dorsal/arxiv
View SchemaNotes on phase space quantization
| Authors | J. Kiukas, P. Lahti, K. Ylinen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0601009 |
| URL | https://arxiv.org/abs/quant-ph/0601009 |
| Journal | J. Math. Phys. 47, 072104 (2006) |
Abstract
We consider questions related to a quantization scheme in which a classical variable f:\Omega\to R on a phase space \Omega is associated with a semispectral measure E^f, such that the moment operators of E^f are required to be of the form \Gamma(f^k), with \Gamma a suitable mapping from the set of classical variables to the set of (not necessarily bounded) operators in some Hilbert space. In particular, we investigate the situation where the map \Gamma is implemented by the operator integral with respect to some fixed positive operator measure. The phase space \Omega is first taken to be an abstract measurable space, then a locally compact unimodular group, and finally R^2, where we determine explicitly the relevant operators \Gamma(f^k) for certain variables f, in the case where the quantization map \Gamma is implemented by a translation covariant positive operator measure. In addition, we consider the question under what conditions a positive operator measure is projection valued.
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"abstract": "We consider questions related to a quantization scheme in which a classical\nvariable f:\\Omega\\to R on a phase space \\Omega is associated with a\nsemispectral measure E^f, such that the moment operators of E^f are required to\nbe of the form \\Gamma(f^k), with \\Gamma a suitable mapping from the set of\nclassical variables to the set of (not necessarily bounded) operators in some\nHilbert space. In particular, we investigate the situation where the map \\Gamma\nis implemented by the operator integral with respect to some fixed positive\noperator measure. The phase space \\Omega is first taken to be an abstract\nmeasurable space, then a locally compact unimodular group, and finally R^2,\nwhere we determine explicitly the relevant operators \\Gamma(f^k) for certain\nvariables f, in the case where the quantization map \\Gamma is implemented by a\ntranslation covariant positive operator measure. In addition, we consider the\nquestion under what conditions a positive operator measure is projection\nvalued.",
"arxiv_id": "quant-ph/0601009",
"authors": [
"J. Kiukas",
"P. Lahti",
"K. Ylinen"
],
"categories": [
"quant-ph"
],
"journal_ref": "J. Math. Phys. 47, 072104 (2006)",
"title": "Notes on phase space quantization",
"url": "https://arxiv.org/abs/quant-ph/0601009"
},
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