dorsal/arxiv
View SchemaClassical Topology and Quantum States
| Authors | A. P. Balachandran |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0002055 |
| URL | https://arxiv.org/abs/quant-ph/0002055 |
| DOI | 10.1007/s12043-001-0120-y |
| Journal | Pramana 56:223-237,2001 |
Abstract
Any two infinite-dimensional (separable) Hilbert spaces are unitarily isomorphic. The sets of all their self-adjoint operators are also therefore unitarily equivalent. Thus if all self-adjoint operators can be observed, and if there is no further major axiom in quantum physics than those formulated for example in Dirac's `Quantum Mechanics', then a quantum physicist would not be able to tell a torus from a hole in the ground. We argue that there are indeed such axioms involving observables with smooth time evolution: they contain commutative subalgebras from which the spatial slice of spacetime with its topology (and with further refinements of the axiom, its $C^K-$ and $C^\infty-$ structures) can be reconstructed using Gel'fand - Naimark theory and its extensions. Classical topology is an attribute of only certain quantum observables for these axioms, the spatial slice emergent from quantum physics getting progressively less differentiable with increasingly higher excitations of energy and eventually altogether ceasing to exist. After formulating these axioms, we apply them to show the possibility of topology change and to discuss quantized fuzzy topologies. Fundamental issues concerning the role of time in quantum physics are also addressed.
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"abstract": "Any two infinite-dimensional (separable) Hilbert spaces are unitarily\nisomorphic. The sets of all their self-adjoint operators are also therefore\nunitarily equivalent. Thus if all self-adjoint operators can be observed, and\nif there is no further major axiom in quantum physics than those formulated for\nexample in Dirac\u0027s `Quantum Mechanics\u0027, then a quantum physicist would not be\nable to tell a torus from a hole in the ground. We argue that there are indeed\nsuch axioms involving observables with smooth time evolution: they contain\ncommutative subalgebras from which the spatial slice of spacetime with its\ntopology (and with further refinements of the axiom, its $C^K-$ and $C^\\infty-$\nstructures) can be reconstructed using Gel\u0027fand - Naimark theory and its\nextensions. Classical topology is an attribute of only certain quantum\nobservables for these axioms, the spatial slice emergent from quantum physics\ngetting progressively less differentiable with increasingly higher excitations\nof energy and eventually altogether ceasing to exist. After formulating these\naxioms, we apply them to show the possibility of topology change and to discuss\nquantized fuzzy topologies. Fundamental issues concerning the role of time in\nquantum physics are also addressed.",
"arxiv_id": "quant-ph/0002055",
"authors": [
"A. P. Balachandran"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"doi": "10.1007/s12043-001-0120-y",
"journal_ref": "Pramana 56:223-237,2001",
"title": "Classical Topology and Quantum States",
"url": "https://arxiv.org/abs/quant-ph/0002055"
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