Toeplitz Quantization for Non-commutating Symbol Spaces such as S U q ( 2 )

Stephen Bruce Sontz

Communications in Mathematics (2016)

  • Volume: 24, Issue: 1, page 43-69
  • ISSN: 1804-1388

Abstract

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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 S U 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.

How to cite

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Sontz, Stephen Bruce. "Toeplitz Quantization for Non-commutating Symbol Spaces such as $SU_q(2)$." Communications in Mathematics 24.1 (2016): 43-69. <http://eudml.org/doc/286710>.

@article{Sontz2016,
abstract = {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.},
author = {Sontz, Stephen Bruce},
journal = {Communications in Mathematics},
keywords = {Toeplitz quantization; non-commutating symbols; creation and annihilation operators; canonical commutation relations; anti-Wick quantization; second quantization of a quantum group},
language = {eng},
number = {1},
pages = {43-69},
publisher = {University of Ostrava},
title = {Toeplitz Quantization for Non-commutating Symbol Spaces such as $SU_q(2)$},
url = {http://eudml.org/doc/286710},
volume = {24},
year = {2016},
}

TY - JOUR
AU - Sontz, Stephen Bruce
TI - Toeplitz Quantization for Non-commutating Symbol Spaces such as $SU_q(2)$
JO - Communications in Mathematics
PY - 2016
PB - University of Ostrava
VL - 24
IS - 1
SP - 43
EP - 69
AB - 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.
LA - eng
KW - Toeplitz quantization; non-commutating symbols; creation and annihilation operators; canonical commutation relations; anti-Wick quantization; second quantization of a quantum group
UR - http://eudml.org/doc/286710
ER -

References

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