The Generic Superintegrable System on the $3$-Sphere and the ${9j}$ Symbols of ${\mathfrak{su}(1,1)}$

Vincent X. Genest ; Luc Vinet

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

The $9j$ symbols of $\mathfrak{su}(1,1)$ are studied within the framework of the generic superintegrable system on the 3-sphere. The canonical bases corresponding to the binary coupling schemes of four $\mathfrak{su}(1,1)$ representations are constructed explicitly in terms of Jacobi polynomials and are seen to correspond to the separation of variables in different cylindrical coordinate systems. A triple integral expression for the $9j$ coefficients exhibiting their symmetries is derived. A double integral formula is obtained by extending the model to the complex three-sphere and taking the complex radius to zero. The explicit expression for the vacuum coefficients is given. Raising and lowering operators are constructed and are used to recover the relations between contiguous coefficients. It is seen that the $9j$ symbols can be expressed as the product of the vacuum coefficients and a rational function. The recurrence relations and the difference equations satisfied by the $9j$ coefficients are derived.

Détail

Keywords: $\mathfrak{su}(1,1)$ algebra; $9j$ symbols; superintegrable systems.

Bibliographie

[1] Ališauskas S., “The triple sum formulas for {$9j$} coefficients of {${\rm SU}(2)$} and {${\rm u}_q(2)$}”, J. Math. Phys., 41 (2000), 7589–7610, arXiv: math.QA/9912142 | DOI | MR | Zbl

[2] Ališauskas S. J., Jucys A. P., “Weight lowering operators and the multiplicity-free isoscalar factors for the group {$R_{5}$}”, J. Math. Phys., 12 (1971), 594–605 | DOI | MR | Zbl

[3] Andrews G. E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[4] Arfken G. B., Weber H. J., Mathematical methods for physicists, 5th ed., Harcourt/Academic Press, Burlington, MA, 2001 | MR

[5] Bonzom V., Fleury P., “Asymptotics of {W}igner {$3nj$}-symbols with small and large angular momenta: an elementary method”, J. Phys. A: Math. Theor., 45 (2012), 075202, 20 pp., arXiv: 1108.1569 | DOI | MR | Zbl

[6] Diaconis P., Griffiths R., “An introduction to multivariate {K}rawtchouk polynomials and their applications”, J. Statist. Plann. Inference, 154 (2014), 39–53, arXiv: 1309.0112 | DOI | MR | Zbl

[7] Dunkl C. F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, 81, Cambridge University Press, Cambridge, 2001 | DOI | MR | Zbl

[8] Genest V. X., Vinet L., Zhedanov A., “The multivariate {K}rawtchouk polynomials as matrix elements of the rotation group representations on oscillator states”, J. Phys. A: Math. Gen., 46 (2013), 505203, 24 pp., arXiv: 1306.4256 | DOI | MR | Zbl

[9] Genest V. X., Vinet L., Zhedanov A., “Superintegrability in two dimensions and the {R}acah–{W}ilson algebra”, Lett. Math. Phys., 104 (2014), 931–952, arXiv: 1307.5539 | DOI | MR | Zbl

[10] Granovskiĭ Ya. I., Zhednov A. S., “New construction of {$3nj$}-symbols”, J. Phys. A: Math. Gen., 26 (1993), 4339–4344 | DOI | MR | Zbl

[11] Haggard H. M., Littlejohn R. G., “Asymptotics of the {W}igner {$9j$}-symbol”, Classical Quantum Gravity, 27 (2010), 135010, 17 pp., arXiv: 0912.5384 | DOI | MR | Zbl

[12] Hoare M. R., Rahman M., “A probabilistic origin for a new class of bivariate polynomials”, SIGMA, 4 (2008), 089, 18 pp., arXiv: 0812.3879 | DOI | MR | Zbl

[13] Kalnins E. G., Kress J. M., Miller W. (Jr.), “Second order superintegrable systems in conformally flat spaces. {IV}: {T}he classical 3{D} {S}täckel transform and 3{D} classification theory”, J. Math. Phys., 47 (2006), 043514, 26 pp. | DOI | MR | Zbl

[14] Kalnins E. G., Miller W. (Jr.), Post S., “Two-variable {W}ilson polynomials and the generic superintegrable system on the 3-sphere”, SIGMA, 7 (2011), 051, 26 pp., arXiv: 1010.3032 | DOI | MR | Zbl

