International Journal of Theoretical Physics

, Volume 38, Issue 1, pp 407–429 | Cite as

Classical and Quantum Probability in the ∈-Model

  • Diederik Aerts
  • Sven Aerts
  • Thomas Durt
  • Olivier Leveque


We describe the probabilistic study of a hiddenvariable model in which the origin of the quantumprobability is due to fluctuations of the internal stateof the measuring apparatus. By varying the intensity of these fluctuations from zero to a maximalvalue, we describe in a heuristic manner the transitionfrom classical behavior to quantum behavior. Wecharacterize this transition in terms of theAccardi–Fedullo inequalities. This is a review article in whichwe gather our recent contributions to the subject, mostof which have not been published in articleform.


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  1. Accardi, L. (1984). The probabilistic roots of the quantum mechanical paradoxes, in The Wave-Particle Dualism, Diner, S., et al., eds, Reidel, Dordrecht.Google Scholar
  2. Accardi, L., and Fedullo, A. (1982). On the statistical meaning of the complex numbers in quantum mechanics, Nuovo Cimento 34, 161.Google Scholar
  3. Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics, J. Math. Phys. 27, 203.Google Scholar
  4. Aerts, D., and D' Hooghe, B. (1996). The operator structure of a non-quantum non-classical entity, Int. J. Theor. Phys. 35, 2241.Google Scholar
  5. Aerts, D., and Durt, T. (1994a). Quantum, classical and intermediate: A measurement model, in Proceedings of the Symposium on the Foundations of Modern Physics, Helsinki, Editions Frontiere, Gif sur Yvette, pp. 101–124.Google Scholar
  6. Aerts, D., and Durt, T. (1994b). Quantum, classical and intermediate. An illustrative example, Found. of Phys. 24, 1407–1421.Google Scholar
  7. Aerts, D., Durt, T., and Van Bogaert, B. (1992). A physical example of quantum fuzzy sets and the classical limit, Tatra Mountains Math. Publ. 1, 5–15.Google Scholar
  8. Aerts, D., Aerts, S., Coecke, B., D' Hooghe, B., Durt, T., and Valckenborgh, F. (1997). A model with varying fluctuations in the measurement context, in New Developments on Fundamental Problems in Quantum Physics, M. Ferrero and A. van der Merwe, eds., Kluwer, Dordrecht.Google Scholar
  9. Aerts, S. (1994). Conditional probabilities and the ε-model, graduation thesis, Free University of Brussels.Google Scholar
  10. Aerts, S. (1996). Conditional probabilities with a quantal and a Kolmogorovian limit, Int. J. Theor. Phys. 35, 2245.Google Scholar
  11. Aerts, S. (1998). Interactive probability models: Inverse problems on the sphere, Int. J. Theor. Phys., to appear.Google Scholar
  12. Bana, G., and Durt, T. (1997). Proof of Kolmogorov censorship, Found. Phys. 27, 1355–1373.Google Scholar
  13. Ballentine, L. E. (1986). Probability theory in quantum mechanics, Am. J. Phys. 54(10), 883.Google Scholar
  14. Bell, J. S. (1964). On the EPR paradox, Physics 1, 195.Google Scholar
  15. Boole, G. (1862). On the theory of probabilities, Phil. Trans. R. Soc. Lond. 152, 225–252.Google Scholar
  16. Corleo, G., Gutkowski, D., and Masotto, G. (1975). Are Bell' s type inequalities sufficient conditions for local hidden-variable theories? Nuovo Cimento 25B, 413.Google Scholar
  17. Czachor, M. (1992). On classical models of spin, Found. Phys. Lett. 5, 249.Google Scholar
  18. Durt, T. (1996a). From quantum to classical, a toy model, Doctoral thesis, V. U. B., Brussels.Google Scholar
  19. Durt, T. (1996b). Why God might play dice, Int. J. Theor. Phys. 35, 2271.Google Scholar
  20. Durt, T. (1997). Three interpretations of the violation of Bell' s inequalities, Found. Phys. 27, 415.Google Scholar
  21. Durt, T. (1998). Do dice remember? Int. J. Theor. Phys., this issue.Google Scholar
  22. Galambos, J., and Simonelli, I. (1996). Bonferroni-Type Inequalities with Applications, Springer-Verlag, Berlin.Google Scholar
  23. Gudder, S. P. (1969). Quantum probability spaces, Proc. Am. Math. Soc. 21, 296.Google Scholar
  24. Gudder, S. P. (1979). Stochastic Methods of Quantum Mechanics, North-Holland, New York.Google Scholar
  25. Gudder, S. P. (1984). Reality, locality, and probability, Found. Phys. 14, 997.Google Scholar
  26. Gudder, S. P. (1988). Quantum Probability, Academic Press, London.Google Scholar
  27. Gutkowski, D., and Masotto, G. (1974). An inequality stronger than Bell' s inequality, Nuovo Cimento 22B, 121.Google Scholar
  28. Lévêque, O. (1995). Du quantique au classique, graduation thesis, Free University of Brussels and Ecole Polytechnique de Lausanne (European exchange program Erasmus).Google Scholar
  29. Mielnik, B. (1968). Geometry of quantum states, Commun. Math. Phys. 9, 55.Google Scholar
  30. Pitowsky, I. (1982). Resolution of the EPR and Bell paradoxes, Phys. Rev. Lett. 48, 1299.Google Scholar
  31. Pitowsky, I. (1989). Quantum Probability, Quantum Logic, Springer-Verlag, Berlin.Google Scholar
  32. Randall, C., and Foulis, D. (1972). Operational statistics I, J. Math. Phys. 13, 1667.Google Scholar
  33. Randall, C., and Foulis, D. (1973). Operational statistics I, J. Math. Phys. 14, 1472.Google Scholar
  34. Randall, C., and Foulis, D. (1976). A mathematical setting for inductive reasoning, in Foundations of Probability Theory, Statistical Inference, and Statistical Theories of Science III, C. Hooker, ed., Reidel, Dordrecht, p. 169.Google Scholar
  35. Suppes, P. (1966). The probabilistic argument for a nonclassical logic for quantum mechanics, Phil. Sci. 33, 14–21.Google Scholar

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© Plenum Publishing Corporation 1999

Authors and Affiliations

  • Diederik Aerts
  • Sven Aerts
  • Thomas Durt
  • Olivier Leveque

There are no affiliations available

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