Quantum Mechanics: Structures, Axioms and Paradoxes

  • Diederik Aerts
Part of the Einstein Meets Magritte: An Interdisciplinary Reflection on Science, Nature, Art, Human Action and Society book series (EMMA, volume 7)


In this article we present an analysis of quantum mechanics and its problems and paradoxes taking into account some of the results and insights that have been obtained during the last two decades by investigations that are commonly classified in the field of ‘quantum structures research’. We will concentrate on these aspects of quantum mechanics that have been investigated in our group at Brussels Free University1. We try to be as clear and self contained as possible: firstly because the article is also aimed at scientists not specialized in quantum mechanics, and secondly because we believe that some of the results and insights that we have obtained present the deep problems of quantum mechanics in a simple way.


Density Operator Atomic State Physical Entity Standard Quantum Mechanic Hide Variable Theory 
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.


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  1. [1]
    von Neumann, J., Mathematische Grundlagen der Quanten-Mechanik, Springer-Verlag, Berlin, 1932.Google Scholar
  2. [2]
    Birkhoff, G. and von Neumann, J., “The logic of quantum mechanics”, Annals of Mathematics, 37, 1936, p. 823.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Foulis, D., “A Half-Century of Quantum Logic—What have we learned?” in Quantum Structures and the Nature of Reality, the Indigo Book of Einstein meets Magritte, eds.,Aerts, D. and Pykacz, J., Kluwer Academic, Dordrecht, 1998.Google Scholar
  4. [4]
    Jammer, M., The Philosophy of quantum mechanics, Wiley and Sons, New York, Sydney, Toronto, 1974.Google Scholar
  5. [5]
    Aerts, D. and Durt, T., “Quantum. Classical and Intermediate, an illustrative example”, Found. Phys. 24, 1994, p. 1353.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Aerts, D., “Foundations of Physics: a general realistic and operational approach”, to be published in International Journal of Theoretical Physics.Google Scholar
  7. [7]
    Aerts, D., “A possible explanation for the probabilities of quantum mechanics and example of a macroscopic system that violates Bell inequalities”, in Recent developments in quantum logic, (eds.), Mittelstaedt, P. and Stachow, E.W., Grundlagen der Exakten Naturwissenschaften, band 6, Wissenschaftverlag, Bibliografisches Institut, Mannheim, 1985.Google Scholar
  8. [8]
    Aerts, D., “A Possible Explanation for the Probabilities of quantum mechanics”, J. Math. Phys., 27, 1986, p. 202.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Aerts, D., “Quantum structures: an attempt to explain their appearance in nature”, Int. J. Theor. Phys., 34, 1995, p. 1165.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Gerlach, F. and Stern, O., “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld”, Zeitschrift für Physik 9, 1922, p. 349.CrossRefGoogle Scholar
  11. [11]
    Bell, J.S., Rev. Mod. Phys., 38, 1966, p. 447.zbMATHCrossRefGoogle Scholar
  12. [12]
    Jauch, J.M. and Piron, C., Helv. Phys. Acta, 36, 1963, p. 827.MathSciNetzbMATHGoogle Scholar
  13. [13]
    Gleason, A.M.J. Math. Mech., 6, 1957, p. 885.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Kochen, S. and Specker, E.P., J. Math. Mech., 17, 1967, p. 59.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Gudder, S.P., Rev. Mod. Phys., 40, 1968, p. 229.CrossRefGoogle Scholar
  16. [16]
    Accardi, L., Rend. Sera. Mat. Univ. Politech. Torino, 1982, p. 241.Google Scholar
  17. [17]
    Accardi, L. and Fedullo, A., Lett. Nuovo Cimento, 34, 1982, p. 161.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Aerts, D., “Example of a macroscopical situation that violates Bell inequalities”, Lett. Nuovo Cimento, 34, 1982, p. 107.CrossRefGoogle Scholar
  19. [19]
    Aerts, D., “The physical origin of the EPR paradox”, in Open questions in quantum physics, (eds.), Tarozzi, G. and van der Merwe, A., Reidel, Dordrecht, 1985.Google Scholar
  20. [20]
    Aerts, D., “The physical origin of the Einstein-Podolsky-Rosen paradox and how to violate the Bell inequalities by macroscopic systems”, in Proceedings of the Symposium on the Foundations of Modern Physics, (eds.), Lahti, P. and Mittelstaedt, P., World Scientific, Singapore, 1985.Google Scholar
  21. [21]
    Aerts, D., “Quantum Structures, Separated Physical Entities and Probability”, Found. Phys. 24, 1994, p. 1227.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Coecke, B., “Hidden Measurement Representation for Quantum Entities Described by Finite Dimensional Complex Hilbert Spaces”, Found. Phys., 25, 1995, p. 203.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Coecke, B., “Generalization of the Proof on the Existence of Hidden Measurements to Experiments with an Infinite Set of Outcomes”, Found. Phys. Lett., 8, 1995, p. 437.MathSciNetCrossRefGoogle Scholar
  24. [24]
    Coecke, B., “New Examples of Hidden Measurement Systems and Outline of a General Scheme”, Tatra Mountains Mathematical Publications, 10, 1996, p. 203.Google Scholar
  25. [25]
    Aerts, D. and Durt, T., “Quantum, classical and intermediate: a measurement model”, in Montonen C. (ed.), Editions Frontieres, Gives Sur Yvettes, France, 1994.Google Scholar
  26. [26]
