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

Toeplitz Quantization for Non-commutating Symbol Spaces such as SU q (2) 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematics de Gruyter

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

Sontz, Stephen Bruce
Communications in Mathematics , Volume 24 (1) – Aug 1, 2016
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References (36)

Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
2336-1298
eISSN
2336-1298
DOI
10.1515/cm-2016-0005
Publisher site
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Abstract

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.

Journal

Communications in Mathematics – de Gruyter

Published: Aug 1, 2016

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