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Topos Perspective on the Kochen-Specker Theorem: I. Quantum States as Generalized Valuations

Christopher John Isham, J. Butterfield

International Journal of Theoretical PhysicsNov 1, 1998231 citations

A Topos Perspective on the Kochen-Specker Theorem II. Conceptual Aspects and Classical Analogues

Jeremy Butterfield, C. J. Isham

International Journal of Theoretical PhysicsMar 1, 1999127 citations

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

Jeremy Butterfield, Christopher John Isham

We extend the topos-theoretic treatment given in previous papers 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 $Δ\subseteq \mathR$. 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).We extend the topos-theoretic treatment given in previous papers 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 $Δ\subseteq \mathR$. 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).

arXiv:quant-ph/0107123Jul 24, 20018 citations

A Topos Perspective on the Kochen-Specker Theorem: III. Von Neumann Algebras as the Base Category

Hamilton, J., Christopher John Isham, Jeremy Butterfield

We extend the topos-theoretic treatment given in previous papers of assigning values to quantities in quantum theory, and of related issues such as the Kochen-Specker theorem. This extension has two main parts: the use of von Neumann algebras as a base category (Section 2); and the relation of our generalized valuations to (i) the assignment to quantities of intervals of real numbers, and (ii) the idea of a subobject of the coarse-graining presheaf (Section 3).

arXiv:quant-ph/9911020Nov 5, 199917 citations

Logical Reflections on the Kochen-Specker Theorem

John L. Bell

Springer eBooksJan 1, 19967 citations

A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations

Christopher John Isham, Jeremy Butterfield

The Kochen-Specker theorem 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 truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint 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 generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.The Kochen-Specker theorem 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 truth-values assigned to propositions are (i) contextual; and (ii) multi-valued, where the space of contexts and the multi-valued logic for each context come naturally from the topos theory of presheaves. The first step in our theory is to demonstrate that the Kochen-Specker theorem is equivalent to the statement that a certain presheaf defined on the category of self-adjoint 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 generalized valuation in quantum theory whose values are sieves of operators. In particular, we show how each quantum state leads to such a generalized valuation.

arXiv:quant-ph/9803055Mar 20, 199827 citations

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

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