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The superposition of the states and the logic approach to quantum mechanics

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Abstract

An axiomatic approach to quantum mechanics is proposed in terms of a “logic” scheme satisfying a suitable set of axioms. In this context the notion of pure, maximal, and characteristic state as well as the superposition relation and the superposition principle for the states are studied. The role the superposition relation plays in the reversible and in the irreversible dynamics is investigated and its connection with the tensor product is studied. Throughout the paper, theW *-algebra model, which satisfies our axioms, is used to exemplify results and properties of the general scheme.

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References

  • Aerts, D., and Daubechies, I. (1978).Helvetica Physica Acta,51, 661.

    Google Scholar 

  • Beltrametti, E. G., and Cassinelli, G. (1976).Rivista del Nuovo Cimento,6, 321.

    Google Scholar 

  • Berzi, V., and Zecca, A. (1974).Communications in Mathematical Physics,35, 93.

    Google Scholar 

  • Birkhoff, G., and von Neumann, J. (1936).Annals of Mathematics,37, 823.

    Google Scholar 

  • Birkhoff, G. (1967).Lattice Theory, Vol. 25. American Mathematical Society, Providence, Rhode Island.

    Google Scholar 

  • Cantoni, V. (1975).Communications in Mathematical Physics,44, 125.

    Google Scholar 

  • Cantoni, V. (1976).Communications in Mathematical Physics,50, 241.

    Google Scholar 

  • Chen, E. (1973).Journal of Mathematical Physics,14, 1462.

    Google Scholar 

  • Cirelli, R., and Gallone, F. (1973).Annales de l'Institute Henry Poincaré,19, 297.

    Google Scholar 

  • Cirelli, R., Gallone, F., and Gubbay, B. (1975).Journal of Mathematical Physics,16, 201.

    Google Scholar 

  • Crawley, P., and Dilworth, R. P. (1973).Algebraic Theory of Lattices. Prentice-Hall, Englewood Cliffs, New Jersey.

    Google Scholar 

  • Davies, E. B. (1974).Communications in Mathematical Physics,39, 91 (1974).

    Google Scholar 

  • Delyannis, P. C. (1976).Journal of Mathematical Physics,17, 248.

    Google Scholar 

  • Dirac, P. A. M. (1947).The Principle of Quantum Mechanics. Oxford, Clarendon Press.

    Google Scholar 

  • Diximier, J. (1969).Les C *-Algebres et leurs Representations. Gauthier-Villars, Paris.

    Google Scholar 

  • Diximier, J. (1969).Rendiconti della Scuola Internazionale di Fisica “Enrico Fermi,” XLV, 1.

    Google Scholar 

  • Frigerio, A., Gorini, V., Kossakowski, A., Sudarshan, E. C. G., and Verri, M., (1978).Reports in Mathematical Physics,13, 149.

    Google Scholar 

  • Gallone, F., and Zecca, A. (1973).International Journal of Theoretical Physics,8, 51.

    Google Scholar 

  • Gleason, A. M. (1957).Journal of Mathematics and Mechanics,6, 885.

    Google Scholar 

  • Gorini, V., and Zecca, A. (1975).Journal of Mathematical Physics,16, 667.

    Google Scholar 

  • Gudder, S. P. (1965).Transactions of the American Mathematical Society,119, 428.

    Google Scholar 

  • Gudder, S. P. (1970).Journal of Mathematical Physics,11, 1037.

    Google Scholar 

  • Gudder, S. P. (1978). InMathematical Foundations of Quantum Theory, A. R. Marlow, ed. Academic, New York.

    Google Scholar 

  • Haag, R., and Kastler, D. (1964).Journal of Mathematical Physics,5, 848.

    Google Scholar 

  • Haake, F. (1973).Springer Tracts in Modern Physics, Vol. 66. Springer Verlag, Berlin.

    Google Scholar 

  • Hepp, K. (1972).Helvetica Physica Acta,45, 237.

    Google Scholar 

  • Hepp, K., and Lieb, E. (1973).Helvetica Physica Acta,46, 573.

    Google Scholar 

  • Holland, S. (1963).Transactions of the American Mathematical Society,108, 66.

    Google Scholar 

  • Hugenholtz, N. M. (1967).Communications in Mathematical Physics,6, 189.

    Google Scholar 

  • Jauch, J. M. (1968).Foundations of Quantum Mechanics. Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Jauch, J. M., and Piron, C. (1963).Helvetica Physica Acta,36, 827.

    Google Scholar 

  • Jauch, J. M., and Piron, C. (1969).Helvetica Physica Acta,42, 842.

    Google Scholar 

  • Kadison, R. D. (1965).Topology,Suppl. 2, 177–198.

    Google Scholar 

  • Lanz, L., Lugiato, L., and Ramella, G. (1971).Physica,54, 94.

    Google Scholar 

  • Ludwig, G. (1974).Foundations of Quantum Mechanics and Ordered Linear Spaces, Marburg 1973. Lecture Notes in Physics 29, Springer Verlag, Berlin.

    Google Scholar 

  • Lugiato, L. (1975).Physica,81A, 565.

    Google Scholar 

  • Lugiato, L. (1976).Physica,82A, 1.

    Google Scholar 

  • Mackey, G. W. (1963).Mathematical Foundations of Quantum Mechanics, Benjamin, New York.

    Google Scholar 

  • Maeda, M., and Maeda, S. (1970).Theory of Symmetric Lattices. Springer Verlag, Berlin.

    Google Scholar 

  • Piron, C. (1964).Helvetica Physica Acta,37, 439.

    Google Scholar 

  • Piron, C. (1976). InQuantum Mechanics. Determinism. Causality and Particles, M. Flato et al., eds. Reidel, Dordrecht, Holland.

    Google Scholar 

  • Piron, C. (1976).Foundations of Quantum Physics. Benjamin, Reading, Massachusetts.

    Google Scholar 

  • Pool, J. C. T. (1968).Communications in Mathematical Physics,9, 118.

    Google Scholar 

  • Pulmannová, S. (1976).Communications in Mathematical Physics,49, 47.

    Google Scholar 

  • Reed, M., and Simon, B. (1972).Methods of Modern Mathematical Physics, Vol. 10. Academic Press, New York.

    Google Scholar 

  • Ruelle, D. (1969).Statistical Mechanics. Benjamin, New York.

    Google Scholar 

  • Sakai, S. (1971). C*-Algebras and W*-Algebras. Springer Verlag, Berlin.

    Google Scholar 

  • Schatten, R. (1960).Norm Ideals of Completely Continuous Operators, Springer Verlag, Berlin.

    Google Scholar 

  • Varadarajan, V. S. (1965).Communications in Pure and Applied Mathematics,18, 767.

    Google Scholar 

  • Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vols. I and II. Van Nostrand, Princeton, New Jersey.

    Google Scholar 

  • von Neumann, J. (1955).Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton, New Jersey.

    Google Scholar 

  • Zecca, A. (1976).International Journal of Theoretical Physics,15, 785.

    Google Scholar 

  • Zecca, A. (1978).Journal of Mathematical Physics,19, 1482.

    Google Scholar 

Download references

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Authors and Affiliations

  1. Istituto di Scienze Fisiche dell'Università, Milano, Italy

    Antonio Zecca

  2. Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Milano, Italy

    Antonio Zecca

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  1. Antonio Zecca
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Zecca, A. The superposition of the states and the logic approach to quantum mechanics. Int J Theor Phys 20, 191–230 (1981). https://doi.org/10.1007/BF00669795

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  • DOI: https://doi.org/10.1007/BF00669795

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