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Journal Articles Communications in Mathematics Year : 2016

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


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.
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hal-03664922 , version 1 (11-05-2022)


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Stephen Bruce Sontz. Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2). Communications in Mathematics, 2016, Volume 24 (2016), Issue 1 (1), pp.43 - 69. ⟨10.1515/cm-2016-0005⟩. ⟨hal-03664922⟩


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