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Observables in the Logico-Algebraic Approach

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Current Research in Operational Quantum Logic

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 111))

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Abstract

The notion of observable is a quantum mechanical variant of that of a random variable (see [32], [69], [20], [29], etc.). It is supposed to help in treating noncompatible events. In the logico-algebraic approach an observable is defined as a σ-additive measure with values in a “quantum logic” (= with values in an abstract σ-complete orthomodular poset). Though an individual observable ranges in a Boolean σ-algebra, non-classical phenomena occur when one develops “noncomutative” probability theory. First, the range Boolean σ-algebra does not have to be set-representable (even in the fundamental case of the Hilbert space logic!). Second, when one treats collections of observables, the respective range Boolean σ-algebras vary and often do not allow for an umbrella Boolean σ-algebra. In this article we want to survey a few lines of recent research on observables. We also want to formulate — in the main text and in the notes at the end (referred to as EquationSource % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa % aaleaacaaIXaaabeaakiaacYcacaWGUbWaaSbaaSqaaiaaikdaaeqa % aOGaeSOjGSeaaa!3B8F! ]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$ {{n}_{1}},{{n}_{2}} \ldots $$) — some open questions which seem related to quantum axiomatics and noncommutative probability theory. An extensive bibliography of papers dealing with observables is also provided.

The work on this survey was supported by the grant GACR 201/96/0117 of the Czech Grant Agency and by the project no. VS 96049 of the Czech Ministry of Education.

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References

  1. Birkhoff, G. and von Neumann, J. (1936) The logic of quantum mechanics, Annals of Mathematics 37, 823–843.

    Article  MathSciNet  Google Scholar 

  2. Brabec, J. and Ptäk, P. (1982) On compatibility in quantum logics, Founda-tions of Physics 12, 207–212.

    Article  ADS  Google Scholar 

  3. Catlin, D. (1968) Spectral theory in quantum logics, International Journal of Theoretical Physics 1, 285–297.

    Article  MathSciNet  ADS  Google Scholar 

  4. Cooke, R., Keane, M. and Moran, W. (1985) An elementary proof of Gleason’s theorem, Mathematical Proceedings of the Cambridge Philosophical Society 98, 117–128.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Cushen, C. and Hudson, R. (1971) A quantum-mechanical central limit the-orem, Journal of Applied Probability 8, 454–469.

    Article  MathSciNet  MATH  Google Scholar 

  6. Davies, R.O. (1971) Measures not approximable or not specifiable by means of balls, Mathematika 18, 157–160.

    Article  MathSciNet  MATH  Google Scholar 

  7. de Lucia, P. and Ptäk, P. (1992) Quantum probability spaces that are nearly classical, Bulletin of the Polish Academy of Sciences — Math. 40, 163–173.

    MATH  Google Scholar 

  8. De Simone, A., Navara, M. and Ptäk, P. (n.d.) On interval homogeneous orthomodular lattices, To appear.

    Google Scholar 

  9. Dorninger, D., Länger, H. and Mgczyfiski, M. (1983) Zur Darstellung von Ob-servablen auf a-stetigen Quantenlogiken, Sitzungsberichte von Osterreichische Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse. II 192, 169–176

    MATH  Google Scholar 

  10. Dravecky, J. and Sipos, J. (1980) On the additivity of Gudder integral, Math-ematica Slovaca 30, 299–303.

    MathSciNet  MATH  Google Scholar 

  11. Dvureéenskij, A. (1979) Laws of large numbers and the central limit theorem on a logic, Mathematica Slovaca 29, 397–410.

    MathSciNet  Google Scholar 

  12. Dvurecenskij, A. and Pulmannovâ, S. (1982) On joint distributions of observables, Mathematica Slovaca 32, 155–166.

    MathSciNet  MATH  Google Scholar 

  13. Dvurecenskij, A. and Pulmannovâ, S. (1984) Connection between joint distributions and compatibility, Reports on Mathematical Physics 19, 349–359.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Dvurecenskij, A. and Pulmannovâ, S. (1989) Type II joint distribution and compatibility of observables, Demonstratio Mathematica 22, 479–497.

    MathSciNet  MATH  Google Scholar 

  15. Foulis, D. and Ptâk, P. (1995) On absolutely compatible elements and hidden variables in quantum logics, Richerche di Matematica 44, Fasc. 1, 19–29.

    MATH  Google Scholar 

  16. Greechie, R. (1971) Orthomodular lattices admitting no states, Journal of Combinatorial Theory 10, 119–132.

    Article  MathSciNet  MATH  Google Scholar 

  17. Gudder, S. (1965) Spectral methods for a generalized probability theory, Transactions of the American Mathematical Society 119, 428–442.

    Article  MathSciNet  MATH  Google Scholar 

  18. Gudder. S. (1966) Uniqueness and existence properties of bounded observables, Pacific Journal of Mathematics 19, 81–93, 588–589.

