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The Screen Representation of Spin Networks: 2D Recurrence, Eigenvalue Equation for 6j Symbols, Geometric Interpretation and Hamiltonian Dynamics

  • Roger W. Anderson
  • Vincenzo Aquilanti
  • Ana Carla Peixoto Bitencourt
  • Dimitri Marinelli
  • Mirco Ragni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7972)

Abstract

This paper treats 6j symbols or their orthonormal forms as a function of two variables spanning a square manifold which we call the “screen”. We show that this approach gives important and interesting insight. This two dimensional perspective provides the most natural extension to exhibit the role of these discrete functions as matrix elements that appear at the very foundation of the modern theory of classical discrete orthogonal polynomials. Here we present 2D and 1D recursion relations that are useful for the direct computation of the orthonormal 6j, which we name U. We present a convention for the order of the arguments of the 6j that is based on their classical and Regge symmetries, and a detailed investigation of new geometrical aspects of the 6j symbols. Specifically we compare the geometric recursion analysis of Schulten and Gordon with the methods of this paper. The 1D recursion relation, written as a matrix diagonalization problem, permits an interpretation as a discrete Shrödinger-like equations and an asymptotic analysis illustrates semiclassical and classical limits in terms of Hamiltonian evolution.

Keywords

Angular Momentum Semiclassical Limit Semiclassical Approximation Spin Network Regge Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bitencourt, A.C.P., Marzuoli, A., Ragni, M., Anderson, R.W., Aquilanti, V.: Exact and asymptotic computations of elementary spin networks: Classification of the quantum–classical boundaries. In: Murgante, B., Gervasi, O., Misra, S., Nedjah, N., Rocha, A.M.A.C., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2012, Part I. LNCS, vol. 7333, pp. 723–737. Springer, Heidelberg (2012); See arXiv:1211.4993[math-ph]CrossRefGoogle Scholar
  2. 2.
    Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevanjee, N., Littlejohn, R.G., Yu, L.: Semiclassical mechanics of the Wigner 6j-symbol. J. Phys. A 45(6), 065209 (2012)Google Scholar
  3. 3.
    Ragni, M., Bitencourt, A.C., Aquilanti, V., Anderson, R.W., Littlejohn, R.G.: Exact computation and asymptotic approximations of 6j symbols: Illustration of their semiclassical limits. Int. J. Quantum Chem. 110(3), 731–742 (2010)CrossRefGoogle Scholar
  4. 4.
    Littlejohn, R.G., Mitchell, K.A., Reinsch, M., Aquilanti, V., Cavalli, S.: Internal spaces, kinematic rotations, and body frames for four-atom systems. Phys. Rev. A 58, 3718–3738 (1998)CrossRefGoogle Scholar
  5. 5.
    Aquilanti, V., Bitencourt, A.C.P., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity. Phys. Scripta 78(5), 058103 (2008)Google Scholar
  6. 6.
    Aquilanti, V., Bitencourt, A., da S. Ferreira, C., Marzuoli, A., Ragni, M.: Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications. Theor. Chem. Acc. 123, 237–247 (2009)Google Scholar
  7. 7.
    Aquilanti, V., Capecchi, G.: Harmonic analysis and discrete polynomials. From semiclassical angular momentum theory to the hyperquantization algorithm. Theor. Chem. Accounts 104, 183–188 (2000)CrossRefGoogle Scholar
  8. 8.
    Aquilanti, V., Marinelli, D., Marzuoli, A.: Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials. J. Phys. A: Math. Theor. 46, 175303 (2013), arXiv:1301.1949v2 [math-ph]Google Scholar
  9. 9.
    Ragni, M., Littlejohn, R.G., Bitencourt, A.C.P., Aquilanti, V., Anderson, R.W.: The screen representation of spin networks. Images of 6j symbols and semiclassical features. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Quang, N.H., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013, Part II. LNCS, vol. 7972, pp. 60–72. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Varshalovich, D., Moskalev, A., Khersonskii, V.: Quantum Theory of Angular Momentum. World Scientific, Singapore (1988)Google Scholar
  11. 11.
    Freidel, L., Louapre, D.: Asymptotics of 6j and 10j symbols. Classical and Quantum Gravity 20, 1267–1294 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Aquilanti, V., Haggard, H.M., Hedeman, A., Jeevangee, N., Littlejohn, R., Yu, L.: Semiclassical mechanics of the Wigner 6j-symbol. J. Phys. A 45(065209) (2012), arXiv:1009.2811v2 [math-ph]Google Scholar
  13. 13.
    Anderson, R.W., Aquilanti, V.: The discrete representation correspondence between quantum and classical spatial distributions of angular momentum vectors. J. Chem. Phys. 124, 214104 (9 pages) (2006)Google Scholar
  14. 14.
    Anderson, R.W., Aquilanti, V., Marzuoli, A.: 3nj morphogenesis and semiclassical disentangling. J. Phys. Chem. A 113, 15106–15117 (2009)CrossRefGoogle Scholar
  15. 15.
    Anderson, R., Aquilanti, V., da S. Ferreira, C.: Exact computation and large angular momentum asymptotics of 3nj symbols: semiclassical disentangling of spin-networks. J. Chem. Phys. 129, 161101 (5 pages) (2008)Google Scholar
  16. 16.
    Neville, D.: A technique for solving recurrence relations approximately and its application to the 3 − j and 6 − j symbols. J. Math. Phys. 12, 2438 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Schulten, K., Gordon, R.: Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta. J. Math. Phys. 16, 1971–1988 (1975)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Braun, P.A.: Discrete semiclassical methods in the theory of Rydberg atoms in external fields. Rev. Mod. Phys. 65, 115–161 (1993)CrossRefGoogle Scholar
  19. 19.
    Braun, P.: WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator. Sov. Phys. Theor. Math. Phys. 16, 1070–1081 (1978)CrossRefGoogle Scholar
  20. 20.
    Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Bloch, F., et al. (eds.) Spectroscopic and Group Theoretical Methods in Physics, pp. 1–58. North-Holland, Amsterdam (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Roger W. Anderson
    • 1
  • Vincenzo Aquilanti
    • 2
    • 3
  • Ana Carla Peixoto Bitencourt
    • 4
  • Dimitri Marinelli
    • 5
    • 6
  • Mirco Ragni
    • 4
  1. 1.Department of ChemistryUniversity of CaliforniaSanta CruzU.S.A.
  2. 2.Dipartimento di ChimicaUniversità di PerugiaItaly
  3. 3.Istituto Metodologie Inorganiche e Plasmi CNRRomaItaly
  4. 4.Departamento de FísicaUniversidade Estadual de Feira de SantanaBrazil
  5. 5.Dipartimento di FisicaUniversità degli Studi di PaviaItaly
  6. 6.INFN, sezione di PaviaPaviaItaly

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