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An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics

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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.

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Authors and Affiliations

  1. Department of Mathematics, Technion, Haifa, 32000, Israel

    Michael Entov

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Leonid Polterovich & Frol Zapolsky

Authors
  1. Michael Entov
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  2. Leonid Polterovich
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  3. Frol Zapolsky
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Correspondence to Michael Entov.

Additional 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.

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Entov, M., Polterovich, L. & Zapolsky, F. An “Anti-Gleason” Phenomenon and Simultaneous Measurements in Classical Mechanics. Found Phys 37, 1306–1316 (2007). https://doi.org/10.1007/s10701-007-9158-0

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  • DOI: https://doi.org/10.1007/s10701-007-9158-0

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