    Google Scholar 

  19. Gudder, S. (1973) Generalized measure theory, Foundations of Physics 3, 399–411.

    Article  MathSciNet  ADS  Google Scholar 

  20. Gudder, S. (1979) Stochastic Methods in Quantum Mechanics, Elsevier & North—Holland, Amsterdam.

    MATH  Google Scholar 

  21. Gudder, S. and Marchand, J. (1972) Noncommutative probability on von Neumann algebras, Journal of Mathematical Physics 13, 799–806.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Gudder, S. and Mullihin, H. (1973) Measure theoretic convergence of observables and operators, Journal of Mathematical Physics 14, 234–242.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. Gudder, S. and Piron, C. (1972) Observables and the field in quantum mechanics, Journal of Mathematical Physics 12, 1583–1588.

    Article  MathSciNet  ADS  Google Scholar 

  24. Harman, B. and Riecan, B. (1982) On the martingale convergence theorem in quantum theory, Transactions of the 9th Prague Conference on Information Theory, Stochastic Decision Functions and Random Processes ( Academia, Prague, 1983 ).

    Google Scholar 

  25. Jackson, S. and Mauldin, R.D. (n.d.) On the a-class generated by open balls, To appear.

    Google Scholar 

  26. Kalmär, I.G. (1978) Atomistic orthomodular lattices and a generalized probability theory, Publicationes Mathematicae Debrecen 25, 139–153

    MathSciNet  Google Scholar 

  27. Keleti, T. and Preiss D. (n.d.) The balls do not generate all Borel sets using complements and countable disjoint union, To appear.

    Google Scholar 

  28. Kronfli, N. (1970) Integration theory of observables, International Journal of Theoretical Physics 3, 199–204.

    Article  MathSciNet  ADS  Google Scholar 

  29. Länger, H. and Mgczynski, M. (1988) An order-theoretical characterization of spectral measures, Contributions to General Algebra 6, 181–188.

    Google Scholar 

  30. Loomis, L. (1947) On the representation of a-complete Boolean algebras, Bulletin of the American Mathematical Society 53, 757–760.

    Article  MathSciNet  MATH  Google Scholar 

  31. Lutterovâ, T. and Pulmannovâ, S. (1985) An individual ergodic theorem on the Hilbert space logic, Mathematica Slovaca 35, 361–371.

    MathSciNet  MATH  Google Scholar 

  32. Mackey, G. (1963) The Mathematical Foundations of Quantum Mechanics, Benjamin, New York.

    MATH  Google Scholar 

  33. Magfk, J. (1956) Baireova a Borelova mfra, Casopis pro péstovdni matematiky 81, 431–450 (appeared in English in Czechoslovak Mathematical Journal in 1957).

    Google Scholar 

  34. Navara, M. (1984) The integral on a-classes is monotonic, Reports on Math-ematical Physics 20, 417–421.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Navara, M. (1989) When is the integral on quantum probability space addi-tive?, Real Analysis Exchange 14, 228–234.

    MathSciNet  MATH  Google Scholar 

  36. Navara, M. (1995) Uniqueness of bounded observables, Annales de l’Institut Henri Poincaré–Theoretical Physics 63, 155–176.

    MathSciNet  MATH  Google Scholar 

  37. Navara, M. and Ptak, P. (1983) Two-valued measures on σ-classes, asopis pro péstovdni matematiky 108, 225–229.

    MathSciNet  MATH  Google Scholar 

  38. Navara, M. and Ptak, P. (1983) On the Radon-Nikodym property for σ-classes, Journal of Mathematical Physics 24, 1450.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  39. Neubrunn, T. (1970) A note on quantum probability spaces, Proceedings of the American Mathematical Society 25, 672–675.

    Article  MathSciNet  MATH  Google Scholar 

  40. Neubrunn, T. (1974) On certain generalized random variables, Acta Facultatis Rerum Naturaliurn Universitatis Comenianae: Mathematica 29, 1–6.

    MathSciNet  MATH  Google Scholar 

  41. Nikodym, O.M. (1957) Critical remarks on some basic notions in Boolean lattices II, Rendiconti del Seminario Matematico della Universita di Padova 27, 193–217.

    MathSciNet  MATH  Google Scholar 

  42. Olerek, V. (1995) The σ–class generated by balls contains all Borel sets, Proceedings of the American Mathematical Society 123, 3665–3675.

    MathSciNet  Google Scholar 

  43. Ovtchinikoff, P. and Sultanbekoff, F. (1996) Notes on Gudder’s integral, Preprint.

    Google Scholar 

  44. Preiss, D. and Tiger, J. (1991) Measures in Banach spaces are determined by their values on balls, Mathematica 38, 391–397.

    MATH  Google Scholar 

  45. Ptak, P. (1984) Spaces of observables, Czechoslovak Mathematical Journal 34, 552–561.

    MathSciNet  Google Scholar 

  46. Ptak, P. (1987) Exotic logics, Colloquium Mathematicum 54, 1–7.

    MathSciNet  MATH  Google Scholar 

  47. Ptak, P. (1987) An observation on observables, Prdce OVUT 4, 81–86.

    MathSciNet  Google Scholar 

  48. Ptak, P. (1990) Measure-determined enlargements of Boolean σ-algebras, Commentationes Mathematicae Universitatis Carolinae 31, 105–107.

