dorsal/arxiv
View SchemaA Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations
| Authors | J. Butterfield, C. J. Isham |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0107123 |
| URL | https://arxiv.org/abs/quant-ph/0107123 |
| Journal | Int.J.Theor.Phys. 41 (2002) 613-639 |
Abstract
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 $\Delta \subseteq \mathR$. 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).
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"abstract": "We extend the topos-theoretic treatment given in previous papers of assigning\nvalues to quantities in quantum theory. In those papers, the main idea was to\nassign a sieve as a partial and contextual truth value to a proposition that\nthe value of a quantity lies in a certain set $\\Delta \\subseteq \\mathR$. Here\nwe relate such sieve-valued valuations to valuations that assign to quantities\nsubsets, rather than single elements, of their spectra (we call these\n`interval\u0027 valuations). There are two main results. First, there is a natural\ncorrespondence between these two kinds of valuation, which uses the notion of a\nstate\u0027s support for a quantity (Section 3). Second, if one starts with a more\ngeneral notion of interval valuation, one sees that our interval valuations\nbased on the notion of support (and correspondingly, our sieve-valued\nvaluations) are a simple way to secure certain natural properties of\nvaluations, such as monotonicity (Section 4).",
"arxiv_id": "quant-ph/0107123",
"authors": [
"J. Butterfield",
"C. J. Isham"
],
"categories": [
"quant-ph",
"gr-qc"
],
"journal_ref": "Int.J.Theor.Phys. 41 (2002) 613-639",
"title": "A Topos Perspective on the Kochen-Specker Theorem: IV. Interval Valuations",
"url": "https://arxiv.org/abs/quant-ph/0107123"
},
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