Abstract
The story of quantum probability theory and quantum logic begins with von Neumann’s recognition1, that quantum mechanics can be regarded as a kind of “probability theory”, if the subspace lattice L(H) of the system’s Hilbert space H plays the role of event algebra and the ‘tr(WE)’-s play the role of probability distributions over these events. This idea had been completed in the Gleason theorem 2:
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This idea appeared in J. von Neumann, Mathematische Grundlagen der Quantenmechanik, (Berlin: Springer, 1932). One can find it in a more explicit and somewhat different form in G. Birkhoff and J. von Neumann, “The logic of quantum mechanics”, Ann. Math. 37 (1936), 823–843.
A. M. Gleason, “Measures on the closed subspaces of a Hilbert space”, J. math. Phys. 6 (1957), 8855–8893.
The subspaces, the corresponding projectors and the corresponding events are denoted by the same letter.
Cf. M. Dummett, The logical basis of metaphysics, ( London: Duckworth, 1995 ).
R. Feynmann and A. Hibbs, Quantum Mechanics and Path Integrals, ( New York: McGraw-Hill, 1965 ).
M. Strauss, “Mathematics as logical syntax–A method to formalize the language of a physical theory”, Erkenntnis, 7 (1937), 147–153.
J. S. Bell, Speakable and unspeakable in quantum mechanics, ( Cambridge: Cambridge University Press, 1987 ), p. 166.
Cf. M. Rédei, “Why John von Neumann did not like the Hilbert space formalism of quantum mechanics (and what he liked instead)”, Studies in the History and Philosophy of Modern Physics, 27 (1996) 493–510.
L. E. Szabó, “On an attempt to resolve the EPR-Bell paradox via Reichenbachian concept of common cause”, forthcoming in Int. J. of Theor. Phys.
Cf. M. Rédei, Von Neumann’s concept of quantum logic and quantum probability, in this volume.
S. P. Gudder, Quantum probability, Academic Press Inc., San Diego, 1988, p. 57.
B. Van Fraassen, Quantum Mechanics - An Empiricist View, ( Oxford: Clarendon Press, 1991 ), p. 111.
L. E. Szabó, “Is quantum mechanics compatible with a deterministic universe? Two interpretations of quantum probabilities”, Foundations of Physics Letters, 8 (1995), 421–440.
Bana, G. and Durt, T., Proof of Kolmogorovian Censorship, Foundations of Physics, 27, 13551373. (1997)
See J. F. Clauser and A. Shimony, “Bell’s theorem: experimental tests and implications”, Rep. Prog. Phys. 41 (1978), 1881–1927. There is, however, an important conceptual disagreement between (16) and the original Clauser-Horne inequalities, see L. E. Szabó, “Quantum mechanics in an entirely deterministic universe”, Int. J. Theor. Phys., 34 (1995), 1751–1766.
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Szabó, L.E. (2001). Critical Reflections on Quantum Probability Theory. In: Rédei, M., Stöltzner, M. (eds) John von Neumann and the Foundations of Quantum Physics. Vienna Circle Institute Yearbook [2000], vol 8. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2012-0_13
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DOI: https://doi.org/10.1007/978-94-017-2012-0_13
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