dorsal/arxiv
View SchemaGroup Theoretical Quantization and the Example of a Phase Space S^1 x R^+
| Authors | Martin Bojowald, Thomas Strobl |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9908079 |
| URL | https://arxiv.org/abs/quant-ph/9908079 |
| DOI | 10.1063/1.533258 |
| Journal | J.Math.Phys. 41 (2000) 2537-2567 |
Abstract
The group theoretical quantization scheme is reconsidered by means of elementary systems. Already the quantization of a particle on a circle shows that the standard procedure has to be supplemented by an additional condition on the admissibility of group actions. A systematic strategy for finding admissible group actions for particular subbundles of cotangent spaces is developed, two-dimensional prototypes of which are T^*R^+ and S^1 x R^+ (interpreted as restrictions of T^*R and T^*S^1 to positive coordinate and momentum, respectively). In this framework (and under an additional, natural condition) an SO_+(1,2)-action on S^1 x R^+ results as the unique admissible group action. For symplectic manifolds which are (specific) parts of phase spaces with known quantum theory a simple projection method of quantization is formulated. For T^*R^+ and S^1 x R^+ equivalent results to those of more established (but more involved) quantization schemes are obtained. The approach may be of interest, e.g., in attempts to quantize gravity theories where demanding nondegenerate metrics of a fixed signature imposes similar constraints.
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"abstract": "The group theoretical quantization scheme is reconsidered by means of\nelementary systems. Already the quantization of a particle on a circle shows\nthat the standard procedure has to be supplemented by an additional condition\non the admissibility of group actions. A systematic strategy for finding\nadmissible group actions for particular subbundles of cotangent spaces is\ndeveloped, two-dimensional prototypes of which are T^*R^+ and S^1 x R^+\n(interpreted as restrictions of T^*R and T^*S^1 to positive coordinate and\nmomentum, respectively). In this framework (and under an additional, natural\ncondition) an SO_+(1,2)-action on S^1 x R^+ results as the unique admissible\ngroup action.\n For symplectic manifolds which are (specific) parts of phase spaces with\nknown quantum theory a simple projection method of quantization is formulated.\nFor T^*R^+ and S^1 x R^+ equivalent results to those of more established (but\nmore involved) quantization schemes are obtained. The approach may be of\ninterest, e.g., in attempts to quantize gravity theories where demanding\nnondegenerate metrics of a fixed signature imposes similar constraints.",
"arxiv_id": "quant-ph/9908079",
"authors": [
"Martin Bojowald",
"Thomas Strobl"
],
"categories": [
"quant-ph",
"gr-qc",
"hep-th"
],
"doi": "10.1063/1.533258",
"journal_ref": "J.Math.Phys. 41 (2000) 2537-2567",
"title": "Group Theoretical Quantization and the Example of a Phase Space S^1 x R^+",
"url": "https://arxiv.org/abs/quant-ph/9908079"
},
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