dorsal/arxiv
View SchemaBringing Up a Quantum Baby
| Authors | A. P. Balachandran |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9702055 |
| URL | https://arxiv.org/abs/quant-ph/9702055 |
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 vectors in the domain of the Hamiltonian: The ``probability densities'' (hermitean forms) \psi^\dagger \chi for \psi,\chi in this domain generate an algebra from which the classical configuration space with its topology (and with further refinements of the axiom, its C^K and C^infinity structures) can be reconstructed using Gel'fand - Naimark theory. Classical topology is an attribute of only certain quantum states for these axioms, the configuration space 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 vectors in the domain of the Hamiltonian: The\n``probability densities\u0027\u0027 (hermitean forms) \\psi^\\dagger \\chi for \\psi,\\chi in\nthis domain generate an algebra from which the classical configuration space\nwith its topology (and with further refinements of the axiom, its C^K and\nC^infinity structures) can be reconstructed using Gel\u0027fand - Naimark theory.\nClassical topology is an attribute of only certain quantum states for these\naxioms, the configuration space emergent from quantum physics getting\nprogressively less differentiable with increasingly higher excitations of\nenergy 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/9702055",
"authors": [
"A. P. Balachandran"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th",
"math.QA",
"q-alg"
],
"title": "Bringing Up a Quantum Baby",
"url": "https://arxiv.org/abs/quant-ph/9702055"
},
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