Toeplitz Quantization for Non-commutating Symbol Spaces such as 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.Keywords: Toeplitz quantization, non-commutating symbols, creation and annihilation operators, canonical commutation relations, anti-Wick quantization, second quantization of a quantum group MSC 2010: 47B35, 81S99 1 IntroductionThe history of Toeplitz operators covers a bit over one hundred years and includes many major works, far too numerous to mention here. For a recent reference that will give the reader some first links to that extensive literature, see Section 3.5 in [18]. Speaking for myself, the papers [6], [7] and [11] have been rather influential. But the study of Toeplitz operators with symbols coming from a non-commutative algebra seems to be limited mostly to cases where the algebra is a matrix algebra or is some other quite specific noncommutative algebra such as in two recent works, [22] and [23], of the author. The papers [22] and [23] can be considered as two rather elaborated examples of the theory presented here. That study is continued in this paper, but in a much more general setting intended to clarify the mathematical structures at play in those two examples. A new example, the quantum group SU q (2) as symbol space, will also be presented here.The paper is organized as follows. After presenting the foundations of this theory in the next section, we define and analyze in Section 3 the Toeplitz quantization. In particular, Toeplitz operators are defined as densely defined operators acting in a quantum Hilbert space. The symbols of these Toeplitz operators come from a possibly non-commutative algebra A, which in physics terminology serves as the phase space for the theory. The common domain of these Toeplitz operators is P, a pre-Hilbert space and sub-a...

Co-Toeplitz operators and their associated quantization

“…Note that R n is indeed a non-zero classical relation. Based on what is true in the Toeplitz setting as is presented in [13], I conjecture that both of the cases R n ∈ R and R n / ∈ R can occur. The intuition here is that the terms R 0 , R 1 , .…”

“…for φ, ψ ∈ P and g ∈ A. This translates directly into T g * ⊂ (T g ) * , an inclusion of densely defined operators acing in H. For more details, including examples, see [13].…”

“…The upshot is that Planck's constant will be introduced into the theory and semi-classical algebras as well as a dequantized (or classical) algebra will be defined. To make this paper more selfcontained we review how the relevant material of [13] applies in the co-Toeplitz setting.…”

“…However, it seems difficult to delve into all this in greater detail at the present abstract level, though these considerations can be brought to bear on specific examples. The reader can consult [13] for more details, including motivation, for the topics of this section.…”

“…We define P := alg{a, c}, the sub-algebra (but not sub- * -algebra) of SU q (2) generated by a and c. This is a sub-algebra of 'holomorphic' elements. This is the same sub-algebra that was used in [13] for a Toeplitz quantization of SU q (2). We can identify P as the free algebra generated by a and c, modulo the relation ac = q ca, and so (as an algebra) P is the complex Manin quantum plane, which is denoted by A 2|0 q in [7].…”