dorsal/arxiv
View SchemaQuantum Quandaries: a Category-Theoretic Perspective
| Authors | John C. Baez |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0404040 |
| URL | https://arxiv.org/abs/quant-ph/0404040 |
| Journal | In Structural Foundations of Quantum Gravity, eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press, Oxford, 2006, pp. 240-265 |
Abstract
General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose objects are (n-1)-dimensional manifolds representing "space" and whose morphisms are n-dimensional cobordisms representing "spacetime". Quantum theory makes heavy use of the category Hilb, whose objects are Hilbert spaces used to describe "states", and whose morphisms are bounded linear operators used to describe "processes". Moreover, the categories nCob and Hilb resemble each other far more than either resembles Set, the category whose objects are sets and whose morphisms are functions. In particular, both Hilb and nCob but not Set are *-categories with a noncartesian monoidal structure. We show how this accounts for many of the famously puzzling features of quantum theory: the failure of local realism, the impossibility of duplicating quantum information, and so on. We argue that these features only seem puzzling when we try to treat Hilb as analogous to Set rather than nCob, so that quantum theory will make more sense when regarded as part of a theory of spacetime.
{
"annotation_id": "d2d380b9-fb81-4103-b66f-c4ad62bef526",
"date_created": "2026-03-02T18:02:06.339000Z",
"date_modified": "2026-03-02T18:02:06.339000Z",
"file_hash": "00ee9e929e840d1a67663fbf5cd5ebcb70121d6669fbb88d8c312cd451f85ab3",
"private": false,
"record": {
"abstract": "General relativity may seem very different from quantum theory, but work on\nquantum gravity has revealed a deep analogy between the two. General relativity\nmakes heavy use of the category nCob, whose objects are (n-1)-dimensional\nmanifolds representing \"space\" and whose morphisms are n-dimensional cobordisms\nrepresenting \"spacetime\". Quantum theory makes heavy use of the category Hilb,\nwhose objects are Hilbert spaces used to describe \"states\", and whose morphisms\nare bounded linear operators used to describe \"processes\". Moreover, the\ncategories nCob and Hilb resemble each other far more than either resembles\nSet, the category whose objects are sets and whose morphisms are functions. In\nparticular, both Hilb and nCob but not Set are *-categories with a noncartesian\nmonoidal structure. We show how this accounts for many of the famously puzzling\nfeatures of quantum theory: the failure of local realism, the impossibility of\nduplicating quantum information, and so on. We argue that these features only\nseem puzzling when we try to treat Hilb as analogous to Set rather than nCob,\nso that quantum theory will make more sense when regarded as part of a theory\nof spacetime.",
"arxiv_id": "quant-ph/0404040",
"authors": [
"John C. Baez"
],
"categories": [
"quant-ph",
"gr-qc",
"math.QA"
],
"journal_ref": "In Structural Foundations of Quantum Gravity, eds. Steven French,\n Dean Rickles and Juha Saatsi, Oxford U. Press, Oxford, 2006, pp. 240-265",
"title": "Quantum Quandaries: a Category-Theoretic Perspective",
"url": "https://arxiv.org/abs/quant-ph/0404040"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "88ae24ce-7922-4a08-9dca-578af2f254c8",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}