dorsal/arxiv
View SchemaExact Evolution Operator on Non-compact Group Manifolds
| Authors | Nurit Krausz, M. S. Marinov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9709050 |
| URL | https://arxiv.org/abs/quant-ph/9709050 |
| DOI | 10.1063/1.533401 |
| Journal | J.Math.Phys. 41 (2000) 5180-5208 |
Abstract
Free quantal motion on group manifolds is considered. The Hamiltonian is given by the Laplace -- Beltrami operator on the group manifold, and the purpose is to get the (Feynman's) evolution kernel. The spectral expansion, which produced a series of the representation characters for the evolution kernel in the compact case, does not exist for non-compact group, where the spectrum is not bounded. In this work real analytical groups are investigated, some of which are of interest for physics. An integral representation for the evolution operator is obtained in terms of the Green function, i.e. the solution to the Helmholz equation on the group manifold. The alternative series expressions for the evolution operator are reconstructed from the same integral representation, the spectral expansion (when exists) and the sum over classical paths. For non-compact groups, the latter can be interpreted as the (exact) semi-classical approximation, like in the compact case. The explicit form of the evolution operator is obtained for a number of non-compact groups.
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"abstract": "Free quantal motion on group manifolds is considered. The Hamiltonian is\ngiven by the Laplace -- Beltrami operator on the group manifold, and the\npurpose is to get the (Feynman\u0027s) evolution kernel. The spectral expansion,\nwhich produced a series of the representation characters for the evolution\nkernel in the compact case, does not exist for non-compact group, where the\nspectrum is not bounded. In this work real analytical groups are investigated,\nsome of which are of interest for physics. An integral representation for the\nevolution operator is obtained in terms of the Green function, i.e. the\nsolution to the Helmholz equation on the group manifold. The alternative series\nexpressions for the evolution operator are reconstructed from the same integral\nrepresentation, the spectral expansion (when exists) and the sum over classical\npaths. For non-compact groups, the latter can be interpreted as the (exact)\nsemi-classical approximation, like in the compact case. The explicit form of\nthe evolution operator is obtained for a number of non-compact groups.",
"arxiv_id": "quant-ph/9709050",
"authors": [
"Nurit Krausz",
"M. S. Marinov"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"doi": "10.1063/1.533401",
"journal_ref": "J.Math.Phys. 41 (2000) 5180-5208",
"title": "Exact Evolution Operator on Non-compact Group Manifolds",
"url": "https://arxiv.org/abs/quant-ph/9709050"
},
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