Isham, Chris and Butterfield, Jeremy
(1998)
A Topos Perspective on the KochenSpecker Theorem: I. Quantum States as Generalised Valuations.
UNSPECIFIED.
Abstract
Any attempt to construct a realist interpretation of quantum theory founders on the KochenSpecker theorem, which asserts the impossibility of assigning values to quantum quantities in a way that preserves functional relations between them. We construct a new type of valuation which is defined on all operators, and which respects an appropriate version of the functional composition principle. The truthvalues assigned to propositions are (i) contextual; and (ii) multivalued, where the space of contexts and the multivalued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the KochenSpecker theorem is equivalent to the statement that a certain presheaf defined on the category of selfadjoint operators has no global elements. We then show how the use of ideas drawn from the theory of presheaves leads to the definition of a generalised valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalised valuation. A key ingredient throughout is the idea that, in a situation where no normal truthvalue can be given to a proposition asserting that the value of a physical quantity A lies in a set D of real numbers , it is nevertheless possible to ascribe a partial truthvalue which is determined by the set of all coarsegrained propositions that assert that some function f(A) lies in f(D), and that are true in a normal sense. The set of all such coarsegrainings forms a sieve on the category of selfadjoint operators, and is hence fundamentally related to the theory of presheave
Item Type: 
Other

Creators: 
Creators  Email  ORCID 

Isham, Chris    Butterfield, Jeremy   

Additional Information: 
Published in: International Journal of Theoretical Physics, volume 37, 1998, pages 26692733 
Keywords: 
KochenSpecker theorem; category theory; topos theory; generalised truthvalues; presheaves 
Subjects: 
Specific Sciences > Physics > Quantum Mechanics 
Depositing User: 
Jeremy Butterfield

Date Deposited: 
29 Aug 2004 
Last Modified: 
07 Oct 2010 15:12 
Item ID: 
1916 
Public Domain: 
No 
URI: 
http://philsciarchive.pitt.edu/id/eprint/1916 
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