An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics
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-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.
Keywords
Cite
@article{arxiv.quant-ph/0606214,
title = {An "anti-Gleason" phenomenon and simultaneous measurements in classical mechanics},
author = {Michael Entov and Leonid Polterovich and Frol Zapolsky},
journal= {arXiv preprint arXiv:quant-ph/0606214},
year = {2009}
}
Comments
Revised version: discussions on Gleason's theorem and the classical limit of a quantum spin system added, bibliography updated. To appear in Foundations of Physics