dorsal/arxiv
View SchemaA topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
| Authors | C. J. Isham, J. Butterfield |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9803055 |
| URL | https://arxiv.org/abs/quant-ph/9803055 |
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.
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"abstract": "The Kochen-Specker theorem asserts the impossibility of assigning values to\nquantum quantities in a way that preserves functional relations between them.\nWe construct a new type of valuation which is defined on all operators, and\nwhich respects an appropriate version of the functional composition principle.\nThe truth-values assigned to propositions are (i) contextual; and (ii)\nmulti-valued, where the space of contexts and the multi-valued logic for each\ncontext come naturally from the topos theory of presheaves.\n The first step in our theory is to demonstrate that the Kochen-Specker\ntheorem is equivalent to the statement that a certain presheaf defined on the\ncategory of self-adjoint operators has no global elements. We then show how the\nuse of ideas drawn from the theory of presheaves leads to the definition of a\ngeneralized valuation in quantum theory whose values are sieves of operators.\nIn particular, we show how each quantum state leads to such a generalized\nvaluation.",
"arxiv_id": "quant-ph/9803055",
"authors": [
"C. J. Isham",
"J. Butterfield"
],
"categories": [
"quant-ph",
"gr-qc"
],
"title": "A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations",
"url": "https://arxiv.org/abs/quant-ph/9803055"
},
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