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Tensor products and probability weights

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Abstract

We study a general tensor product for two collections of related physical operations or observations. This is a free product, subject only to the condition that the operations in the first collection fail to have any influence on the statistics of operations in the second collection and vice versa. In the finite-dimensional case, it is shown that the vector space generated by the probability weights on the general tensor product is the algebraic tensor product of the vector spaces generated by the probability weights on the components. The relationship between the general tensor product and the tensor product of Hilbert spaces is examined in the light of this result.

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References

  1. Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823–843.

  2. Cook, T. (1985). Banach spaces of weights on quasi manuals,International journal of Theoretical Physics,24, 1113–1131.

  3. Dirac, P. (1930).The Principles of Quantum Mechanics, Oxford-Clarendon Press, Oxford.

  4. Foulis, D., and Randall, C. (1980). Empirical logic and tensor products, inInterpretations and Foundations of Quantum Theory, pp. 9–20, H. Neumann, ed., Wissenschaftsverlag, Bibliographisches Institut, Mannheim.

  5. Foulis, D., and Randall, C. (1985). Dirac revisited, inSymposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt, eds., pp. 97–112, World Scientific, Singapore.

  6. Foulis, D., Piron, C., and Randall, C. (1983). Realism, operationalism, and quantum mechanics,Foundations of Physics,13, 813–842.

  7. Gleason, A. (1957). Measures on the closed subspaces of a Hilbert space,Journal of Mathematics and Mechanics,6, 885–893.

  8. Greechie, R., and Miller, F. (1970). Structures Related to States on an Empirical Logic I: Weights on Finite Spaces, Kansas State University Technical Report Number 14, Manhattan, Kansas.

  9. Groenewold, H. (1985). The elusive quantal individual,Physics Reports,127, 379–401.

  10. Gudder, S. (1986). Logical cover spaces,Annales de l'Institut Henri Poincaré,45, 327–337.

  11. Gudder, S., Kläy, M., and Rüttimann, G. (1987). States on hypergraphs,Demonstratio Mathematica, to appear.

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

  13. Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, New York.

  14. Kläy, M. (1985). Stochastic Models on Empirical Systems, Empirical Logics and Quantum Logics, and States on Hypergraphs, Ph.D. thesis, University of Bern, Switzerland.

  15. Randall, C., and Foulis, D. (1973). Operational statistics II: Manuals of operations and their logics,Journal of Mathematical Physics,14, 1472–1480.

  16. Randall, C., and Foulis, D. (1978). The operational approach to quantum mechanics, inThe Logico-Algebraic Approach to Quantum Mechanics, Volume III, C. Hooker, ed., pp. 167–201, D. Reidel, Boston.

  17. Randall, C., and Foulis, D. (1980). Operational statistics and tensor products, inInterpretations and Foundations of Quantum Theory, H. Neumann, ed., pp. 21–28, Wissenschaftsverlag, Bibliographisches Institut, Mannheim.

  18. Randall, C., and Foulis, D. (1985). Stochastic entities, inRecent Developments in Quantum Logic, P. Mittelstaedt and E. Stachow, eds., pp. 265–284, Wissenschaftsverlag, Bibliographisches Institut, Mannheim.

  19. Rüttimann, G. (1985a). Expectation functionals of observables and counters,Reports on Mathematical Physics,21, 213–222.

  20. Rüttimann, G. (1985b). Facial sets of probability measures,Probability and Mathematical Statistics,6, 187–215.

  21. Schrödinger, E. (1935). Discussion of probability relations between separated systems,Proceedings of the Cambridge Philosophical Society,31, 555–562.

  22. Schrödinger, E. (1936). Discussion of probability relations between separated systems,Proceedings of the Cambridge Philosophical Society,32, 446–452.

  23. Schultz, F. (1984). A characterization of state spaces of orthomodular lattices,Journal of Combinatorial Theory (A),17, 317–328.

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

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Kläy, M., Randall, C. & Foulis, D. Tensor products and probability weights. Int J Theor Phys 26, 199–219 (1987). https://doi.org/10.1007/BF00668911

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Keywords

  • Hilbert Space
  • Field Theory
  • Vector Space
  • Elementary Particle
  • Quantum Field Theory