construction of this\nalgebra from $P$.",
"arxiv_id": "quant-ph/9603005",
"authors": [
"N. P. Landsman"
],
"categories": [
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"doi": "10.1142/S0129055X97000038",
"journal_ref": "Rev.Math.Phys. 9 (1997) 29-58",
"title": "Poisson spaces with a transition probability",
"url": "https://arxiv.org/abs/quant-ph/9603005"
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