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Embedding quantum logics in Hilbert space

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Abstract

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.

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References

  1. Birkhoff, G., Lattice Theory, Amer. Math. Soc. Colloquium Publ., Vol. 25, Providence, R.I., 1967.

  2. BirkhoffG., and vonNeumannJ., ‘The logic of quantum mechanics’, Ann. Math. 37, 823–843 (1936).

    Google Scholar 

  3. DaviesE.B., Quantum Theory of Open Systems, Academic Press, N.Y., 1976.

    Google Scholar 

  4. DaviesE.B., and LewisJ., ‘An operational approach to quantum probability’, Commun. Math. Phys. 17, 239–260 (1970).

    Google Scholar 

  5. EmchG., Algebraic Methods in Statistical Mechanics and Quantum Field Theory, Wiley-Interscience, N.Y., 1972.

    Google Scholar 

  6. FoulisD., ‘Baer *-semigroups’, Proc. Amer. Math. Soc. 11, 648–654 (1960).

    Google Scholar 

  7. FoulisD., ‘A note on orthomodular lattices’, Portugal Math. 21, 65–72 (1962).

    Google Scholar 

  8. GreechieR., ‘An orthomodular lattice with a full set of states not embeddable in Hilbert space’, Carib. J. Sci. Math. 2, 15–26 (1969).

    Google Scholar 

  9. GreechieR., ‘Orthomodular lattices admitting no states’, J. Comb. Theory 10, 119–132 (1971).

    Google Scholar 

  10. GudderS., Stochastic Methods in Quantum Mechanics, Elsevier-North Holland, N.Y., 1979.

    Google Scholar 

  11. HaagR., and KastlerD., ‘An algebraic approach to quantum field theory’, J. Math. Phys. 5, 848–861 (1964).

    Google Scholar 

  12. HewittE., and ZuckermanH., ‘The l 1-algebra of a commutative semigroup’, Trans. Amer. Math. Soc. 83, 70–97 (1956).

    Google Scholar 

  13. JauchJ., Foundations of Quantum Mechanics, Addison-Wesley, Reading, Mass., 1968.

    Google Scholar 

  14. MackeyG., The Mathematical Foundations of Quantum Mechanics, Benjamin, N.Y., 1963.

    Google Scholar 

  15. Michel, J., ‘Hilbert space representations of Baer *-semigroups and quantum logics’, Dissertation, Univ. of Wisconsin, 1974.

  16. NaimarkM., Normed Algebras, Noordhoff, Groningen, The Netherlands, 1972.

    Google Scholar 

  17. PironC., Foundations of Quantum Physics, W.A. Benjamin Inc., Reading, Massachusetts, 1976.

    Google Scholar 

  18. PoolJ., ‘Baer *-semigroups and the logic of quantum mechanics’, Commun. Math. Phys. 9, 118–141 (1968).

    Google Scholar 

  19. PoolJ., ‘Semi-modularity and the logic of quantum mechanics’, Commun. Math. Phys., 9, 212–228 (1968).

    Google Scholar 

  20. RüttimanG., ‘On the logical structure of quantum mechanics’, Found. Phys. 1, 173–182 (1972).

    Google Scholar 

  21. VaradarajanV., Geometry of Quantum Theory, Vols. 1 and 2, Van Nostrand, Princeton, N.J., 1968 and 1970.

    Google Scholar 

  22. ZierlerN., ‘Axioms for non-relativistic quantum mechanics’, Pac. J. Math. 11, 1151–1169 (1961).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, 80208, Denver, Colorado, USA

    Stanley P. Gudder

  2. Department of Mathematics, Marietta College, 45750, Marietta, Ohio, USA

    John R. Michel

Authors
  1. Stanley P. Gudder
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  2. John R. Michel
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Gudder, S.P., Michel, J.R. Embedding quantum logics in Hilbert space. Lett Math Phys 3, 379–386 (1979). https://doi.org/10.1007/BF00397211

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  • DOI: https://doi.org/10.1007/BF00397211

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