| Title:
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Toeplitz Quantization for Non-commutating Symbol Spaces such as $SU_q(2)$ (English) |
| Author:
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Sontz, Stephen Bruce |
| Language:
|
English |
| Journal:
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Communications in Mathematics |
| ISSN:
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1804-1388 |
| Volume:
|
24 |
| Issue:
|
1 |
| Year:
|
2016 |
| Pages:
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43-69 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
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Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group $SU_q(2)$ is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck's constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum) objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck's constant and a Hilbert space where natural, densely defined operators act. (English) |
| Keyword:
|
Toeplitz quantization |
| Keyword:
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non-commutating symbols |
| Keyword:
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creation and annihilation operators |
| Keyword:
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canonical commutation relations |
| Keyword:
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anti-Wick quantization |
| Keyword:
|
second quantization of a quantum group |
| MSC:
|
47B35 |
| MSC:
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81S99 |
| idZBL:
|
Zbl 1369.47037 |
| idMR:
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MR3546806 |
| . |
| Date available:
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2016-08-26T11:21:15Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145805 |
| . |
| Reference:
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