1. Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2)
- Author
-
Stephen Bruce Sontz
- Subjects
Pure mathematics ,General Mathematics ,01 natural sciences ,symbols.namesake ,second quantization of a quantum group ,0103 physical sciences ,anti-Wick quantization ,0101 mathematics ,[MATH]Mathematics [math] ,Quantum ,Commutative property ,Mathematics ,creation and annihilation operators ,010308 nuclear & particles physics ,Quantum group ,lcsh:Mathematics ,Quantization (signal processing) ,010102 general mathematics ,Toeplitz quantization ,Hilbert space ,Creation and annihilation operators ,lcsh:QA1-939 ,non-commutating symbols ,Second quantization ,Toeplitz matrix ,symbols ,canonical commutation relations - 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 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.
- Published
- 2016