2013/08/25 by Stephen Bruce Sontz, Sontz, Stephen Bruce · 1 citation
Mathematics · Physics and Astronomy · #47B35 #81S99 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Quantum Physics (quant-ph) #math-ph #math.MP #math.OA #msc:47B35 #msc:81S99 #quant-ph
paper · pdf · doi:10.48550/arxiv.1308.5454
35 pages, a new title and a new example using the quantum group $SU_q(2)$
openalex publication_date 2013/08/25 · arxiv created 2016/04/28 · arxiv updated 2016/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.