dorsal/arxiv
View SchemaFeatures of Moyal Trajectories
| Authors | Nuno Costa Dias, Joao Nuno Prata |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0604167 |
| URL | https://arxiv.org/abs/quant-ph/0604167 |
| DOI | 10.1063/1.2409495 |
| Journal | J. Math. Phys. 48, 012109 (2007). |
Abstract
We study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of starfunction and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formulation to study the semiclassical expansion of Moyal trajectories, which is cast in terms of a (order by order in $\hbar$) recursive hierarchy of i) first order partial differential equations as well as ii) systems of first order ordinary differential equations. The latter formulation is derived independently for analytic Hamiltonians as well as for the more general case of smooth local integrable ones. We present various examples illustrating these results.
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"abstract": "We study the Moyal evolution of the canonical position and momentum\nvariables. We compare it with the classical evolution and show that, contrary\nto what is commonly found in the literature, the two dynamics do not coincide.\nWe prove that this divergence is quite general by studying Hamiltonians of the\nform $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are\nthen suggested. We introduce the concept of starfunction and use it to\nreformulate the Moyal equations in terms of a system of ordinary differential\nequations on the noncommutative Moyal plane. We then use this formulation to\nstudy the semiclassical expansion of Moyal trajectories, which is cast in terms\nof a (order by order in $\\hbar$) recursive hierarchy of i) first order partial\ndifferential equations as well as ii) systems of first order ordinary\ndifferential equations. The latter formulation is derived independently for\nanalytic Hamiltonians as well as for the more general case of smooth local\nintegrable ones. We present various examples illustrating these results.",
"arxiv_id": "quant-ph/0604167",
"authors": [
"Nuno Costa Dias",
"Joao Nuno Prata"
],
"categories": [
"quant-ph"
],
"doi": "10.1063/1.2409495",
"journal_ref": "J. Math. Phys. 48, 012109 (2007).",
"title": "Features of Moyal Trajectories",
"url": "https://arxiv.org/abs/quant-ph/0604167"
},
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