Jeremy Butterfield

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

AbstractAny 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

Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity

\Delta \subseteq \mathbb{R}

A topos perspective on the kochen-Specker theorem: III. Von Neumann algebras as the base category

, 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.

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

In a previous paper, we proposed assigning asthe value of a physical quantity in quantum theory acertain kind of set (a sieve) of quantities that arefunctions of the given quantity. The motivation was in part physical — such a valuationilluminates the Kochen–Specker theorem — andin part mathematical — the valuation arisesnaturally in the topos theory of presheaves. This paperdiscusses the conceptual aspects of this proposal. We also undertake two othertasks. First, we explain how the proposed valuationscould arise much more generally than just in quantumphysics; in particular, they arise as naturally in classical physics. Second, we give anothermotivation for such valuations (that applies equally toclassical and quantum physics). This arises fromapplying to propositions about the values of physical quantities some general axioms governingpartial truth for any kind of proposition.

Against Pointillisme about Mechanics

We discuss some ways in which topos theory (a branch of category theory) can be applied to interpretative problems in quantum theory and quantum gravity. In Sec.1, we introduce these problems. In Sec.2, we introduce topos theory, especially the idea of a topos of presheaves. In Sec.3, we discuss several possible applications of topos theory to the problems in Sec.1. In Sec.4, we draw some conclusions.

Stochastic Einstein Locality Revisited

This is a collection of eleven original essays in analytical philosophy by British and American philosophers, centering on the connection between mind and language. Two themes predominate: how it is that thoughts and sentences can represent the world; and what having a thought - a belief, for instance - involves. Developing from these themes are the questions: what does having a belief require of the believer, and of the way he or she relates to the environment? In particular, does having a belief require speaking a language? The volume concludes the informal series stemming from the meetings sponsored by the Thyssen Foundation.

The Oxford Questions on the foundations of quantum physics

We extend the topos-theoretic treatment given in previous papers of assigningvalues to quantities in quantum theory, and of related issues such as theKochen–Specker theorem. This extension has two main parts: the use of vonNeumann algebras as a base category and the relation of our generalized valuationsto (i) the assignment to quantities of intervals of real numbers and (ii) the ideaof a subobject of the coarse-graining presheaf.

Emergence and Reduction Combined in Phase Transitions

AbstractWe 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