Products of Logics

  • Antonio Zecca

Abstract

As pointed out by Gudder,1 the problem of providing a definition of tensor product for general quantum logics seems to be unavoidable if a theory of quantum measurement is addressed and developed in the context of quantum logics. More specifically, suppose we have two physical systems Σ and \(\tilde \Sigma\) with corresponding logics L and \(\tilde L\). For instance Σ could be the physical system under study and \(\tilde \Sigma\) a measure-merit apparatus. If one wants to study the compound physical system \(\Sigma \; + \;\tilde \Sigma\) with corresponding logic L, one has to provide a definition of (tensor) product of L with \(\tilde L\) in order to obtain L. A solution of the general problem should contain as special cases the following standard situations. Suppose first that both Σ and E are classical systems with phase space A and à respectively and associated logics L = P(A), L = P(Ã) the power set of their phase spaces. In this case L is provided by the power set of the cartesian product of the phase spaces, namely by L = P(A × Ã).

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References

  1. 1.
    S.P. Gudder, in “Mathematical Foundations of Quantum Theory”, A.R. Marlow ed., Academic Press, New York (1978).Google Scholar
  2. 2.
    K.E. Hellwig and D. Krausser, Int. J. Theor. Phys. 10 (1974), p. 261.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    K.E. Hellwig and D. Krausser, Int. J. Theor. Phys. 16 (1977), p. 775.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    A. Zecca, J. Math. Phys. 19 (1978), p. 1482.MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    D. Aerts and I Daubechies, Helv. Phys. Acta 51 (1978), p. 661.MathSciNetGoogle Scholar
  6. 6.
    D.J. Foulis and C.H. Randall, Empirical logic and tensor product, in “Proceedings of the Colloquium on the Interpretations and Foundations of Quantum Theories, Marburg 1979”, Holger Neumann ed., to appear.Google Scholar
  7. 7.
    C.H. Randall and D.J. Foulis, Operational statistics and tensor product, in “Proc. Colloq. on the Interpretations and Foundations of Quantum Theories, Marburg 1979”, Holger Neumann ed., to appear.Google Scholar
  8. 8.
    F. Maeda and S. Maeda, “Theory of Symmetric Lattices”, Springer Verlag, Berlin (1970).MATHGoogle Scholar
  9. 9.
    C. Piron, “Foundations of Quantum Physics”, Benjamin, London (1976).MATHGoogle Scholar
  10. 10.
    S.S. Holland, Trans. A.M.S. 108 (1963), p. 66.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    A. Zecca, “The Superposition of the States and the Logic Approach to Quantum Mechanics”, IFUM 229/FT, Milano (March 1979).Google Scholar
  12. 12.
    M. Hughenholtz, Commun. Math. Phys. 6 (1967), p. 189.ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1981

Authors and Affiliations

  • Antonio Zecca
    • 1
    • 2
  1. 1.Istituto di Scienze FisicheMilanoItaly
  2. 2.INFN Sezione di MilanoItaly

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