[15] Kalnins E. G., Miller W. (Jr.), Tratnik M. V., “Families of orthogonal and biorthogonal polynomials on the {$N$}-sphere”, SIAM J. Math. Anal., 22 (1991), 272–294 | DOI | MR | Zbl

[16] Koekoek R., Lesky P. A., Swarttouw R. F., Hypergeometric orthogonal polynomials and their {$q$}-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010 | DOI | MR | Zbl

[17] Koelink H. T., Van der Jeugt J., “Convolutions for orthogonal polynomials from {L}ie and quantum algebra representations”, SIAM J. Math. Anal., 29 (1998), 794–822, arXiv: q-alg/9607010 | DOI | MR | Zbl

[18] Lievens S., Van der Jeugt J., “{$3nj$}-coefficients of {${\rm su}(1,1)$} as connection coefficients between orthogonal polynomials in {$n$} variables”, J. Math. Phys., 43 (2002), 3824–3849 | DOI | MR | Zbl

[19] Lievens S., Van der Jeugt J., “Realizations of coupled vectors in the tensor product of representations of {$\mathfrak{su}(1,1)$} and {$\mathfrak{su}(2)$}”, J. Comput. Appl. Math., 160 (2003), 191–208 | DOI | MR | Zbl

[20] Miller W. (Jr.), Lie theory and special functions, Mathematics in Science and Engineering, 43, Academic Press, New York–London, 1968 | MR | Zbl

[21] Nikiforov A. F., Suslov S. K., Uvarov V. B., Classical orthogonal polynomials of a discrete variable, Springer Series in Computational Physics, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[22] Regge T., Williams R. M., “Discrete structures in gravity”, J. Math. Phys., 41 (2000), 3964–3984, arXiv: gr-qc/0012035 | DOI | MR | Zbl

[23] Rosengren H., “On the triple sum formula for {W}igner {$9j$}-symbols”, J. Math. Phys., 39 (1998), 6730–6744 | DOI | MR | Zbl

[24] Rosengren H., “Another proof of the triple sum formula for {W}igner {$9j$}-symbols”, J. Math. Phys., 40 (1999), 6689–6691 | DOI | MR | Zbl

[25] Rosengren H., “Multivariable orthogonal polynomials and coupling coefficients for discrete series representations”, SIAM J. Math. Anal., 30 (1999), 232–272 | DOI | MR

[26] Rudzikas Z., Theoretical atomic spectroscopy, Cambridge Monographs on Atomic Molecular and Chemical Physics, 7, Cambridge University Press, Cambridge, 2007

[27] Srinivasa Rao K., Rajeswari V., “A note on the triple sum series for the {$9j$} coefficient”, J. Math. Phys., 30 (1989), 1016–1017 | DOI | MR | Zbl

[28] Suhonen J., From nucleons to nucleus. Concepts of microscopic nuclear theory, Theoretical and Mathematical Physics, Springer, Berlin, 2007 | DOI | MR | Zbl

[29] Tratnik M. V., “Some multivariable orthogonal polynomials of the {A}skey tableau-discrete families”, J. Math. Phys., 32 (1991), 2337–2342 | DOI | MR | Zbl

[30] Van der Jeugt J., “Coupling coefficients for {L}ie algebra representations and addition formulas for special functions”, J. Math. Phys., 38 (1997), 2728–2740 | DOI | MR | Zbl

[31] Van der Jeugt J., “Hypergeometric series related to the {$9$}-{$j$} coefficient of {${\mathfrak su}(1,1)$}”, J. Comput. Appl. Math., 118 (2000), 337–351 | DOI | MR | Zbl

[32] Van der Jeugt J., “{$3nj$}-coefficients and orthogonal polynomials of hypergeometric type”, Orthogonal Polynomials and Special Functions ({L}euven, 2002), Lecture Notes in Math., 1817, eds. E. Koelink, W. Van Assche, Springer, Berlin, 2003, 25–92 | DOI | MR | Zbl

[33] Vilenkin N. Ja., Klimyk A. U., Representation of {L}ie groups and special functions, Mathematics and its Applications, 316, Kluwer Academic Publishers Group, Dordrecht, 1995 | DOI | MR

[34] Yu L., Littlejohn R. G., “Semiclassical analysis of the Wigner $9j$ symbol with small and large angular momenta”, Phys. Rev. A, 83 (2011), 052114, 14 pp., arXiv: 1104.1499 | DOI