International Journal of Theoretical Physics

, Volume 38, Issue 1, pp 289–358 | Cite as

Foundations of Quantum Physics: A General Realistic and Operational Approach

  • Diederik Aerts


We present a general formalism with the aim ofdescribing the situation of an entity, how it is, how itreacts to experiments, how we can make statistics withit, and how it ‘changes’ under the influence of the rest of the universe. Therefore we baseour formalism on the following basic notions: (1) thestates of the entity, which describe the modes of beingof the entity, (2) the experiments that can be performed on the entity, which describe how weact upon and collect knowledge about the entity, (3) theoutcomes of our experiments, which describe how theentity and the experiments "are" and “behave” together, (4) theprobabilities, which describe our repeated experimentsand the statistics of these repeated experiments, and(5) the symmetries, which describe the interactions ofthe entity with the external world without beingexperimented upon. Starting from these basic notions weformulate the necessary derived notions: mixed states,mixed experiments and events, an eigenclosure structure describing the properties of theentity, an orthoclosure structure introducing anorthocomplementation, outcome determination, experimentdetermination, state determination, and atomicity giving rise to some of the topological separationaxioms for the closures. We define the notion ofsubentity in a general way and identify the morphisms ofour structure. We study specific examples in detail in the light of this formalism: a classicaldeterministic entity and a quantum entity described bythe standard quantum mechanical formalism. We present apossible solution to the problem of the description of subentities within the standard quantummechanical procedure using the tensor product of theHilbert spaces, by introducing a new completed quantummechanics in Hilbert space, were new ‘pure’states are introduced, not represented by rays of theHilbert space.


