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A Hilbert Space Realization of Nonlinear Quantum Mechanics as Classical Extension of Its Linear Counterpart

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Abstract

This paper studies the state-effect-probability structure associated with thequantum mechanics of nonlinear (homogeneous, in general nonadditive) operatorson a Hilbert space. Its aim is twofold: to provide a concrete representation ofthe features of nonlinear quantum mechanics on a Hilbert space, and to showthat the properties of the nonlinear version of quantum mechanics here describedhave the structure of a classical logic.

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REFERENCES

  1. E. G. Beltrametti and S. Bugajski, A classical extension of quantum mechanics, J. Phys. A 28 (1995), 3329-3343.

    Google Scholar 

  2. E. G. Beltrametti and S. Bugajski, Quantum observables in classical frameworks, Int. J. Theor. Phys. 34 (1995), 1221-1229.

    Google Scholar 

  3. E. G. Beltrametti and S. Bugajski, Effect algebras and statistical physical theories, J. Math. Phys. 38 (1997), 3020-3030.

    Google Scholar 

  4. E. G. Beltrametti and S. Bugajski, Classical extensions of operational statistical theories, in Waves, Information and Foundations of Physics (R. Pratesi and L. Ronchi, eds.), Società Italiana di Fisica, Bologna, 1998, pp. 333-343.

    Google Scholar 

  5. S. Bugajski, Nonlinear quantum mechanics is a classical theory, Int. J. Theor. Phys. 30 (1992), 961-971.

    Google Scholar 

  6. G. Cattaneo, M. L. Dalla Chiara, and R. Giuntini, Some algebraic structures for many-valued logics, Tatra Mountains Math. Pub. 15 (1998), 173-196.

    Google Scholar 

  7. G. Cattaneo, C. Garola, and G. Nisticò, Preparation-effect versus question-proposition structures, Phys. Essays 2 (1989), 197-216.

    Google Scholar 

  8. G. Cattaneo, R. Giuntini, and R. Pilla, BZMVdMM and Stonian MV algebras (applications, to fuzzy sets and rough approximations), Fussy Sets Systs., 108 (1999), 201-222.

    Google Scholar 

  9. G. Cattaneo and S. Gudder, Algebraic structures arising in axiomatic unsharp quantum mechanics, Found. Phys. 29 (1999), 1607-1637.

    Google Scholar 

  10. G. Cattaneo and G. Nisticò, Orthogonality and orthocomplementations in the axiomatic approach to quantum mechanics: Remarks about some critiques, J. Math. Phys. 25 (1984), 513-531.

    Google Scholar 

  11. R. Haag and U. Bannier, Comments on Mielnik's generalized (non linear) quantum mechanics, Commun. Math. Phys. 60 (1978), 1-6.

    Google Scholar 

  12. B. Mielnik, Generalized quantum mechanics, Commun. Math. Phys. 37 (1974), 221-256.

    Google Scholar 

  13. B. Mielnik, Quantum logic: is it necessarily orthocomplemented, in Quantum Mechanics, Determinism, Causality and Particle (M. Flato et al., ed.), Reidel, Dordrecht, Holland, 1976.

    Google Scholar 

  14. B. Mielnik, Mobility of nonlinear systems, J. Math. Phys. 21 (1980), 44-54.

    Google Scholar 

  15. B. Mielnik, Phenomenon of mobility in nonlinear theories, Commun. Math. Phys. 101 (1985), 323-339.

    Google Scholar 

  16. C. Piron, Foundations of Quantum Physics, Benjamin, Reading, Massachusetts, 1976.

    Google Scholar 

  17. S. Weinberg, Precision tests of quantum mechanics, Phys. Rev. Lett. 62 (1989), 485-488.

    Google Scholar 

  18. S. Weinberg, Testing quantum mechanics, Ann. Phys. 194 (1989), 336-386.

    Google Scholar 

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Authors and Affiliations

  1. Department of Philosophy, California State University, Chico

    Marco Del Seta

  2. London School of Economics and Political science, CPNSS, London, U.K

    Marco Del Seta

  3. Dipartimento di Informatica, Sistematica e Comunicazione, Universitá di Milano-Bicocca, 20126, Milano, Italy

    Gianpiero Cattaneo

Authors
  1. Marco Del Seta
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  2. Gianpiero Cattaneo
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Del Seta, M., Cattaneo, G. A Hilbert Space Realization of Nonlinear Quantum Mechanics as Classical Extension of Its Linear Counterpart. International Journal of Theoretical Physics 39, 621–640 (2000). https://doi.org/10.1023/A:1003685620562

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