Advertisement

International Journal of Theoretical Physics

, Volume 41, Issue 4, pp 613–639 | Cite as

Topos Perspective on the Kochen–Specker Theorem: IV. Interval Valuations

  • J. ButterfieldEmail author
  • C. J. Isham
Article

Abstract

We extend the topos-theoretic treatment given in previous papers (Butterfield, J. and Isham, C. J. (1999). International Journal of Theoretical Physics38, 827–859; Hamilton, J., Butterfield, J., and Isham, C. J. (2000). International Journal of Theoretical Physics39, 1413–1436; Isham, C. J. and Butterfield, J. (1998). International Journal of Theoretical Physics37, 2669–2733) of assigning values to quantities in quantum theory. In those papers, the main idea was to assign a sieve as a partial and contextual truth value to a proposition that the value of a quantity lies in a certain set \(\Delta \subseteq \mathbb{R}\). Here we relate such sieve-valued valuations to valuations that assign to quantities subsets, rather than single elements, of their spectra (we call these “interval” valuations). There are two main results. First, there is a natural correspondence between these two kinds of valuation, which uses the notion of a state's support for a quantity (Section 3). Second, if one starts with a more general notion of interval valuation, one sees that our interval valuations based on the notion of support (and correspondingly, our sieve-valued valuations) are a simple way to secure certain natural properties of valuations, such as monotonicity (Section 4).

Kochen–Specker theorem topos theory valuations supports quantum logic 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. Butterfield, J. (2001). Topos theory as aframework for partial truth. In Proceedings of the 11th International Congress of Logic Methodology and Philosophy of Science, P. Gardenfors, K. Kijania-Placek, and J. Wolenski, eds., Kluwer Academic, Norwell, MA. Available at: http://philsciarchive. pitt.edu, ID code D PITT-PHIL-SCI00000192.Google Scholar
  2. Butterfield, J. and Isham, C. J. (1999). Atoposperspective on theKochen-Specker theorem: II. Conceptual aspects, and classical analogues. International Journal of Theoretical Physics 38, 827–859.Google Scholar
  3. Hamilton, J., Butterfield, J., and Isham, C. J.(2000). A topos perspective on the Kochen-Specker theorem: III. Von Neumann algebras as the base category. International Journal of Theoretica Physics 39, 1413–1436.Google Scholar
  4. Isham, C. J. and Butterfield, J. (1998). Atopos perspective on theKochen-Specker theorem: I. Quantum states as generalised valuations. International Journal of Theoretical Physics 37, 2669–2733.Google Scholar
  5. Kadison, R. V. and Ringrose, J. R. (1983). Fundamentals of theTheory of Operator Algebras Vol. 1: Elementary Theory, Academic Press, New York.Google Scholar
  6. Kochen, S. and Specker, E. P. (1967). The problem of hiddenvariables in quantum mechanics. Journal of Mathematics and Mechanics 17, 59–87.Google Scholar
  7. Vermaas, P. (2000). A Philosopher's Understanding of QuantumMechanics, Cambridge University Press, Cambridge, UK.Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  1. 1.All Souls CollegeOxfordUnited Kingdom
  2. 2.Technology and Medicine, South KensingtonThe Blackett Laboratory, Imperial College of ScienceLondonUnited Kingdom

Personalised recommendations