Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 108, 28 pages      arXiv:1404.0876

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

Vincent X. Genest and Luc Vinet
Centre de Recherches Mathématiques, Université de Montréal, P.O. Box 6128, Centre-ville Station, Montréal, QC, Canada, H3C 3J7

Received August 15, 2014, in final form November 25, 2014; Published online December 05, 2014

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.

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

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  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, math.QA/9912142.
  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.
  3. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
  4. Arfken G.B., Weber H.J., Mathematical methods for physicists, 5th ed., Harcourt/Academic Press, Burlington, MA, 2001.
  5. Bonzom V., Fleury P., Asymptotics of Wigner $3nj$-symbols with small and large angular momenta: an elementary method, J. Phys. A: Math. Theor. 45 (2012), 075202, 20 pages, arXiv:1108.1569.
  6. Diaconis P., Griffiths R., An introduction to multivariate Krawtchouk polynomials and their applications, J. Statist. Plann. Inference 154 (2014), 39-53, arXiv:1309.0112.
  7. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  8. Genest V.X., Vinet L., Zhedanov A., The multivariate Krawtchouk polynomials as matrix elements of the rotation group representations on oscillator states, J. Phys. A: Math. Gen. 46 (2013), 505203, 24 pages, arXiv:1306.4256.
  9. Genest V.X., Vinet L., Zhedanov A., Superintegrability in two dimensions and the Racah-Wilson algebra, Lett. Math. Phys. 104 (2014), 931-952, arXiv:1307.5539.
  10. Granovskiǐ Ya.I., Zhednov A.S., New construction of $3nj$-symbols, J. Phys. A: Math. Gen. 26 (1993), 4339-4344.
  11. Haggard H.M., Littlejohn R.G., Asymptotics of the Wigner $9j$-symbol, Classical Quantum Gravity 27 (2010), 135010, 17 pages, arXiv:0912.5384.
  12. Hoare M.R., Rahman M., A probabilistic origin for a new class of bivariate polynomials, SIGMA 4 (2008), 089, 18 pages, arXiv:0812.3879.
  13. Kalnins E.G., Kress J.M., Miller Jr. W., Second order superintegrable systems in conformally flat spaces. IV. The classical 3D Stäckel transform and 3D classification theory, J. Math. Phys. 47 (2006), 043514, 26 pages.
  14. Kalnins E.G., Miller Jr. W., Post S., Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere, SIGMA 7 (2011), 051, 26 pages, arXiv:1010.3032.
  15. Kalnins E.G., Miller Jr. W., Tratnik M.V., Families of orthogonal and biorthogonal polynomials on the $N$-sphere, SIAM J. Math. Anal. 22 (1991), 272-294.
  16. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  17. Koelink H.T., Van der Jeugt J., Convolutions for orthogonal polynomials from Lie and quantum algebra representations, SIAM J. Math. Anal. 29 (1998), 794-822, q-alg/9607010.
  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.
  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.
  20. Miller Jr. W., Lie theory and special functions, Mathematics in Science and Engineering, Vol. 43, Academic Press, New York - London, 1968.
  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.
  22. Regge T., Williams R.M., Discrete structures in gravity, J. Math. Phys. 41 (2000), 3964-3984, gr-qc/0012035.
  23. Rosengren H., On the triple sum formula for Wigner $9j$-symbols, J. Math. Phys. 39 (1998), 6730-6744.
  24. Rosengren H., Another proof of the triple sum formula for Wigner $9j$-symbols, J. Math. Phys. 40 (1999), 6689-6691.
  25. Rosengren H., Multivariable orthogonal polynomials and coupling coefficients for discrete series representations, SIAM J. Math. Anal. 30 (1999), 232-272.
  26. Rudzikas Z., Theoretical atomic spectroscopy, Cambridge Monographs on Atomic Molecular and Chemical Physics, Vol. 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.
  28. Suhonen J., From nucleons to nucleus. Concepts of microscopic nuclear theory, Theoretical and Mathematical Physics, Springer, Berlin, 2007.
  29. Tratnik M.V., Some multivariable orthogonal polynomials of the Askey tableau-discrete families, J. Math. Phys. 32 (1991), 2337-2342.
  30. Van der Jeugt J., Coupling coefficients for Lie algebra representations and addition formulas for special functions, J. Math. Phys. 38 (1997), 2728-2740.
  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.
  32. Van der Jeugt J., $3nj$-coefficients and orthogonal polynomials of hypergeometric type, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Lecture Notes in Math., Vol. 1817, Editors E. Koelink, W. Van Assche, Springer, Berlin, 2003, 25-92.
  33. Vilenkin N.Ja., Klimyk A.U., Representation of Lie groups and special functions, Mathematics and its Applications, Vol. 316, Kluwer Academic Publishers Group, Dordrecht, 1995.
  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 pages, arXiv:1104.1499.

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