    MathSciNet  MATH  Google Scholar 

  49. Ptak, P. (1981) Realcompactness and the notion of observable, Journal of the London Mathematical Society 23, 534–536.

    Article  MathSciNet  MATH  Google Scholar 

  50. Ptak, P. and Pulmannova, S. (1991) Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers.

    Google Scholar 

  51. Ptak, P. and Rogalewicz, V. (1983) Regularly full logics and the uniqueness problem for observables, Annales de l’Institut Henri Poincaré A 38, 69–74.

    MathSciNet  MATH  Google Scholar 

  52. Ptak, P. and Rogalewicz, V. (1983) Measures on orthomodular partially or-dered sets, Journal of Pure and Applied Algebra 28, 75–80.

    Article  MathSciNet  MATH  Google Scholar 

  53. Ptâk, P. and Tkadlec, J. (1988) A note on determining of states, Casopis pro péstovânI matematiky 113, 435–436.

    MATH  Google Scholar 

  54. Ptâk, P. and Wright, J.D.M. (1985) On the concreteness of quantum logics, Aplikace Matematiky 30, 274–285.

    MathSciNet  MATH  Google Scholar 

  55. Pulmannovâ, S. (1981) On the observables on quantum logics, Foundations of Physics 11, 127–136.

    Article  MathSciNet  ADS  Google Scholar 

  56. Pulmannovâ, S. (1984) On a characterization of linear subspaces of observables, Demonstratio Mathematica 17, 1073–1078.

    MathSciNet  MATH  Google Scholar 

  57. Pulmannovâ, S. (1986) Joint distributions of observables on spectral logics, Reports on Mathematical Physics 26, 67–71.

    Article  ADS  Google Scholar 

  58. Pulmannovâ. S. and Dvurecenskij, A. (1985) Uncertainty principle and joint distributions of observables, Annales de l’Institut Henri Poincaré A 42, 253–265.

    MATH  Google Scholar 

  59. Pulmannovâ, S. and Stehlikovâ, B. (1986) Strong law of large numbers and central limit theorem on a Hilbert space, Reports on Mathematical Physics 23, 99–107.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  60. Rogalewicz, V. (1984) A note on the uniqueness problem for observables, Acta Polytechnica (Prague) VI, 107–111.

    MathSciNet  Google Scholar 

  61. Rogalewicz, V. (1984) On the uniqueness problem for quite full logics, Annales de l’Institut Henri Poincaré, Sect. A 41, 445–451.

    MathSciNet  MATH  Google Scholar 

  62. Rüttimann, G. (1977) Jauch-Piron states, Journal of Mathematical Physics 18, 189–193.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. Shultz, F.W. (1977) Events and observables in axiomatic quantum mechanics, International Journal of Theoretical Physics, 16, 259–272.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  64. Sikorski, R. (1949) On inducing of homomorphisms by mappings, Fundamenta Mathematicae 36, 7–22.

    MathSciNet  MATH  Google Scholar 

  65. Sikorski, R. (1964) Boolean Algebras, Springer-Verlag, Heidelberg.

    MATH  Google Scholar 

  66. Sipos, J. (1979) Integral with respect to a pre-measure, Mathematica Slovaca 29, 141–155.

    MathSciNet  MATH  Google Scholar 

  67. Urbanik, K. (1961) Joint probability distributions of observables in quantum mechanics, Studia Mathematica 21, 117–133.

    MathSciNet  MATH  Google Scholar 

  68. Urbanik, K. (1985) Joint distributions and compatibility of observables, Demonstratio Mathematica 18, 31–41.

    MathSciNet  MATH  Google Scholar 

  69. Varadarajan, V. (1968–1970) Geometry of Quantum Theory I & II, Van Nostrand, Princeton.

    Google Scholar 

  70. Zerbe, J. and Gudder, S. (1985) Additivity of integrals on generalized measure spaces, Journal of Combinatorial Theory A 39, 42–51.

    Article  MathSciNet  MATH  Google Scholar 

  71. Zierler, U. (1967) Order properties of bounded observables, Proceedings of the American Mathematical Society 20, 272–280.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Czech Technical University, 166 27, Prague 6, Czech Republic

    Pavel Pták

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  1. Pavel Pták
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Editors and Affiliations

  1. Department of Mathematics, FUND, Free University of Brussels, Brussels, Belgium

    Bob Coecke

  2. Department of Theoretical Physics, University of Geneva, Geneva, Switzerland

    David Moore

  3. Department of Mathematics and Computer Science, Juniata College, Huntingdon, Pennsylvania, USA

    Alexander Wilce

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Pták, P. (2000). Observables in the Logico-Algebraic Approach. In: Coecke, B., Moore, D., Wilce, A. (eds) Current Research in Operational Quantum Logic. Fundamental Theories of Physics, vol 111. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1201-9_3

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  • DOI: https://doi.org/10.1007/978-94-017-1201-9_3

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