International Journal of Theoretical Physics

, Volume 39, Issue 3, pp 497–502 | Cite as

Quantum Axiomatics and a Theorem of M. P. Solèr

  • Diederik Aerts
  • Bart Van Steirteghem
Article

Abstract

Three of the traditional quantum axioms (orthocomplementation, orthomodularity,and the covering law) show incompatibilities with two products introduced byAerts for the description of joint entities. Inspired by Solèr's theorem and Holland'sAUG axiom, we propose a property of 'plane transitivity,' which also characterizesclassical Hilbert spaces among infinite-dimensional orthomodular spaces, as apossible partial substitute for the 'defective' axioms.

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REFERENCES

  1. Aerts, D. (1982). Description of many physical entities without the paradoxes encountered in quantum mechanics, Found. Phys. 12, 1131-1170.Google Scholar
  2. Aerts, D. (1983). Classical theories and non classical theories as a special case of a more general theory, J. Math. Phys. 24, 2441-2452.Google Scholar
  3. Aerts, D. (1984). Construction of the tensor product for the lattices of properties of physical entities, J. Math. Phys. 25, 1434-1441.Google Scholar
  4. 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. 38, 359-385.Google Scholar
  5. Chevalier, G. (1988). Orthomodular spaces and Baer *-rings, in Proceedings of the First Winter School on Measure Theory, Liptovsky´ Ja´n, January 10-15, pp. 7-14.Google Scholar
  6. Dikranjan, D., Giuli, E., and Tozzi, A. (1988). Topological categories and closure operators, Quaestiones Mathematicae 11, 323-337.Google Scholar
  7. Faure, C.-A. (1994). Categories of closure spaces and corresponding lattices, Cahier Top. geom. Diff. Categ. 35, 309-319.Google Scholar
  8. Faure, C.-A., and Fro¨licher, A. (1995). Dualities for infinite-dimensional projective geometries, Geom. Ded. 56, 225-236.Google Scholar
  9. Holland, S. S., Jr. (1995). Orthomodularity in infinite dimensions; a theorem of M. Sole`r, Bull. Am. Math. Soc. 32, 205-234.Google Scholar
  10. Maeda, F., and Maeda, S. (1970). Theory of Symmetric Lattices, Springer-Verlag, Berlin.Google Scholar
  11. Mayet, R. (1998). Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms, Int. J. Theor. Phys. 37, 109-114.Google Scholar
  12. Moore, D. J. (1995). Categories of representations of physical systems, Helv. Phys. Acta 68, 658-678.Google Scholar
  13. Moore, D. J. (1999). On state spaces and property lattices, Stud. Hist. Phil. Mod. Phys., to appear.Google Scholar
  14. Piron, C. (1976). Foundations of Quantum Physics, Benjamin, New York.Google Scholar
  15. Piron, C. (1989). Recent developments in quantum mechanics, Helv. Phys. Acta 62, 82-90.Google Scholar
  16. Piron, C. (1990). Me´canique quantique bases et applications, Presses Polytechniques et Universitaires Romandes, Lausanne.Google Scholar
  17. Pulmannova`, S. (1983). Coupling of quantum logics, Int. J. Theor. Phys. 22, 837.Google Scholar
  18. Pulmannova`, S. (1985). Tensor product of quantum logics, J. Math. Phys. 26, 1.Google Scholar
  19. Pulmannova`, S. (1994). Quantum logics and Hilbert spaces, Found. Phys. 24, 403.Google Scholar
  20. Pulmannova`, S. (1996). Axiomatization of quantum logics, Int. J. Theor. Phys. 35, 2309-2319.Google Scholar
  21. Sole`r, M.P. (1995). Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra 23, 219-243.Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Diederik Aerts
    • 1
  • Bart Van Steirteghem
    • 1
  1. 1.FUND, Department of MathematicsBrussels Free UniversityBrussels

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