dorsal/arxiv
View SchemaOn Quantizing $T^*S^1$
| Authors | Mark J. Gotay, Hendrik B. Grundling |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9609025 |
| URL | https://arxiv.org/abs/quant-ph/9609025 |
| DOI | 10.1016/S0034-4877(97)85622-4 |
| Journal | Rept.Math.Phys. 40 (1997) 107-123 |
Abstract
In this paper we continue our study of Groenewold-Van Hove obstructions to quantization. We show that there exists such an obstruction to quantizing the cylinder $T^*S^1.$ More precisely, we prove that there is no quantization of the Poisson algebra of $T^*S^1$ which is irreducible on a naturally defined $e(2) \times R$ subalgebra. Furthermore, we determine the maximal ``polynomial'' subalgebras that can be consistently quantized, and completely characterize the quantizations thereof. This example provides support for one of the conjectures in Gotay et al 1996, but disproves part of another. Passing to coverings, we also derive a no-go result for $R^2$ which is comparatively stronger than those originally found by Groenewold and Van Hove.
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"abstract": "In this paper we continue our study of Groenewold-Van Hove obstructions to\nquantization. We show that there exists such an obstruction to quantizing the\ncylinder $T^*S^1.$ More precisely, we prove that there is no quantization of\nthe Poisson algebra of $T^*S^1$ which is irreducible on a naturally defined\n$e(2) \\times R$ subalgebra. Furthermore, we determine the maximal\n``polynomial\u0027\u0027 subalgebras that can be consistently quantized, and completely\ncharacterize the quantizations thereof. This example provides support for one\nof the conjectures in Gotay et al 1996, but disproves part of another. Passing\nto coverings, we also derive a no-go result for $R^2$ which is comparatively\nstronger than those originally found by Groenewold and Van Hove.",
"arxiv_id": "quant-ph/9609025",
"authors": [
"Mark J. Gotay",
"Hendrik B. Grundling"
],
"categories": [
"quant-ph",
"dg-ga",
"hep-th",
"math.DG"
],
"doi": "10.1016/S0034-4877(97)85622-4",
"journal_ref": "Rept.Math.Phys. 40 (1997) 107-123",
"title": "On Quantizing $T^*S^1$",
"url": "https://arxiv.org/abs/quant-ph/9609025"
},
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