Toeplitz Quantization without Measure or Inner Product

Sontz, Stephen Bruce

doi:10.1007/978-3-319-06248-8_5

Cited by 2 publications

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References 7 publications

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“…In a series of recent papers the author has introduced a theory of Toeplitz operators having symbols in a not necessarily commutative algebra with a * -operation (also called a conjugation). See [11] for the general theory and [8], [9] and [10] for various examples of that theory. The associated Toeplitz quantization is also described in those papers.…”

Section: Introductionmentioning

confidence: 99%

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Co-Toeplitz operators and their associated quantization

Sontz

2020

Adv. Oper. Theory

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We define co-Toeplitz operators, a new class of Hilbert space operators, in order to define a co-Toeplitz quantization scheme that is dual to the Toeplitz quantization scheme introduced by the author in the setting of symbols that come from a possibly non-commutative algebra with unit. In the present dual setting the symbols come from a possibly non-co-commutative co-algebra with co-unit. However, this co-Toeplitz quantization is a usual quantization scheme in the sense that to each symbol we assign a densely defined linear operator acting in a fixed Hilbert space. Creation and annihilation operators are also introduced as certain types of co-Toeplitz operators, and then their commutation relations provide the way for introducing Planck's constant into this theory. The domain of the co-Toeplitz quantization is then extended as well to a set of co-symbols, which are the linear functionals defined on the co-algebra. A detailed example based on the quantum group (and hence co-algebra) SU q (2) as symbol space is presented.

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Section: Introductionmentioning

confidence: 99%

“…There are at least three aspects of the theory in [11] that make it relevant to quantum physics. First, the Toeplitz operators are densely defined linear operators, all acting in the same Hilbert space, and so the self-adjoint extensions of the symmetric Toeplitz operators can be interpreted as being physical observables.…”

Section: Introductionmentioning

confidence: 99%

Co-Toeplitz operators and their associated quantization

Sontz

2020

Adv. Oper. Theory

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“…The most important structure on A is strangely enough the conjugation operation. Even the multiplication on A (as an algebra) is not a critically important structure and can be dispensed with as is done in [24]. …”

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“…This map would have to satisfy some other conditions as well to make the theory work out. We will not go into further details about this more general approach, which is discussed in [24]. We next wish to use the kernel K to extend the identity map on P to a projection map P K : A → A.…”

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confidence: 99%

“…As a further step in the theory, Planck's constant is introduced into the quantum side of this theory via the canonical commutation relations that certain Toeplitz operators (those of creation and annihilation) satisfy. The bridge between the classical side of the theory and the quantum side is provided by P. Two algebraic structures, namely the multiplication and inner product for A, are auxiliary to this basic outline and can be modified without changing the basic theory as indicated in [24].…”

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

Sontz

2016

Communications in Mathematics

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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...

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Toeplitz Quantization without Measure or Inner Product

Cited by 2 publications

References 7 publications

Co-Toeplitz operators and their associated quantization

Co-Toeplitz operators and their associated quantization

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

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Toeplitz Quantization without Measure or Inner Product

Cited by 2 publications

References 7 publications

Co-Toeplitz operators and their associated quantization

Co-Toeplitz operators and their associated quantization

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

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