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References (17)
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Acknowledgements Chris Isham gratefully acknowledges support by the EPSRC grant GR/R36572. Jeremy Butterfield thanks Hans Halvorson for very helpful discussions -
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Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized ValuationsInternational Journal of Theoretical Physics, 37
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But again the conclusion-that taking R as subsethood in these schemas is sufficient for these propertiesreflects the correspondence in Section 3 between sieve-valued and intervalvalued valuations -
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Topos Perspective on the Kochen=nSpeckerTheorem: III. Von Neumann Algebras as theBase CategoryInternational Journal of Theoretical Physics, 39
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The analogue of the proviso above -
3) for all bounded Borel functions f and allÂ, we have the equality f (σ(Â)) = σ(f (Â)) (as always occurs if has pure discrete spectrum), not merely f (σ(Â)) ⊆ σ(f (Â)) as in Eq -
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A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical AnaloguesInternational Journal of Theoretical Physics, 38
- Publisher
- Springer Journals
- Copyright
- Copyright © 2002 by Plenum Publishing Corporation
- Subject
- Physics; Physics, general; Quantum Physics; Elementary Particles, Quantum Field Theory; Theoretical, Mathematical and Computational Physics
- ISSN
- 0020-7748
- eISSN
- 1572-9575
- DOI
- 10.1023/A:1015276209768
- Publisher site
- See Article on Publisher Site
Abstract
We extend the topos-theoretic treatment given in previous papers (Butterfield, J. and Isham, C. J. (1999). International Journal of Theoretical Physics 38, 827–859; Hamilton, J., Butterfield, J., and Isham, C. J. (2000). International Journal of Theoretical Physics 39, 1413–1436; Isham, C. J. and Butterfield, J. (1998). International Journal of Theoretical Physics 37, 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).
Journal
International Journal of Theoretical Physics – Springer Journals
Published: Sep 30, 2004