Abstract
We study some q-analogs of Racah polynomials and some of their applications in the theory of representation of quantum algebras. Possible implementations in quantum optics are discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. A. Malkin and V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).
J Guerrero, V. I. Man’ko, G. Marmo, and A. Simoni, “Geometrical aspects of the Lie group representation and their optical applications,” J. Russ. Laser Res., 23, 49–80 (2002).
B. Gruber, Yu. F. Smirnov, and Yu. I. Kharitonov, “Quantum algebra U q(gl(3)) and nonlinear optics,” J. Russ. Laser Res., 24, 56–68 (2003).
L. D. Landau and E. M. Lifshits, Quantum Mechanics. Nonrelativistic Theory [in Russian], Fizmatlit, Moscow (2002).
R. Askey and R. Wilson, “A set of orthogonal polynomials that generalize Racah coefficients or 6j symbols,” SIAM J. Math. Anal., 10, 1008–1020 (1979).
R. Koekoek and R. F. Swarttouw, “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue,” Reports of the Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft (1998), No. 98-17.
R. Askey and R. Wilson, “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials,” Mem. Amer. Math. Soc., Providence, Rhode Island, 319 (1985).
A. F. Nikiforov and V. B. Uvarov, “Classical orthogonal polynomials in a discrete variable on nonuniform lattices,” Preprint No. 17 of the M. V. Keldysh Institute of Applied Mathematics [in Russian], Moscow (1983).
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable. Springer Series in Computational Physics, Springer Verlag, Berlin (1991) [Russian Edition, Nauka, Moscow (1985)].
V.G. Drinfel’d, “Quantum groups,” in: Proceedings of the International Congress of Mathematicians (Berkeley 1986), American Mathematical Society, Providence, R. I. (1987), pp. 798–820.
L.D. Faddev and L. A. Takhtajan, Lect. Notes Phys., 246, 183 (1986).
M. Jimbo, “Quantum R matrix for the generalized Toda system,” Commun. Math. Phys., 102, 537–547 (1986).
P. P. Kulish and N. Yu. Reshetikhin, Zapiski Nauchnykh Seminarov LOMI [in Russian], 101 (1981)
E. K. Sklyanin, “Some algebraic structures connected with the Yang-Baxter equation,” Funct. Anal. Appl., 16, 263–270 (1983).
A. N. Kirillov and N. Yu. Reshetikhin, LOMI Preprint E-9-88, Leningrad (1988).
R. Álvarez-Nodarse, “q-Analog of the vibrational IBM and the quantum algebra SU q(1, 1)” [in Russian], Master’s Thesis, M. V. Lomonosov Moscow State University (November 1992).
R. Álvarez-Nodarse and Yu. F. Smirnov, “q-Dual Hahn polynomials on the nonuniform lattice x(s) = [s]q[s + 1]q and the q-algebras SU q(1, 1) and SU q(2),” J. Phys. A: Math. Gen., 29, 1435–1451 (1996).
A. A. Malashin, “q-Analog of Racah polynomials on the lattice x(s) = [s]q[s+1]q and its connections with 6j-symbols for the SU q(2) and SU q(1, 1) quantum algebras” [in Russian], Master’s Thesis, M. V. Lomonosov Moscow State University (January 1992).
H. Rosengren, “An elementary approach to 6j-symbols (classical, quantum, rational and elliptic),” Los Alamos ArXiv math.CA/0312310 (2003).
Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, “Method of projection operators and the q-analog of the quantum theory of angular momentum. Clebsch-Gordan coefficients and irreducible tensor operators,” Sov. J. Nucl. Phys., 53, 593–605 (1991).
Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, “The projection-operator method and the q-analog of the quantum theory of angular momentum. Racah coefficients, 3j-and 6j-symbols, and their symmetry properties,” Sov. J. Nucl. Phys., 53, 1069–1086 (1991).
Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, “Tree technique and irreducible tensor operators for the SU q(2) quantum algebra. 9j-Symbols,” Sov. J. Nucl. Phys., 55, 1599–1604 (1992).
Yu. F. Smirnov, V. N. Tolstoy, and Yu. I. Kharitonov, “The tree technique and irreducible tensor operators for the SU q(2) quantum algebra. The algebra of irreducible tensor operators,” Phys. Atom. Nucl., 56, 690–700 (1993).
H. T. Koelink, “Askey-Wilson polynomials and the quantum SU(2) group: Survey and applications,” Acta Appl. Math., 44, 295–352 (1996).
N. Ja. Vilenkin and A. U. Klimyk, Representations of Lie Groups and Special Functions, Kluwer Academic Publishers, Dordrecht (1992), Vols. II, III.
R. Álvarez-Nodarse, Polinomios Hipergemétricos y q-Polinomios [in Spanish], Monografías del Seminario García Galdeano, Universidad de Zaragoza, Prensas Universitarias de Zaragoza, Zaragoza, Spain (2003), Vol. 26.
N. M. Atakishiyev, M. Rahman, and S. K. Suslov, “On classical orthogonal polynomials,” Constr. Approx., 11, 181–226 (1995).
A. F. Nikiforov and V. B. Uvarov, “Polynomial solutions of hypergeometric-type difference equations and their classification,” Integral Transform. Spec. Funct., 1, 223–249 (1993).
A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable [in Russian], Nauka, Moscow (1985).
R. Álvarez-Nodarse, “Polinomios generalizados y q-polinomios: propiedades espectrales y aplicaciones” [in Spanish], Tesis Doctoral, Universidad Carlos III de Madrid, Madrid (1996).
G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press (1990).
P. M. Asherova, Yu. F. Smirnov, and V. N. Tolstoi, “Weyl q-coefficients for u q(3) and Racah q-coefficients for su q(2),” Sov. J. Nucl. Phys., 58, 1859–1872 (1996).
A. A. Malashin, Yu. F. Smirnov, and Yu.I. Kharitonov, Sov. J. Nucl. Phys., 53, 1105–1119 (1995).
Author information
Authors and Affiliations
Additional information
Manuscript submitted by the authors in English November 25, 2005.
Rights and permissions
About this article
Cite this article
Álvarez-Nodarse, R., Smirnov, Y.F. & Costas-Santos, R.S. A q-analog of Racah polynomials and q-algebra SU q(2) in quantum optics. J Russ Laser Res 27, 1–32 (2006). https://doi.org/10.1007/s10946-006-0001-4
Issue Date:
DOI: https://doi.org/10.1007/s10946-006-0001-4