dorsal/arxiv
View SchemaSymplectic quantization, inequivalent quantum theories, and Heisenberg's principle of uncertainty
| Authors | Merced Montesinos, G. F. Torres del Castillo |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0407051 |
| URL | https://arxiv.org/abs/quant-ph/0407051 |
| DOI | 10.1103/PhysRevA.70.032104 |
| Journal | Phys.Rev. A70 (2004) 032104 |
Abstract
We analyze the quantum dynamics of the non-relativistic two-dimensional isotropic harmonic oscillator in Heisenberg's picture. Such a system is taken as toy model to analyze some of the various quantum theories that can be built from the application of Dirac's quantization rule to the various symplectic structures recently reported for this classical system. It is pointed out that that these quantum theories are inequivalent in the sense that the mean values for the operators (observables) associated with the same physical classical observable do not agree with each other. The inequivalence does not arise from ambiguities in the ordering of operators but from the fact of having several symplectic structures defined with respect to the same set of coordinates. It is also shown that the uncertainty relations between the fundamental observables depend on the particular quantum theory chosen. It is important to emphasize that these (somehow paradoxical) results emerge from the combination of two paradigms: Dirac's quantization rule and the usual Copenhagen interpretation of quantum mechanics.
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"abstract": "We analyze the quantum dynamics of the non-relativistic two-dimensional\nisotropic harmonic oscillator in Heisenberg\u0027s picture. Such a system is taken\nas toy model to analyze some of the various quantum theories that can be built\nfrom the application of Dirac\u0027s quantization rule to the various symplectic\nstructures recently reported for this classical system. It is pointed out that\nthat these quantum theories are inequivalent in the sense that the mean values\nfor the operators (observables) associated with the same physical classical\nobservable do not agree with each other. The inequivalence does not arise from\nambiguities in the ordering of operators but from the fact of having several\nsymplectic structures defined with respect to the same set of coordinates. It\nis also shown that the uncertainty relations between the fundamental\nobservables depend on the particular quantum theory chosen. It is important to\nemphasize that these (somehow paradoxical) results emerge from the combination\nof two paradigms: Dirac\u0027s quantization rule and the usual Copenhagen\ninterpretation of quantum mechanics.",
"arxiv_id": "quant-ph/0407051",
"authors": [
"Merced Montesinos",
"G. F. Torres del Castillo"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"doi": "10.1103/PhysRevA.70.032104",
"journal_ref": "Phys.Rev. A70 (2004) 032104",
"title": "Symplectic quantization, inequivalent quantum theories, and Heisenberg\u0027s principle of uncertainty",
"url": "https://arxiv.org/abs/quant-ph/0407051"
},
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