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Communications in Mathematics

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

Stephen Bruce Sontz
Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/cm-2016-0005


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 SUq(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.

Keywords: Toeplitz quantization; non-commutating symbols; creation and annihilation operators; canonical commutation relations; anti-Wick quantization; second quantization of a quantum group


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About the article

Received: 2016-07-18

Accepted: 2016-07-30

Published Online: 2016-08-20

Published in Print: 2016-08-01

Citation Information: Communications in Mathematics, Volume 24, Issue 1, Pages 43–69, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0005.

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© 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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