Skip to main content
SpringerLink
Log in

Quantum Observables and Effect Algebras

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

We study observables on monotone σ-complete effect algebras. We find conditions when a spectral resolution implies existence of the corresponding observable. We characterize sharp observables of a monotone σ-complete homogeneous effect algebra using its orthoalgebraic skeleton. In addition, we study compatibility in orthoalgebras and we show that every orthoalgebra satisfying RIP is an orthomodular poset.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauer, H.: Probability theory. de Gruyter, Berlin (1996)

    Book  MATH  Google Scholar 

  2. Buhagiar, D., Chetcuti, E., Dvurečenskij, A.: Loomis-Sikorski representation of monotone σ-complete effect algebras. Fuzzy Sets Syst. 157, 683–690 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Dvurečenskij, A.: Measures and ⊥-decomposable measures on effects of a Hilbert space. Atti Sem. Mat. Fis. Univ. Modena 45, 259–288 (1997)

    MathSciNet  MATH  Google Scholar 

  4. Dvurečenskij, A.: Loomis–Sikorski theorem for σ-complete MV-algebras and ℓ-groups. J. Austral. Math. Soc. Ser A 68, 261–277 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dvurečenskij, A.: On effect algebras which can be covered by MV-algebras. Inter. J. Theor. Phys. 41, 221–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dvurečenskij, A.: Central elements and Cantor–Bernstein’s theorem for pseudo-effect algebras. J. Austral. Math Soc. 74, 121–143 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dvurečenskij, A.: Representable effect algebras and observables. Inter. J. Theor. Phys. 53, 2855–2866 (2014). https://doi.org/10.1007/s10773-014-2083-z

    Article  MathSciNet  MATH  Google Scholar 

  8. Dvurečenskij, A.: On orders of observables on effect algebras, Inter. J. Theor. Phys. https://doi.org/10.1007/s10773-017-3472-x

  9. Dvurečenskij, A., Kuková, M.: Observables on quantum structures. Inf. Sci. 262, 215–222 (2014). https://doi.org/10.1016/j.ins.2013.09.014

    Article  MathSciNet  MATH  Google Scholar 

  10. Dvurečenskij, A., Pták, P.: Quantum logics with the Riesz interpolation property. Math. Nachr. 271, 10–14 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dvurečenskij, A., Pulmannová, S.: New Trends in quantum structures, p 541 + xvi. Kluwer Academic Publication, Dordrecht, Ister Science, Bratislava (2000)

    Book  MATH  Google Scholar 

  12. Foulis, D.J., Bennett, M.K.: Effect algebras and unsharp quantum logics. Found. Phys. 24, 1325–1346 (1994)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Foulis, D.J., Greechie, R.J., Rüttimann, G.T.: Filters and supports in orthoalgebras. Inter. J. Theor. Phys. 31, 789–807 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gudder, S.P.: Uniqueness and existence properties of bounded observables. Pac. J. Math. 19, 81–93 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Halmos, P.R.: Measure theory. Springer-Verlag, Berlin (1974)

    MATH  Google Scholar 

  16. Jenča, G.: Blocks of homogeneous effect algebras. Bull. Austral. Math. Soc. 64, 81–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jenča, G.: Sharp and meager elements in orthocomlete homogeneous effect algebras. Order 27, 41–61 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jenča, G., Riečanová, Z.: On sharp elements in lattice ordered effect algebras. Busefal 80, 24–29 (1999)

    Google Scholar 

  19. Jenčová, A., Pulmannová, S., Vinceková, E.: Sharp and fuzzy observables on effect algebras. Inter. J. Theor. Phys. 47, 125–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jenčová, A., Pulmannová, S., Vinceková, E.: Observables on σ-MV-algebras and σ-lattice effect algebras. Kybernetika 47, 541–559 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Ji, W.: Characterization of homogeneity in orthocomplete atomic effect algebras. Fuzzy Sets Syst. 236, 113–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kallenberg, O.: Foundations of modern probability. Springer-Verlag, Berlin Heidelberg (1997)

    MATH  Google Scholar 

  23. Mundici, D.: Tensor products and the LoomisSikorski theorem for MV-algebras. Advan. Appl Math. 22, 227–248 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  24. Niederle, J., Paseka, J.: Homogeneous orthocomplete effect algebras are covered by MV-algebras. Fuzzy Sets Syst. 210, 89–101 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Pták, P., Pulmannová, S.: Orthomodular structures as quantum logics. VEDA and Kluwer Academic Publication, Bratislava and Dordrecht (1991)

    MATH  Google Scholar 

  26. Pulmannová, A.: A quantum logics description of quantum measurements. Tatra Mountains Math. Publ. 3, 89–100 (1993)

    MathSciNet  MATH  Google Scholar 

  27. Ravindran, K.: On a structure theory of effect algebras. PhD thesis, Kansas State University, Manhattan (1996)

    Google Scholar 

  28. Schröder, B.S.W.: On three notions of orthosummability in orthoalgebras. Inter. J. Theor. Phys. 38, 3305–3313 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author is very indebted to anonymous referees for their careful reading and suggestions which helped us to improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73, Bratislava, Slovakia

    Anatolij Dvurečenskij

  2. Department Algebra Geometry, Palacký University, 17. listopadu 12, CZ-771 46, Olomouc, Czech Republic

    Anatolij Dvurečenskij

Authors
  1. Anatolij Dvurečenskij
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anatolij Dvurečenskij.

Additional information

The paper has been supported by grant of the Slovak Research and Development Agency under contract APVV-16-0073, by the grant VEGA No. 2/0069/16 SAV and GAČR 15-15286S.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dvurečenskij, A. Quantum Observables and Effect Algebras. Int J Theor Phys 57, 637–651 (2018). https://doi.org/10.1007/s10773-017-3594-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-017-3594-1

Keywords

Search

Navigation