An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics

Michael Entov; Leonid Polterovich; Frol Zapolsky

doi:10.1007/s10701-007-9158-0

An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics

Michael Entov, Leonid Polterovich, Frol Zapolsky

Research output: Contribution to journal › Article › peer-review

Abstract

We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories-symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.

Original language	English
Pages (from-to)	1306-1316
Number of pages	11
Journal	Foundations of Physics
Volume	37
Issue number	8
DOIs	https://doi.org/10.1007/s10701-007-9158-0
State	Published - Aug 2007
Externally published	Yes

Bibliographical note

Funding Information:
M. Entov partially supported by E. and J. Bishop Research Fund and by the Israel Science Foundation grant # 881/06. L. Polterovich partially supported by the Israel Science Foundation grant # 11/03.

ASJC Scopus subject areas

General Physics and Astronomy

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10.1007/s10701-007-9158-0

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abstract = "We report on an {"}anti-Gleason{"} phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories-symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.",

author = "Michael Entov and Leonid Polterovich and Frol Zapolsky",

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AB - We report on an "anti-Gleason" phenomenon in classical mechanics: in contrast with the quantum case, the algebra of classical observables can carry a non-linear quasi-state, a monotone functional which is linear on all subspaces generated by Poisson-commuting functions. We present an example of such a quasi-state in the case when the phase space is the 2-sphere. This example lies in the intersection of two seemingly remote mathematical theories-symplectic topology and the theory of topological quasi-states. We use this quasi-state to estimate the error of the simultaneous measurement of non-commuting Hamiltonians.

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An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics

Abstract

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