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

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

2006-06-26

arxiv.org/abs/quant-ph/0606214v2

Physics

Quantum Physics

Revised version: discussions on Gleason's theorem and the classical limit of a quantum spin system added, bibliography updated

Scientific paper

10.1007/s10701-007-9158-0

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-commutative 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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