Abstract
Any attempt to construct a realistinterpretation of quantum theory founders on theKochen–Specker theorem, which asserts theimpossibility of assigning values to quantum quantitiesin a way that preserves functional relations between them. We constructa new type of valuation which is defined on alloperators, and which respects an appropriate version ofthe functional composition principle. The truth-values assigned to propositions are (i) contextual and(ii) multivalued, where the space of contexts and themultivalued logic for each context come naturally fromthe topos theory of presheaves. The first step in our theory is to demonstrate that theKochen–Specker theorem is equivalent to thestatement that a certain presheaf defined on thecategory of self-adjoint operators has no globalelements. We then show how the use of ideas drawn from the theory ofpresheaves leads to the definition of a generalizedvaluation in quantum theory whose values are sieves ofoperators. In particular, we show how each quantum state leads to such a generalized valuation. Akey ingredient throughout is the idea that, in asituation where no normal truth-value can be given to aproposition asserting that the value of a physical quantity A lies in a subset \(\Delta \subseteq \mathbb{R}\), it is nevertheless possible toascribe a partial truth-value which is determined by theset of all coarse-grained propositions that assert thatsome function f(A) lies in f(Δ), and that are true in a normalsense. The set of all such coarse-grainings forms asieve on the category of self-adjoint operators, and ishence fundamentally related to the theory ofpresheaves.
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REFERENCES
Bacciagaluppi, G., and Hemmo, M. (1996). Modal interpretations, decoherence and measurements, Studies in the History and Philosophy of Modern Physics, 27B, 239-278.
Bell, J. S. (1987). On the problem of hidden variables in quantum mechanics, in Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, pp. 1-13.
Brown, H. R. (1992). Bell's other theorem and its connection with non-locality. Part 1, in Bell's Theorem and the Foundations of Modern Physics, A. van der Merwe, F. Selleri, and G. Tarozzi, eds., World Scientific, Singapore.
Bub, J. (1997). Interpreting the Quantum World, Cambridge University Press, Cambridge.
Bub, J., and Clifton, C. (1996). A uniqueness theorem for “no collapse” interpretations of quantum mechanics, Studies in the History and Philosophy of Modern Physics, 27B, 181-219.
Busch, P., Grabowski, M., and Lahti, P. J. (1995). Operational Quantum Physics, Springer-Verlag, Berlin.
Butterfield, J., and Isham, C. J. (1998). A topos perspective on the Kochen–Specker theorem: II. Conceptual aspects and Classical Analogues, quant-ph/9808067, forthcoming in the International Journal of Theoretical Physics, 38.
Clifton, R. (1995). Independently motivating the Kochen-Dieks modal interpretation of quantum mechanics, British Journal for the Philosophy of Science, 46, 33-57.
Dieks, D. (1995). Physical motivation of the modal interpretation of quantum mechanics, Physics Letters A, 197, 367-371.
Dummett, M. (1978). Is logic empirical? In Truth and Other Enigmas, Duckworth, London.
Dunford, N., and Schwartz, J. T. (1964). Linear Operators Part II: Spectral Theory, Interscience Publishers, New York.
Goldblatt, R. (1984). Topoi: The Categorial Analysis of Logic, North-Holland, Amsterdam.
Healey, R. (1989). The Philosophy of Quantum Mechanics, Cambridge University Press, Cambridge.
Isham, C. J. (1997). Topos theory and consistent histories: The internal logic of the set of all consistent sets, International Journal of Theoretical Physics, 36, 785-814.
Kochen, S. (1985). A new interpretation of quantum mechanics, in Symposium on the Foundations of Modern Physics, P. Lahti and P. Mittelstaedt, eds., World Scientific, Singapore, pp. 151-170.
Kochen, S., and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics, Journal of Mathematics and Mechanics, 17, 59-87.
MacLane, S., and Moerdijk, I. (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, Berlin.
Putnam, H. (1975). The logic of quantum mechanics, in Mathematics, Matter and Method, Cambridge University Press, Cambridge, pp. 174-197.
Singer, I. (1978). Some remarks on the Gribov ambiguity, Communications in Mathematical Physics, 60, 7-12.
van Fraassen, B. C. (1981). A modal interpretation of quantum mechanics, in Current Issues in Quantum Logic, E. Beltrametti and B. C. van Fraassen, eds., Plenum Press, New York, pp. 229-258.
van Fraassen, B. C. (1991). Quantum Mechanics: An Empiricist View, Clarendon Press, Oxford.
Vermaas, P. E., and Dieks, D. (1995). The modal interpretation of quantum theory, Foundations of Physics, 25, 145-158.
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Isham, C.J., Butterfield, J. Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations. International Journal of Theoretical Physics 37, 2669–2733 (1998). https://doi.org/10.1023/A:1026680806775
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DOI: https://doi.org/10.1023/A:1026680806775