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

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Abstract

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.

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REFERENCES

  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).

  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. 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. D. Aerts, “The description of one and many physical systems,” in Foundations of Quantum Mechanics, C. Gruber, ed. (A.V.C.P., Lausanne, 1983b).

  5. D. Aerts, “A possible explanation for the probabilities of quantum mechanics,” J. Math. Phys. 27, 202 (1985).

    Google Scholar 

  6. D. Aerts, “Quantum structures, separated physical entities and probability,” Found. Phys. 24, 1227 (1994).

    Google Scholar 

  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. F. J. Belinfante, A Survey of Hidden Variable Theories (Pergamon, Oxford, 1973).

    Google Scholar 

  9. D. Bohm, “A suggested interpretation of quantum theory in terms of hidden variables,” Phys. Rev. 85, 166 (1952).

    Google Scholar 

  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. B. D'Hooghe and J. Pykacz, “On some new operations on orthomodular lattices,” to be published in Internat. J. Theoret. Phys.

  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. A. M. Gleason, “Measures on the closed subspaces of a Hilbert space,” J. Math. Mech. 6, 885 (1957).

    Google Scholar 

  14. S. P. Gudder, “On hidden-variable theories,” J. Math. Phys. 11 (2), 431 (1970).

    Google Scholar 

  15. J. M. Jauch and C. Piron, “Can hidden variables be excluded in quantum mechanics,” Helvetica Phys. Acta 36, 827 (1963).

    Google Scholar 

  16. J. M. Jauch, Foundations of Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1968).

    Google Scholar 

  17. G. Kalmbach, Orthomodular Lattices (Academic, London, 1983).

    Google Scholar 

  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. C. Piron, “Axiomatique quantique,” Helvetica Phys. Acta 37, 439 (1964).

    Google Scholar 

  20. C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976).

    Google Scholar 

  21. C. Piron, “Recent developments on quantum physics,” Helvetica Phys. Acta 62, 82 (1989).

    Google Scholar 

  22. C. Piron, M–</del>écanique Quantique, Bases et Applications (Presses Polytechniques et Univer-sitaire Romandes, 1990).

  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. J. Pykacz, “Fuzzy quantum logics as a basis for quantum probability theory,” Internat. J. Theor. Phys. 37 (1998).

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Broekaert, J., D'Hooghe, B. A Model with Quantum Logic, but Non-Quantum Probability: The Product Test Issue. Foundations of Physics 30, 1481–1501 (2000). https://doi.org/10.1023/A:1026410002431

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