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  1. Aerts, D. (1981). The one and the many, Doctoral Thesis, Free University of Brussels, Brussels.Google Scholar
  2. Aerts, D. (1982). Description of many physical entities without the paradoxes encountered in quantum mechanics, Found. Phys. 12, 1131.Google Scholar
  3. Aerts, D. (1983a). Classical theories and non classical theories as a special case of a more general theory, J. Math. Phys. 24, 2441.Google Scholar
  4. Aerts, D. (1983b). The description of one and many physical systems, in Foundations of Quantum Mechanics, C. Gruber, ed., A.V.C.P., Lausanne, p. 63.Google Scholar
  5. Aerts, D. (1984a). Construction of a structure which makes it possible to describe the joint system of a classical and a quantum system, Rep. Math. Phys. 20, 421.Google Scholar
  6. Aerts, D. (1984b). Construction of the tensor product for lattices of properties of physical entities, J. Math. Phys. 25, 1434.Google Scholar
  7. Aerts, D. (1986). A possible explanation for the probabilities of quantum mechanics, J. Math. Phys. 27, 202.Google Scholar
  8. Aerts, D. (1991). A mechanistic classical laboratory situation violating the Bell inequalities with √2, exactly ‘in the same way’ as its violations by the EPR experiments, Helv. Phys. Acta 64, 1.Google Scholar
  9. Aerts, D. (1994). Quantum structures, separated physical entities and probability, Found. Phys. 24, 1227.Google Scholar
  10. Aerts, D. (1995). Quantum structures: An attempt to explain their appearance in nature, Int. J. Theor. Phys. 34, 1165.Google Scholar
  11. Aerts, D. (1999). A possible solution of the sub entity problem of standard quantum mechanics leading to a new type of Hilbert space quantum mechanics, Preprint FUND-CLEA, Free University of Brussels.Google Scholar
  12. Aerts, D. and Daubechies, I. (1978). Physical justification for using the tensor product to describe two quantum systems as one joint system, Helv. Phys. Acta 51, 661.Google Scholar
  13. Aerts, D., and Durt, T. (1994). Quantum, classical and intermediate, an illustrative example, Found. Phys. 24, 1353.Google Scholar
  14. Aerts, D., and Valckenborgh, F. (1999). Lattice extensions and the description of compound entities, FUND, Brussels Free University, Preprint.Google Scholar
  15. Aerts, D., and Van Steirteghem, B., (1999). Quantum axiomatics and a theorem of M.P. Solèr, International Journal of Theoretical Physics, submitted.Google Scholar
  16. Aerts, D., Coecke, B., Durt, T., and Valckenborgh, F. (1997a). Quantum, classical and intermediate; a model on the Poincarésphere, Tatra Mountains Math. Publ. 10, 225.Google Scholar
  17. Aerts, D., Coecke, B., Durt, T., and Valckenborgh, F. (1997b). Quantum, classical and intermediate; the vanishing vector space structure, Tatra Mountains Math. Publ. 10, 241.Google Scholar
  18. Aerts, D., Colebunders, E., Van der Voorde, A., and Van Steirteghem, B. (1999). State property systems and closure spaces: A study of categorical equivalence, Int. J. Theor. Phys., this issue.Google Scholar
  19. Birkhoff, G., (1973). Lattice Theory, Amer. Math. Soc., Colloq. Publ. Vol. XXV, Providence.Google Scholar
  20. Birkhoff, G., and Von Neumann, J. (1936). The logic of quantum mechanics, Ann. Math. 37, 823.Google Scholar
  21. Beltrametti, E., and Cassinelli, G. (1981). The logic of quantum mechanics, Reading, Mass. Addison-Wesley.Google Scholar
  22. Cohen-Tannoudji, C., Diu, B., and Laloë, F. (1973). Mécanique Quantique, Vol. I, Hermann, Paris.Google Scholar
  23. Foulis, D., and Randall, C. (1981). What are quantum logics and what ought they to be? in Current Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds., Plenum Press, New York, p. 35.Google Scholar
  24. Foulis, D., Piron, C., and Randall, C. (1983). Realism, operationalism, and quantum mechanics, Found. Phys. 13, 813.Google Scholar
  25. Holland, S. (1995). Bull. of the Amer. Math. Soc., 32, 205.Google Scholar
  26. Jauch, J. (1968). Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.Google Scholar
  27. Keller, H. A. (1980), Math. Z., 172, 41.Google Scholar
  28. Ludwig, G. (1983), Foundations of Quantum Mechanics, New York and Berlin, Springer-Verlag.Google Scholar
  29. Ludwig, G. (1985), An Axiomatic Basis for Quantum Mechanics, New York and Berlin, Springer-Verlag.Google Scholar
  30. Mackey, G. W. (1963). Mathematical Foundations of Quantum Mechanics, Benjamin, Reading, Massachusetts.Google Scholar
  31. Piron, C. (1964). Axiomatique quantique, Helv. Phys. Acta 37, 439.Google Scholar
  32. Piron, C. (1976). Foundations of Quantum Physics, Benjamin, Reading, Massachusetts.Google Scholar
  33. Piron, C. (1989). Recent developments in quantum mechanics, Helv. Phys. Acta 62, 82.Google Scholar
  34. Piron, C. (1990). Mècanique Quantique: Bases et Applications, Press Polytechnique de Lausanne, Lausanne, Switzerland.Google Scholar
  35. Pulmannova, S. (1983). Coupling of quantum logics, Int. J. Theor. Phys. 22, 837.Google Scholar
  36. Pulmannova, S. (1984). On the product of quantum logics, Suppl. Circ. Mat. Palermo Ser. II, 231.Google Scholar
  37. Pulmannova, S. (1985). Tensor products of quantum logics, J. Math. Phys. 26, 1.Google Scholar
  38. Pulmannova, S. (1996). Int. J. Theor. Phys., 35, 2309.Google Scholar
  39. 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
  40. Randall, C., and Foulis, D. (1978). The operational approach to quantum mechanics, in Physical Theories as Logico-Operat ional Structures, C. A. Hooker, ed., Reidel, Dordrecht, p. 167.Google Scholar
  41. Randall, C., and Foulis, D. (1981). Operational statistics and tensor products, in Interpretations and Foundations of Quantum Theory, H. Neumann, ed., B. I. Wissenschaftsversl ag, Bibliographisches Institut, Mannheim, p. 21.Google Scholar
  42. Randall, C., and Foulis, D. (1983). Properties and operational propositions in quantum mechanics, Found. Phys. 13, 835.Google Scholar
  43. Solèr, M. P. (1995), Comm. Alg., 23, 219.Google Scholar
  44. Von Neumann, J. (1932). Mathematische Grundlagen der Quanten-Mechanik, Springer-Verlag, Berlin.Google Scholar
  45. Varadarajan, V. (1968). Geometry of Quantum Theory, von Nostrand, Princeton, New Jersey.Google Scholar
  46. Wilbur, (1977). Trans. Amer. Math. Soc., 233, 265.Google Scholar
  47. Zierler, N. (1961). Axioms for non-relativistic quantum mechanics, Pac. J. Math. 11, 1151.Google Scholar

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© Plenum Publishing Corporation 1999

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  • Diederik Aerts

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