    Aerts, D., Durt, T. and Van Bogaert, B., “A physical example of quantum fuzzy sets, and the classical limit”, in the proceedings of the International Conference on Fuzzy Sets, Liptovsky, Tatra mountains, 1993, p. 5.Google Scholar
  27. [27]
    Aerts, D., Durt, T. and Van Bogaert, B., “Quantum Probability, the Classical Limit and Non-Locality”, in the proceedings of the International Symposium on the Foundations of Modern Physics 1992, Helsinki, Finland, ed. T. Hyvonen, World Scientific, Singapore, 1993, p. 35.Google Scholar
  28. [28]
    Randall, C. and Foulis, D., “ Properties and operational propositions in quantum mechanics”, Found. Phys., 13, 1983, p. 835.CrossRefGoogle Scholar
  29. [29]
    Foulis, D., Piron, C. and Randall, C., “Realism, operationalism, and quantum mechanics”, Found. Phys., 13, 1983, p. 813.MathSciNetCrossRefGoogle Scholar
  30. [30]
    Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B., “State property systems and closure spaces: a study of categorical equivalence”, Int. J. Theor. Phys., to appear 1998.Google Scholar
  31. [31]
    Aerts, D., Colebunders, E., Van der Voorde, A. and Van Steirteghem, B., “Categorical study of the state property systems and closure spaces”, preprint, FUND - TOPO, Brussels Free University.Google Scholar
  32. [32]
    Aerts, D. and Van Steirteghem, B., “Quantum Axiomatics and a Theorem of M.P. Solèr”, preprint, FUND, Brussels Free University.Google Scholar
  33. [33]
    Van Steirteghem, B., “Quantum Axiomatics: Investigation of the structure of the category of physical entities and Solér’s theorem”, graduation thesis, FUND, Brussels Free University.Google Scholar
  34. [34]
    Piron, C., “Axiomatique Quantique”, Helv. Phys. Acta, 37, 1964, p. 439.MathSciNetzbMATHGoogle Scholar
  35. [35]
    Amemiya, I. and Araki, H., “A remark of Piron’s paper”, Publ. Res. Inst. Math. Sci., A2, 1966, p. 423.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [36]
    Zierler, N., “Axioms for non-relativistic quantum mechanics”, Pac. J. Math., 11, 1961, p. 1151.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    Varadarajan, V., Geometry of Quantum Theory, Van Nostrand, Princeton, New Jersey, 1968.Google Scholar
  38. [38]
    Piron, C., Foundations of Quantum Physics, Benjamin, Reading, Massachusetts, 1976.Google Scholar
  39. [39]
    Wilbur, W., “On characterizing the standard quantum logics”, Trans. Am. Math. Soc., 233, 1977, p. 265.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [40]
    Keller, H.A., “Ein nicht-klassicher Hilbertsher Raum”, Math. Z., 172, 1980, p. 41.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [40]
    Aerts, D., “The one and the many”, Doctoral Thesis, Brussels Free University, Brussels, 1981.Google Scholar
  42. [41]
    Aerts, D., “Description of many physical entities without the paradoxes encountered in quantum mechanics”, Found. Phys., 12, 1982, p. 1131.MathSciNetCrossRefGoogle Scholar
  43. [42]
    Aerts, D., “Classical theories and Non Classical Theories as a Special Case of a More General Theory”, J. Math. Phys., 24, 1983, p. 2441.MathSciNetCrossRefGoogle Scholar
  44. [43]
    Valckenborgh, F., “Closure Structures and the Theorem of Decomposition in Classical Components”, Tatra Mountains Mathematical Publications, 10, 1997, p. 75.MathSciNetzbMATHGoogle Scholar
  45. [44]
    Aerts, D., “The description of one and many physical systems”, in Foundations of quantum mechanics, eds. C. Gruber, A.V.C.P., Lausanne, 1983, p. 63.Google Scholar
  46. [45]
    Aerts, D., “Construction of a structure which makes it possible to describe the joint system of a classical and a quantum system”, Rep. Math. Phys., 20, 1984, p. 421.MathSciNetCrossRefGoogle Scholar
  47. [46]
    Piron, C., Mécanique Quantique: Bases et applications, Presse Polytechnique et Universitaire Romandes, Lausanne, 1990.zbMATHGoogle Scholar
  48. [47]
    Pulmannovâ, S., “Axiomatization of Quantum Logics”, Int. J. Theor. Phys., 35, 1995, p. 2309.CrossRefGoogle Scholar
  49. [48]
    Solèr, M.P., “Characterization of Hilbert spaces with Orthomodular spaces”, Comm. Algebra, 23, 1995, p. 219.MathSciNetzbMATHCrossRefGoogle Scholar
  50. [49]
    Holland Jr, S.S., “Orthomodularity in Infinite Dimensions: a theorem of M. Solèr”, Bull. Aner. Math. Soc., 32, 1995, p. 205.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [50]
    Aerts, D., “Construction of the tensor product for lattices of properties of physical entities”, J. Math. Phys., 25, 1984, p. 1434.MathSciNetCrossRefGoogle Scholar
  52. [51]
    Aerts, D. and Daubechies, I., “Physical justification for using the tensor product to describe two quantum systems as one joint system”, Hely. Phys. Acta 51, 1978, p. 661.MathSciNetGoogle Scholar
  53. [52]
    Jauch, J., Foundations of quantum mechanics, Addison-Wesley, Reading, Mass, 1968.Google Scholar
  54. [53]
    Cohen-Tannoudji, C., Diu, B. and Laloë, F., Mécanique Quantique, Tome I, Hermann, Paris, 1973.Google Scholar
  55. [54]
    Aerts, D., “A mechanistic classical laboratory situation violating the Bell inequalities with \, exactly in the same way’ as its violations by the EPR experiments”, Hely. Phys. Acta, 64, 1991, p. 1.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1999

Authors and Affiliations

  • Diederik Aerts
    • 1
  1. 1.CLEA-FUNDBrussels Free UniversityBrusselsBelgium

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