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

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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.

[1] S.T. Ali, M. Englis: Berezin-Toeplitz quantization over matrix domains. In: Contributions in Mathematical Physics: A Tribute to Gerard G. Emch, Eds. S.T. Ali and K.B. Sinha. Hindustan Book Agency, New Delhi, India (2007).

[2] S.T. Ali, M. Englis: Matrix-valued Berezin-Toeplitz quantization. J. Math. Phys. 48 (5) (2007) 053504, (14 pages). arXiv: math-ph/0611082

[3] V. Bargmann: On a Hilbert space of analytic functions and its associated integral transform, Part I. Commun. Pure Appl. Math. 14 (3) (1961) 187-214.

[4] M. El Baz, R. Fresneda, J.-P. Gazeau, Y. Hassouni: Coherent state quantization of paragrassmann algebras. J. Phys. A: Math. Theor. 43 (38) (2010) 385202 (15pp). Also see the Erratum for this article in arXiv:1004.4706v3

[5] F.A. Berezin: General Concept of Quantization. Commun. Math. Phys. 40 (1975) 153-174.

[6] C.A. Berger, L.A. Coburn: Toeplitz operators and quantum mechanics. J. Funct. Anal. 68 (1986) 273-299.

[7] C.A. Berger, L.A. Coburn: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301 (1987) 813-829.

[8] D. Borthwick, S. Klimek, A. Lesniewski, M. Rinaldi: Matrix Cartan superdomains, super Toeplitz operators, and quantization. J. Funct. Anal. 127 (1995) 456-510. arXiv: hep-th/9406050

[9] A. Böttcher and B. Silbermann: Analysis of Toeplitz Operators. Springer (2006).

[10] J.-P. Gazeau: Coherent States in Quantum Physics. Wiley-VCH (2009).

[11] B.C. Hall: Holomorphic methods in analysis and mathematical physics, First Summer School in Analysis and Mathematical Physics, Eds. S. Pérez-Esteva and C. Villegas-Blas. In: Contemp. Math.. Am. Math. Soc. (2000) 1-59.

[12] C. Iuliu-Lazaroiu, D. McNamee, C. Sämann: Generalized Berezin-Toeplitz quantization of Kähler supermanifolds. J. High Energy Phys. 2009 (05) (2009). 055, arXiv: 0811.4743v2

[13] A. Yu. Karlovich: Higher order asymptotic formulas for Toeplitz matrices with symbols in generalized Hölder spaces. In: Operator Algebras, Operator Theory and Applications, Eds. Maria Amélia Bastos et al. Birkhäuser (2008) 207-228. arXiv: 0705.0432

[14] A. Yu. Karlovich: Asymptotics of Toeplitz Matrices with Symbols in Some Generalized Krein Algebras. In: Modern Anal. Appl. Springer (2009) 341-359. arXiv: 0803.3767

[15] R. Kerr: Products of Toeplitz Operators on a Vector Valued Bergman Space. Integral Equations Operator Theory 66 (3) (2010) 571-584. arXiv:0804.4234

[16] E.H. Lieb: The classical limit of quantum spin systems. Commun. Math. Phys. 31 (4) (1973) 327-340.

[17] Yu.I. Manin: Topics in Noncommutative Geometry. Princeton University Press (1991).

[18] R.A. Martínez-Avendaño and P. Rosenthal: An Introduction to Operators on the Hardy-Hilbert space. Springer (2007).

[19] M. Reed and B. Simon: Mathematical Methods of Modern Physics, Vol. I: Functional Analysis. Academic Press (1972).

[20] M. Reed and B. Simon: Mathematical Methods of Modern Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic Press (1975).

[21] S.B. Sontz: A Reproducing Kernel and Toeplitz Operators in the Quantum Plane. Communications in Mathematics 21 (2) (2013) 137-160. arXiv:1305.6986

[22] S.B. Sontz: Paragrassmann Algebras as Quantum Spaces, Part I: Reproducing Kernels. In: Geometric Methods in Physics. XXXI Workshop 2012. Trends in Mathematics, Eds. P. Kielanowski et al.. Birkhäuser (2013) 47-63.

[23] S.B. Sontz: Toeplitz Quantization without Measure or Inner Product. In: Geometric Methods in Physics. XXXII Workshop 2013. Trends in Mathematics. (2014) 57-66.

[24] S.B. Sontz: Paragrassmann Algebras as Quantum Spaces, Part II: Toeplitz Operators. Journal of Operator Theory 71 (2014) 411-426. arXiv:1205.5493, doi: http://dx.doi.org/10.7900/jot.2012may24.1969

[25] T. Timmermann: An invitation to quantum groups and duality: From Hopf algebras to multiplicative unitaries and beyond. Euro. Math. Soc. (2008).

Journal Information

CiteScore 2017: 0.33

SCImago Journal Rank (SJR) 2017: 0.128
Source Normalized Impact per Paper (SNIP) 2017: 0.476

Mathematical Citation Quotient (MCQ) 2017: 0.43

Target Group

researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory

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