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A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations | C.J. Isham
; J. Butterfield
; | Date: |
20 Mar 1998 | Subject: | quant-ph gr-qc | Abstract: | 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. | Source: | arXiv, quant-ph/9803055 | Services: | Forum | Review | PDF | Favorites |
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