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Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

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Abstract

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ∗-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.

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References

  1. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol 1. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  2. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol 2. Springer, Berlin (1997)

    Book  Google Scholar 

  3. Bush, P., Grabowski, M, Lahti, P.J.: Operational Quantum Physics. Springer, Berlin (1995)

    MATH  Google Scholar 

  4. Döring, A., Isham, C.J.: A topos foundations for theories of physics: I, Formal languages for physics. J. Math. Phys. 49, 0534515 (2008)

    MATH  Google Scholar 

  5. Döring, A., Isham, C.J: A topos foundations for theories of physics: II, Daseinisation and the liberation of quantum theory. J. Math. Phys. 49, 0534515 (2008)

    MATH  Google Scholar 

  6. Döring, A., Isham, C.J.: A topos foundations for theories of physics: III, Quantum theory and the representation of physical quantities with arrows, \(\breve {A}: \underline {\Sigma }\to P{\mathbb {R}}\). J. Math. Phys. 49, 0534515 (2008)

    Google Scholar 

  7. Döring, A., Barbosa, R.S: Usharp Values, Domains and Topoi, Chapter in Quantum Field Theory and Gravity, pp. 65–95. Springer, Berlin (2012)

    Google Scholar 

  8. Döring, A., Harding, J.: Abelian subalgebras and the Jordan structure of von Neumann algebras. Houston J. Math., to appear

  9. Dye, H.A.: On the geometry of projections in certain operator algebras. Ann. Math. 61(1), 73–89 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  10. Halvorson, H. (ed.): Deep Beauty: Understanding the Quantum World Through Mathematical Innovation. Cambridge University Press, Cambridge (2011)

  11. Hamhalter, J.: Isomorphisms of ordered structures of abelian C ∗-subalgebras of C ∗-algebras. J. Math. Anal. Appl. 383, 391–399 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamhalter, J., Turilova, E.: Structure of associative subalgebras of Jordan operator algebras. Q. J. Math. Oxford Press 64(2), 397–408 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hamhalter, J., Turilova, E.: Automorphisms of ordered structures of abelian parts of operator algebras and their role in quantum theory. Int. J. Theor. Phys. 53, 3333–3345 (2014). doi:10.1007/s10773-013-1691-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Harding, J., Navara, M.: Subalgebras of orthomodular lattices. Order 28, 549–563 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Heunen, C., Landsman, N.P., Spitters, B.: Bohrification of operator algebras and quantum logic. Synthese, 3 186, 719–752 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Heunen, C., Landsman, N.P., Spitters, B.: A topos for algebraic quantum theory. Commun. Math. Phys. 291, 63–110 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  17. Heunen, C., Lindenhovius, B.: Domains of commutative C*-subalgebras. arXiv:1504.02730, 25 April 2015

  18. Heunen, C. The many classical faces of quantum structures. arXiv:1412.2177, 25 March 2015

  19. Kadison, R.V., Ringrose, J.R.: Theory of Operator Alegebras I, II. Academic, New York (1986)

    Google Scholar 

  20. Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral Representation Theory, Applications to Convexity, Banach Spaces, and Potential Theory. de Gruyter, Berlin (2010)

    MATH  Google Scholar 

  21. Kalmbach, G.: Orthomodular Lattices. London Mathematical Society Monographs (Book 18). Academic, New York (1983)

    Google Scholar 

  22. Takesaki, M.: Theory of Operator Algebras I, II, III. Springer, Berlin (2001)

    Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referees for their helpful comments.

The work of the first author was supported by the “Grant Agency of the Czech Republic” grant number P201/12/0290, “Topological and geometrical properties of Banach spaces and operator algebras”.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technicka 2, 166 27, Prague 6, Czech Republic

    Jan Hamhalter

  2. Department of Mathematical Statistics, Institute of Computer Mathematics and Information Technologies, Kazan Federal University, Kremlevskaya 18, Kazan, Russia

    Ekaterina Turilova

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  1. Jan Hamhalter
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  2. Ekaterina Turilova
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Correspondence to Jan Hamhalter.

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Hamhalter, J., Turilova, E. Orthogonal Measures on State Spaces and Context Structure of Quantum Theory. Int J Theor Phys 55, 3353–3365 (2016). https://doi.org/10.1007/s10773-016-2964-4

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  • DOI: https://doi.org/10.1007/s10773-016-2964-4

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