dorsal/arxiv
View SchemaQuantum States from Tangent Vectors
| Authors | J. M. Isidro |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0407159 |
| URL | https://arxiv.org/abs/quant-ph/0407159 |
| DOI | 10.1142/S0217732304015634 |
| Journal | Mod.Phys.Lett. A19 (2004) 2339-2352 |
Abstract
We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces C, by explicitly constructing Hilbert-space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors to C. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic(C), the Picard group of C. In this way Pic(C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when C is complex projective space.
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"abstract": "We argue that tangent vectors to classical phase space give rise to quantum\nstates of the corresponding quantum mechanics. This is established for the case\nof complex, finite-dimensional, compact, classical phase spaces C, by\nexplicitly constructing Hilbert-space vector bundles over C. We find that these\nvector bundles split as the direct sum of two holomorphic vector bundles: the\nholomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum\nstates (except the vacuum) appear as tangent vectors to C. The vacuum state\nappears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are\nclassified by the elements of Pic(C), the Picard group of C. In this way Pic(C)\nappears as the parameter space for nonequivalent vacua. Our analysis is\nmodelled on, but not limited to, the case when C is complex projective space.",
"arxiv_id": "quant-ph/0407159",
"authors": [
"J. M. Isidro"
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"doi": "10.1142/S0217732304015634",
"journal_ref": "Mod.Phys.Lett. A19 (2004) 2339-2352",
"title": "Quantum States from Tangent Vectors",
"url": "https://arxiv.org/abs/quant-ph/0407159"
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