Foundations of Physics

, Volume 48, Issue 11, pp 1546–1556 | Cite as

Heisenberg Uncertainty Relations as Statistical Invariants

  • Aniello FedulloEmail author


For a simple set of observables we can express, in terms of transition probabilities alone, the Heisenberg uncertainty relations, so that they are proven to be not only necessary, but sufficient too, in order for the given observables to admit a quantum model. Furthermore distinguished characterizations of strictly complex and real quantum models, with some ancillary results, are presented and discussed.


Heisenberg uncertainty relations Statistical invariants Quantum models 


  1. 1.
    Accardi, L., Fedullo, A.: On the statistical meaning of complex numbers in quantum mechanics. Lett. Nuovo Cim. 34(7), 161–172 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Accardi, L.: Some trends and problems in quantum probability. In: Accardi, L., Frigerio, A., Gorini, V. (eds.) Quantum probability and applications to the quantum theory of irreversible processes, pp. 1–19. Lecture Notes in Mathematics, Vol. 1055, Springer, Berlin (1984)CrossRefGoogle Scholar
  3. 3.
    Klein, F.: Vergleichende Betrachtungen über neuere geometrische Forschungen. Math. Ann. 43, 63–100 (1893). Gesammelte Abh., Springer, 1, 460–497 (1921). English translation: a comparative review of recent researches in geometry, by Mellen Haskell, Bull. N. Y. Math. Soc., 2 (10), 215–249 (1893)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gudder, S., Zanghì, N.: Probability models. Il Nuovo Cim. 79 B(2), 291–301 (1982)ADSMathSciNetGoogle Scholar
  5. 5.
    Fedullo, A.: On the existence of a Hilbert space model for finite valued observables. Il Nuovo Cim. 107 B(12), 1413–1426 (1992)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Jauch, J.M.: Foundations of Quantum Mechanics. Addison-Wesley Publishing Company, Boston (1968)zbMATHGoogle Scholar
  7. 7.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik, Die Grundlehren der Mathematischen Wissenschaften, Band 38. Springer, Berlin (1932). English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press (1971)Google Scholar
  8. 8.
    Heisenberg, W.: Uber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Z. Phys. 43(3–4), 172–198 (1927). English translation in [13, 62–84]ADSCrossRefGoogle Scholar
  9. 9.
    Sen, D.: The uncertainty relations in quantum mechanics. Curr. Sci. 107(2), 203–218 (2014)Google Scholar
  10. 10.
    Robertson, H.P.: The uncertainty principle. Phys. Rev. 34, 573–574 (1929). Reprinted in [13, 127–128]CrossRefGoogle Scholar
  11. 11.
    Schrödinger, E.: The uncertainty relations in quantum mechanics. Zum Heisenbergschen Unschärfeprinzip, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse 14, 296–303 (1930)Google Scholar
  12. 12.
    Griffiths, D.: Quantum Mechanics. Pearson, Upper Saddle River (2005)Google Scholar
  13. 13.
    Wheeler, J.A., Zurek, W.H. (eds.): Quantum Theory and Measurement. Princeton University Press, Princeton (1983)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Physics “E. R. Caianiello”University of SalernoSalernoItaly

Personalised recommendations