Foundations of Physics

, Volume 38, Issue 3, pp 241–256 | Cite as

An Approach to Quantum Mechanics via Conditional Probabilities

  • Gerd NiesteggeEmail author


The well-known proposal to consider the Lüders-von Neumann measurement as a non-classical extension of probability conditionalization is further developed. The major results include some new concepts like the different grades of compatibility, the objective conditional probabilities which are independent of the underlying state and stem from a certain purely algebraic relation between the events, and an axiomatic approach to quantum mechanics. The main axioms are certain postulates concerning the conditional probabilities and own intrinsic probabilistic interpretations from the very beginning. A Jordan product is derived for the observables, and the consideration of composite systems leads to operator algebras on the Hilbert space over the complex numbers, which is the standard model of quantum mechanics. The paper gives an expository overview of the results presented in a series of recent papers by the author. For the first time, the complete approach is presented as a whole in a single paper. Moreover, since the mathematical proofs are already available in the original papers, the present paper can dispense with the mathematical details and maximum generality, thus addressing a wider audience of physicists, philosophers or quantum computer scientists.


Quantum probability Quantum measurement Operator algebras Jordan algebras 


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  1. 1.
    Alfsen, E.M., Shultz, F.W., Størmer, E.: A Gelfand-Neumark theorem for Jordan algebras. Adv. Math. 28, 11–56 (1978) zbMATHCrossRefGoogle Scholar
  2. 2.
    Alfsen, E.M., Shultz, F.W.: On non-commutative spectral theory and Jordan algebras. Proc. Lond. Math. Soc. 38, 497–516 (1979) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beltrametti, E.G., Cassinelli, G.: The Logic of Quantum Mechanics. Addison-Wesley, Reading (1981) zbMATHGoogle Scholar
  4. 4.
    Bub, J.: Von Neumann’s projection postulate as a probability conditionalization rule in quantum mechanics. J. Philos. Log. 6, 381–390 (1977) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Bunce, L.J., Wright, J.D.M.: Quantum measures and states on Jordan algebras. Commun. Math. Phys. 98, 187–202 (1985) zbMATHCrossRefADSMathSciNetGoogle Scholar
  6. 6.
    Bunce, L.J., Wright, J.D.M.: Continuity and linear extensions of quantum measures on Jordan operator algebras. Math. Scand. 64, 300–306 (1989) zbMATHMathSciNetGoogle Scholar
  7. 7.
    Christensen, E.: Measures on projections and physical states. Commun. Math. Phys. 86, 529–538 (1982) zbMATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Gleason, A.M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 885–893 (1957) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Gudder, S.P.: Quantum Probability. Academic Press, Orlando (1988) zbMATHGoogle Scholar
  10. 10.
    Gunson, J.: On the algebraic structure of quantum mechanics. Commun. Math. Phys. 6, 262–285 (1967) zbMATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Guz, W.: Conditional probability and the axiomatic structure of quantum mechanics. Fortschr. Phys. 29, 345–379 (1981) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitmann, Boston (1984) zbMATHGoogle Scholar
  13. 13.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934) CrossRefGoogle Scholar
  14. 14.
    Keller, H.: Ein nicht-klassischer Hilbertscher Raum. Math. Z. 172, 41–49 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Maeda, S.: Probability measures on projectors in von Neumann algebras. Rev. Math. Phys. 1, 235–290 (1990) CrossRefGoogle Scholar
  16. 16.
    Mittelstaedt, P.: Time dependent propositions and quantum logic. J. Philos. Log. 6, 463–472 (1977) CrossRefMathSciNetGoogle Scholar
  17. 17.
    Niestegge, G.: Non-Boolean probabilities and quantum measurement. J. Phys. A 34, 6031–6042 (2001) zbMATHCrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Niestegge, G.: Why do the quantum observables form a Jordan operator algebra? Int. J. Theor. Phys. 43, 35–46 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Niestegge, G.: Composite systems and the role of the complex numbers in quantum mechanics. J. Math. Phys. 45, 4714–4725 (2004) zbMATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Niestegge, G.: Conditional expectations associated with quantum states. J. Math. Phys. 46, 043507 (2005), 8 pp. CrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Niestegge, G.: Different types of conditional expectation and the Lüders-von Neumann quantum measurement. Int. J. Theor. Phys. 46, 1823–1835 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Piron, C.: Axiomatique quantique. Helv. Phys. Acta 37, 439–468 (1964) zbMATHMathSciNetGoogle Scholar
  23. 23.
    Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991) zbMATHGoogle Scholar
  24. 24.
    Segal, I.E.: Postulates for general quantum mechanics. Ann. Math. 48, 930–938 (1947) CrossRefGoogle Scholar
  25. 25.
    Sherman, S.: On Segal’s postulates for general quantum mechanics. Ann. Math. 64, 593–601 (1956) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Stachow, E.W.: Logical foundation of quantum mechanics. Int. J. Theor. Phys. 19, 251–304 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Varadarajan, V.S.: Geometry of Quantum Theory I. Van Nostrand, New York (1968) zbMATHGoogle Scholar
  28. 28.
    Varadarajan, V.S.: Geometry of Quantum Theory II. Van Nostrand, New York (1970) zbMATHGoogle Scholar
  29. 29.
    Yeadon, F.J.: Measures on projections in W *-algebras of type II. Bull. Lond. Math. Soc. 15, 139–145 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Yeadon, F.J.: Finitely additive measures on projections in finite W *-algebras. Bull. Lond. Math. Soc. 16, 145–150 (1984) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.MunichGermany

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