zbMATH — the first resource for mathematics

An “Anti-gleason” phenomenon and simultaneous measurements in classical mechanics. (English) Zbl 1129.81007
Summary: 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.

81P15 Quantum measurement theory, state operations, state preparations
Full Text: DOI arXiv
[1] Aarnes, J.F.: Quasi-states and quasi-measures. Adv. Math. 86(1), 41–67 (1991) · Zbl 0744.46052
[2] Aarnes, J.F., Rustad, A.B.: Probability and quasi-measures–a new interpretation. Math. Scand. 85, 278–284 (1999) · Zbl 0967.28014
[3] Ballentine, L.E.: The statistical interpretation of quantum mechanics. Rev. Mod. Phys. 42, 358–381 (1970) · Zbl 0203.27801
[4] Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966) · Zbl 0152.23605
[5] Beltrametti, E.G., Bugajski, S.: The Bell phenomenon in classical frameworks. J. Phys. A 29, 247–261 (1996) · Zbl 0914.46060
[6] Berezin, F.A.: General concept of quantization. Commun. Math. Phys. 40, 153–174 (1975) · Zbl 1272.53082
[7] Berndt, R.: An Introduction to Symplectic Geometry. Graduate Studies in Mathematics, vol. 26. American Mathematical Society, Providence (2000)
[8] Busch, P.: Private communication
[9] Cole-McLaughlin, K., Edelsbrunner, H., Harer, J., Natarajan, V., Pascucci, V.: Loops in Reeb graphs of 2-manifolds. Discret. Comput. Geom. 32, 231–244 (2004) · Zbl 1071.57017
[10] Entov, M., Polterovich, L.: Calabi quasimorphism and quantum homology. Int. Math. Res. Not. 30, 1635–1676 (2003) · Zbl 1047.53055
[11] Entov, M., Polterovich, L.: Quasi-states and symplectic intersections. Commun. Math. Helv. 81(1), 75–99 (2006) · Zbl 1096.53052
[12] Entov, M., Polterovich, L., Zapolsky, F.: Quasi-morphisms and the Poisson bracket. Preprint arXiv math.SG/0605406, to appear in Pure Appl. Math. Quat., a special issue in honor of G. Margulis · Zbl 1143.53070
[13] Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) · Zbl 0078.28803
[14] Holland, P.R.: The Quantum Theory of Motion. An Account of the de Broglie–Bohm Causal Interpretation of Quantum Mechanics. Cambridge University Press, Cambridge (1995) · Zbl 0854.00009
[15] Horwitz, L.: Private communication
[16] Jauch, J.M.: Foundations of Quantum Mechanics. Addison–Wesley, Reading (1968) · Zbl 0166.23301
[17] Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–87 (1967) · Zbl 0156.23302
[18] Madore, J.: The fuzzy sphere. Class. Quantum Gravity 9, 69–87 (1992) · Zbl 0742.53039
[19] McDuff, D., Salamon, D.: J-Holomorphic Curves and Symplectic Topology. American Mathematical Society, Providence (2004) · Zbl 1064.53051
[20] Perelomov, A.: Generalized Coherent States and Their Applications. Texts and Monographs in Physics. Springer, Berlin (1986) · Zbl 0605.22013
[21] Peres, A.: Quantum Theory: Concepts and Methods. Fundamental Theories of Physics, vol. 57. Kluwer Academic, Dordrecht (1993) · Zbl 0820.00011
[22] Reeb, G.: Sur les points singuliers d’une forme de Pfaff completement integrable ou d’une fonction numerique. C. R. Acad. Sci. Paris 222, 847–849 (1946) · Zbl 0063.06453
[23] von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1955). Translation of Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932) · JFM 58.0929.06
[24] Zapolsky, F.: Quasi-states and the Poisson bracket on surfaces. J. Mod. Dyn. 1(3), 465–475 (2007) · Zbl 1131.53046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.