International Journal of Theoretical Physics

, Volume 47, Issue 1, pp 104–113 | Cite as

Planck’s Constant in the Light of Quantum Logic

Article

Abstract

The goal of quantum logic is the “bottom-top” reconstruction of quantum mechanics. Starting from a weak quantum ontology, a long sequence of arguments leads to quantum logic, to an orthomodular lattice, and to the classical Hilbert spaces. However, this abstract theory does not yet contain Planck’s constant . We argue, that can be obtained, if the empty theory is applied to real entities and extended by concepts that are usually considered as classical notions. Introducing the concepts of localizability and homogeneity we define objects by symmetry groups and systems of imprimitivity. For elementary systems, the irreducible representations of the Galileo group are projective and determined only up to a parameter z, which is given by z=m/, where m is the mass of the particle and Planck’s constant. We show that has a meaning within quantum mechanics, irrespective of use the of classical concepts in our derivation.

Keywords

Planck’s constant Quantum logic Classical physics 

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References

  1. 1.
    Busch, P.: Indeterminacy relations and simultaneous measurements in quantum theory. Int. J. Theor. Phys. 24, 63–92 (1985) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Busch, P., Grabowski, M., Lahti, P.: Operational Quantum Physics. Springer, Heidelberg (1995) MATHGoogle Scholar
  3. 3.
    Dalla Chiara, M.L., Giuntini, R.: Quantum logics. arXiv:quant-ph/0101028 v2 (2001) Google Scholar
  4. 4.
    Dalla Chiara, M.L.: Unsharp quantum logics. Int. J. Theor. Phys. 34, 1331–1336 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 1331–1352 (1994) Google Scholar
  6. 6.
    Heinonen, T.: Imprecise Measurements in Quantum Mechanics. Turun Yliopisto, Turku (2005) Google Scholar
  7. 7.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison–Wesley, Reading (1968) MATHGoogle Scholar
  8. 8.
    MacLaren, M.D.: Nearly modular orthocomplemented lattices. Trans. Am. Math. Soc. 114, 401–416 (1965) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Mittelstaedt, P.: Constitution of objects in classical mechanics and in quantum mechanics. Int. J. Theor. Phys. 34, 1615–1626 (1995) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Mittelstaedt, P.: Quantum physics and classical physics—in the light of quantum logic. Int. J. Theor. Phys. 44, 771–781 (2005) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Piron, C.: Axiomatique quantique. Helv. Phys. Acta 37, 439–468 (1964) MATHMathSciNetGoogle Scholar
  12. 12.
    Piron, C.: Foundations of Quantum Physics. Benjamin, Reading (1976) MATHGoogle Scholar
  13. 13.
    Solèr, M.P.: Characterisation of Hilbert spaces by orthomodular lattices. Commun. Algebra 23(1), 219–243 (1995) MATHCrossRefGoogle Scholar
  14. 14.
    Varadarajan, V.S.: Geometry of Quantum Theory, vol. 1. Van. Nostrand, Princeton (1968) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.University of CologneCologneGermany

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