Skip to main content

A Generalized Quantum Theory


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.

This is a preview of subscription content, access via your institution.


  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. Alfsen, E.M., Shultz, F.W.: Geometry of State Spaces of Operator Algebras. Mathematics: Theory & Applications. Birkhäuser, Boston (2003)

    Book  Google Scholar 

  3. Baez, J.C.: The octonions. Bull. Am. Math. Soc. 39, 145–205 (2001)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Connes, A.: Characterisation des espaces vectoriels ordonnes sousjacent aux algebres de von Neumann. Ann. Inst. Fourier (Grenoble) 24, 121–155 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  6. Evans, D., Hanche-Olsen, H.: The generators of positive semigroups. J. Funct. Analy. 32, 207–212 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  7. Iochum, B., Shultz, F.W.: Normal state spaces of Jordan and von Neumann algebras. J. Funct. Analy. 50, 317–328 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)

    Article  Google Scholar 

  9. Niestegge, G.: Non-Boolean probabilities and quantum measurement. J. Phys. A 34, 6031–6042 (2001)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  10. Niestegge, G.: A representation of quantum mechanics in order-unit spaces. Found. Phys. 38, 783–795 (2008)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Niestegge, G.: Conditional probability, three-slit experiments, and the Jordan algebra structure of quantum mechanics. Adv. Math. Phys. 2012, 156573 (2012)

    Article  MathSciNet  Google Scholar 

  13. Niestegge, G.: Three-slit experiments and quantum nonlocality. Found. Phys. 43, 805–812 (2013)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  15. Sorkin, R.D.: Quantum mechanics as quantum measure theory. Mod. Phys. Lett. A 9, 3119–3127 (1994)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  16. Ududec, C., Barnum, H., Emerson, J.: Three slit experiments and the structure of quantum theory. Found. Phys. 41, 396–405 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gerd Niestegge.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Niestegge, G. A Generalized Quantum Theory. Found Phys 44, 1216–1229 (2014).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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