dorsal/arxiv
View SchemaIntegrals over Products of Distributions and Coordinate Independence of Zero-Temperature Path Integrals
| Authors | H. Kleinert, A. Chervyakov |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0204067 |
| URL | https://arxiv.org/abs/quant-ph/0204067 |
| DOI | 10.1016/S0375-9601(02)01801-7 |
| Journal | Phys.Lett. A308 (2003) 85-95 |
Abstract
In perturbative calculations of quantum-statistical zero-temperature path integrals in curvilinear coordinates one encounters Feynman diagrams involving multiple temporal integrals over products of distributions, which are mathematically undefined. In addition, there are terms proportional to powers of Dirac delta-functions at the origin coming from the measure of path integration. We give simple rules for integrating products of distributions in such a way that the results ensure coordinate independence of the path integrals. The rules are derived by using equations of motion and partial integration, while keeping track of certain minimal features originating in the unique definition of all singular integrals in $1 - \epsilon$ dimensions. Our rules yield the same results as the much more cumbersome calculations in 1- epsilon dimensions where the limit epsilon --> 0 is taken at the end. They also agree with the rules found in an independent treatment on a finite time interval.
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"abstract": "In perturbative calculations of quantum-statistical zero-temperature path\nintegrals in curvilinear coordinates one encounters Feynman diagrams involving\nmultiple temporal integrals over products of distributions, which are\nmathematically undefined. In addition, there are terms proportional to powers\nof Dirac delta-functions at the origin coming from the measure of path\nintegration. We give simple rules for integrating products of distributions in\nsuch a way that the results ensure coordinate independence of the path\nintegrals. The rules are derived by using equations of motion and partial\nintegration, while keeping track of certain minimal features originating in the\nunique definition of all singular integrals in $1 - \\epsilon$ dimensions. Our\nrules yield the same results as the much more cumbersome calculations in 1-\nepsilon dimensions where the limit epsilon --\u003e 0 is taken at the end. They also\nagree with the rules found in an independent treatment on a finite time\ninterval.",
"arxiv_id": "quant-ph/0204067",
"authors": [
"H. Kleinert",
"A. Chervyakov"
],
"categories": [
"quant-ph",
"hep-th"
],
"doi": "10.1016/S0375-9601(02)01801-7",
"journal_ref": "Phys.Lett. A308 (2003) 85-95",
"title": "Integrals over Products of Distributions and Coordinate Independence of Zero-Temperature Path Integrals",
"url": "https://arxiv.org/abs/quant-ph/0204067"
},
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