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

cm:9451 - Communications in Mathematics, August 20, 2016, Volume 24 (2016), Issue 1 - https://doi.org/10.1515/cm-2016-0005
Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2)Article

Authors: Stephen Bruce Sontz 1

  • 1 Centro de Investigación en Matemáticas

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.


Volume: Volume 24 (2016), Issue 1
Published on: August 20, 2016
Imported on: May 11, 2022
Keywords: [MATH]Mathematics [math]

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