Spectral Resolutions and Quantum Observables

Abstract

An n-dimensional quantum observable in quantum structures is a kind of a σ-homomorphism defined on the Borel σ-algebra of \(\mathbb R^{n}\) with values in a monotone σ-complete effect algebra or in a σ-complete MV-algebra. It defines an n-dimensional spectral resolution that is a mapping from \(\mathbb R^{n}\) into the quantum structure which is a monotone, left-continuous mapping with non-negative increments and which is going to 0 if one variable goes to \(-\infty \) and it goes to 1 if all variables go to \(+\infty \). The basic question is to show when an n-dimensional spectral resolution entails an n-dimensional quantum observable. We show cases when this is possible and we apply the result to existence of three different kinds of joint observables.

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

References

  1. 1.

    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Catlin, D.: Spectral theory in quantum logics. Inter. J. Theor. Phys. 1, 285–297 (1968)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chang, C.C.: Algebraic analysis of many valued logics. Trans. Amer. Math. Soc. 88, 467–490 (1958)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Kluwer Academic Publ., Dordrecht (2000)

    Google Scholar 

  5. 5.

    Di Nola, A., Dvurečenskij, A.: Product MV-algebras. Multi. Val. Logic 6, 193–215 (2001)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Dvurečenskij, A.: On joint distribution in quantum logics. I. Compatible observables. Aplik. matem. 32, 427–435 (1987)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Dvurečenskij, A.: On joint distribution in quantum logics. II, Noncompatible observables. Aplik. matem. 32, 436–450 (1987)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Dvurečenskij, A.: MV-observables and MV-algebras. J. Math. Anal. Appl. 259, 413–428 (2001)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dvurečenskij, A.: Quantum observables and effect algebras. Inter. J. Theor. Phys. 57, 637–651 (2018). https://doi.org/10.1007/s10773-017-3594-1

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

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

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Dvurečenskij, A., Lachman, D.: Two-dimensional observables and spectral resolutions. Rep. Math Phys. 85, 163–191 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Dvurečenskij, A., Pulmannová, S.: New Trends in Quantum Structures, Kluwer Academic Publ., Dordrecht, Ister Science, Bratislava, 2000, 541 + xvi pp

  13. 13.

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

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Fremlin, D.H.: Topological Riesz Spaces and Measure Theory. Cambridge Univ. Press (1974)

  15. 15.

    Goodearl, K.R.: Partially Ordered Abelian Groups with Interpolation, Math Surveys and Monographs No 20. Amer. Math. Soc., Providence (1986)

    Google Scholar 

  16. 16.

    Halmos, P.R.: Measure Theory. Springer, Berlin (1974)

    Google Scholar 

  17. 17.

    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 

  18. 18.

    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)

    Google Scholar 

  19. 19.

    Mundici, D.: Interpretation of AF c-algebras in Łukasiewicz sentential calculus. J. Funct. Anal. 65, 15–63 (1986)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Mundici, D.: Advanced ŁUkasiewicz Calculus and MV-Algebras Trends in Logic, vol. 35. Springer, Dordrecht (2011)

    Google Scholar 

  21. 21.

    Pulmannová, S.: Compatibility and decomposition of effects. J. Math. Phys. 43, 2817–2830 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Ravindran, K.: On a Structure Theory of Effect Algebras, PhD Thesis. Kansas State Univ., Manhattan (1996)

    Google Scholar 

  23. 23.

    Riečan, B., Neubrunn, T.: Integral, Measure, and Ordering. Kluwer Acad. Publ., Dordrecht (1997)

    Google Scholar 

  24. 24.

    Varadarajan, V.S.: Geometry of Quantum Theory, vol. 1. Van Nostrand, Princeton (1968)

    Google Scholar 

Download references

Acknowledgements

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

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anatolij Dvurečenskij.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The paper has been supported by the grant of the Slovak Research and Development Agency under contract APVV-16-0073 and the grant VEGA No. 2/0069/16 SAV (the first author), and by grant CZ.02.2.69/0.0/0.0/16-027/0008482 SPP 8197200115 (the second one)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dvurečenskij, A., Lachman, D. Spectral Resolutions and Quantum Observables. Int J Theor Phys (2020). https://doi.org/10.1007/s10773-020-04507-z

Download citation

Keywords

  • n-dimensional spectral resolution
  • n-dimensional observable
  • Joint observable
  • MV-algebra
  • Effect algebra
  • State
  • Effect-tribe