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

TL;DR

This paper introduces a Toeplitz quantization framework for non-commutative algebras like $SU_q(2)$, enabling quantization without measures and incorporating Planck's constant, thus extending quantum theory to more complex algebraic structures.

Contribution

It develops a new Toeplitz quantization method for non-commutative symbol spaces, including quantum groups, without requiring measures and introduces a second quantization perspective.

Findings

Defined annihilation and creation operators as Toeplitz operators

Established commutation relations for these operators

Demonstrated the theory's applicability to $SU_q(2)$

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 $SU_q(2)$ 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.