Foundations of Physics

, Volume 30, Issue 9, pp 1481–1501 | Cite as

A Model with Quantum Logic, but Non-Quantum Probability: The Product Test Issue

  • Jan Broekaert
  • Bart D'Hooghe


We introduce a model with a set of experiments of which the probabilities of the outcomes coincide with the quantum probabilities for the spin measurements of a quantum spin-\( \frac{1}{2} \) particle. Product tests are defined which allow simultaneous measurements of incompatible observables, which leads to a discussion of the validity of the meet of two propositions as the algebraic model for conjunction in quantum logic. Although the entity possesses the same structure for the logic of its experimental propositions as a genuine spin-\( \frac{1}{2} \) quantum entity, the probability measure corresponding with the meet of propositions using the Hilbert space representation and quantum rules does not render the probability of the conjunction of the two propositions. Accordingly, some fundamental concepts of quantum logic, Piron-products, “classical” systems and the general problem of hidden variable theories for quantum theory are discussed.


Hilbert Space Probability Measure Quantum Theory Space Representation Fundamental Concept 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. Aerts, “The One and the Many: Towards a Unification of the Quantum and the Classical Description of One and Many Physical Entities,” doctoral dissertation, VUB (1981).Google Scholar
  2. 2.
    D. Aerts, “Description of many separated physical entities without the paradoxes encoun-tered in quantum mechanics,” Found. Phys. 12 (12), 1131 (1982).Google Scholar
  3. 3.
    D. Aerts, “Classical theories and non-classical theories as a special case of a more general theory,” J. Math. Phys. 24, 2441 (1983a).Google Scholar
  4. 4.
    D. Aerts, “The description of one and many physical systems,” in Foundations of Quantum Mechanics, C. Gruber, ed. (A.V.C.P., Lausanne, 1983b).Google Scholar
  5. 5.
    D. Aerts, “A possible explanation for the probabilities of quantum mechanics,” J. Math. Phys. 27, 202 (1985).Google Scholar
  6. 6.
    D. Aerts, “Quantum structures, separated physical entities and probability,” Found. Phys. 24, 1227 (1994).Google Scholar
  7. 7.
    D. Aerts, B. Coecke, B. D'Hooghe, and F. Valckenborgh, “A mechanistic macroscopic physical entity with a three-dimensional Hilbert space description,” Helv. Phys. Acta 70, 793 (1997).Google Scholar
  8. 8.
    F. J. Belinfante, A Survey of Hidden Variable Theories (Pergamon, Oxford, 1973).Google Scholar
  9. 9.
    D. Bohm, “A suggested interpretation of quantum theory in terms of hidden variables,” Phys. Rev. 85, 166 (1952).Google Scholar
  10. 10.
    B. Coecke, “Generalization of the proof on the existence of hidden measurements to experiments with an infinite set of outcomes,” Found. Phys. Lett. 8 (5), 437 (1995).Google Scholar
  11. 11.
    B. D'Hooghe and J. Pykacz, “On some new operations on orthomodular lattices,” to be published in Internat. J. Theoret. Phys.Google Scholar
  12. 12.
    D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, “Bell's theorem without inequalities,” Am. J. Phys. 58 (12), 1131 (1990).Google Scholar
  13. 13.
    A. M. Gleason, “Measures on the closed subspaces of a Hilbert space,” J. Math. Mech. 6, 885 (1957).Google Scholar
  14. 14.
    S. P. Gudder, “On hidden-variable theories,” J. Math. Phys. 11 (2), 431 (1970).Google Scholar
  15. 15.
    J. M. Jauch and C. Piron, “Can hidden variables be excluded in quantum mechanics,” Helvetica Phys. Acta 36, 827 (1963).Google Scholar
  16. 16.
    J. M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1968).Google Scholar
  17. 17.
    G. Kalmbach, Orthomodular Lattices (Academic, London, 1983).Google Scholar
  18. 18.
    S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” The Logico-Algebraic Approach to Quantum Mechanics, II, C. A. Hooker, ed. (Reidel, Dordrecht, 1967), p. 293.Google Scholar
  19. 19.
    C. Piron, “Axiomatique quantique,” Helvetica Phys. Acta 37, 439 (1964).Google Scholar
  20. 20.
    C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976).Google Scholar
  21. 21.
    C. Piron, “Recent developments on quantum physics,” Helvetica Phys. Acta 62, 82 (1989).Google Scholar
  22. 22.
    C. Piron, M–</del>écanique Quantique, Bases et Applications (Presses Polytechniques et Univer-sitaire Romandes, 1990).Google Scholar
  23. 23.
    J. Pykacz, “Quantum logics as families of fuzzy subsets of the set of physical states,” in Preprints of the Second IFSA Congress (Tokyo, 1987), 2, 437.Google Scholar
  24. 24.
    J. Pykacz, “Fuzzy quantum logics as a basis for quantum probability theory,” Internat. J. Theor. Phys. 37 (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Jan Broekaert
    • 1
  • Bart D'Hooghe
    • 1
  1. 1.Departement wiskundeVrije Universiteit BrusselBrussel;

Personalised recommendations