Advertisement

Foundations of Physics

, Volume 44, Issue 11, pp 1216–1229 | Cite as

A Generalized Quantum Theory

  • Gerd Niestegge
Article

Abstract

In quantum mechanics, the selfadjoint Hilbert space operators play a triple role as observables, generators of the dynamical groups and statistical operators defining the mixed states. One might expect that this is typical of Hilbert space quantum mechanics, but it is not. The same triple role occurs for the elements of a certain ordered Banach space in a much more general theory based upon quantum logics and a conditional probability calculus (which is a quantum logical model of the Lüders-von Neumann measurement process). It is shown how positive groups, automorphism groups, Lie algebras and statistical operators emerge from one major postulate—the non-existence of third-order interference [third-order interference and its impossibility in quantum mechanics were discovered by Sorkin (Mod Phys Lett A 9:3119–3127, 1994)]. This again underlines the power of the combination of the conditional probability calculus with the postulate that there is no third-order interference. In two earlier papers, its impact on contextuality and nonlocality had already been revealed.

Keywords

Foundations of quantum mechanics Dynamical groups   Positive groups Lie algebras Operator algebras 

References

  1. 1.
    Alfsen, E.M., Shultz, F.W.: State Spaces of Operator Algebras: Basic Theory, Orientations and C*-products. Mathematics: Theory & Applications. Birkhäuser, Boston (2001)Google Scholar
  2. 2.
    Alfsen, E.M., Shultz, F.W.: Geometry of State Spaces of Operator Algebras. Mathematics: Theory & Applications. Birkhäuser, Boston (2003)CrossRefGoogle Scholar
  3. 3.
    Baez, J.C.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2001)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chu, C.H., Wright, J.D.M.: A theory of types for convex sets and ordered Banach spaces. Proc. Lond. Math. Soc. 36, 494–517 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Connes, A.: Characterisation des espaces vectoriels ordonnes sousjacent aux algebres de von Neumann. Ann. Inst. Fourier (Grenoble) 24, 121–155 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Evans, D., Hanche-Olsen, H.: The generators of positive semigroups. J. Funct. Analy. 32, 207–212 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Iochum, B., Shultz, F.W.: Normal state spaces of Jordan and von Neumann algebras. J. Funct. Analy. 50, 317–328 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)CrossRefGoogle Scholar
  9. 9.
    Niestegge, G.: Non-Boolean probabilities and quantum measurement. J. Phys. A 34, 6031–6042 (2001)MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. 10.
    Niestegge, G.: A representation of quantum mechanics in order-unit spaces. Found. Phys. 38, 783–795 (2008)MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. 11.
    Niestegge, G.: hierarchy of compatibility and comeasurability levels in quantum logics with unique conditional probabilities. Commun. Theor. Phys. (Beijing, China) 54, 974–980 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Niestegge, G.: Conditional probability, three-slit experiments, and the Jordan algebra structure of quantum mechanics. Adv. Math. Phys. 2012, 156573 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Niestegge, G.: Three-slit experiments and quantum nonlocality. Found. Phys. 43, 805–812 (2013)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Niestegge, G.: Super quantum probabilities and three-slit experiments—Wright’s pentagon state and the Popescu–Rohrlich box require third-order interference. Phys. Scr. T160, 014034 (2014)CrossRefADSGoogle Scholar
  15. 15.
    Sorkin, R.D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9, 3119–3127 (1994)MathSciNetCrossRefzbMATHADSGoogle Scholar
  16. 16.
    Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. Found. Phys. 41, 396–405 (2011)MathSciNetCrossRefzbMATHADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Fraunhofer ESKMünchenGermany

Personalised recommendations