Sample records for 3j-symbols

  1. 3j Symbols: To Normalize or Not to Normalize? (United States)

    van Veenendaal, Michel


    The systematic use of alternative normalization constants for 3j symbols can lead to a more natural expression of quantities, such as vector products and spherical tensor operators. The redefined coupling constants directly equate tensor products to the inner and outer products without any additional square roots. The approach is extended to…

  2. The screen representation of vector coupling coefficients or Wigner 3j symbols: exact computation and illustration of the asymptotic behavior

    CERN Document Server

    Bitencourt, Ana Carla P; Littlejohn, Robert G; Anderson, Roger; Aquilanti, Vincenzo


    The Wigner $3j$ symbols of the quantum angular momentum theory are related to the vector coupling or Clebsch-Gordan coefficients and to the Hahn and dual Hahn polynomials of the discrete orthogonal hyperspherical family, of use in discretization approximations. We point out the important role of the Regge symmetries for defining the screen where images of the coefficients are projected, and for discussing their asymptotic properties and semiclassical behavior. Recursion relationships are formulated as eigenvalue equations, and exploited both for computational purposes and for physical interpretations.

  3. Boundary conditions in conformal and integrable theories

    CERN Document Server

    Petkova, V B


    The study of boundary conditions in rational conformal field theories is not only physically important. It also reveals a lot on the structure of the theory ``in the bulk''. The same graphs classify both the torus and the cylinder partition functions and provide data on their hidden ``quantum symmetry''. The Ocneanu triangular cells -- the 3j-symbols of these symmetries, admit various interpretations and make a link between different problems.

  4. General formulae for the su{sub q}(2) 6-j symbols simply obtained thanks to trivial q-identities

    Energy Technology Data Exchange (ETDEWEB)

    Brehamet, L. [Centre d`Etudes Nucleaires de Saclay, Gif-sur-Yvette (France)


    The analytical formulae for the su{sub q}(2) 6-j symbols are easily obtained, without the use of any su{sub q}(2)3-j symbol formula. With respect to the checking up on two compatible sets of trivial identities, already successful in the su(2) case , the process becomes still simpler because each set reduces into a single q-numbers identity.

  5. The many faces of Ocneanu cells

    CERN Document Server

    Petkova, V B


    We define generalised chiral vertex operators covariant under the Ocneanu ``double triangle algebra'' {\\cal A}, a novel quantum symmetry intrinsic to a given rational 2-d conformal field theory. This provides a chiral approach, which, unlike the conventional one, makes explicit various algebraic structures encountered previously in the study of these theories and of the associated critical lattice models, and thus allows their unified treatment. The triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\\cal A}, reappear in several guises. With {\\cal A} and its dual algebra {hat A} one associates a pair of graphs, G and {\\tilde G}. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs {\\tilde G} classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining {\\hat A}. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the f...

  6. The Quantum Group Structure of 2D Gravity and Minimal Models; 2, The Genus-Zero Chiral Bootstrap

    CERN Document Server

    Cremmer, E; Roussel, J F; Gervais, Jean-Loup


    The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with L...

  7. Sound scattering by bubble clouds near the sea surface (United States)

    Gaunaurd; Huang


    The classical exact formulation required to evaluate the form function (or the scattering cross-section, SCS) of a single, ideal, air bubble in a boundless liquid is briefly recalled. It is then immediately generalized to the case of a round cloud of many possibly interacting such bubbles of known volume concentration, contained within the same boundless medium. This is further generalized to the case when the bubble cloud is near a free surface. The presence of the nearby pressure release surface, assumed flat, substantially alters the cloud's scattering cross-section relative to its value in the absence of boundaries. We then use an earlier technique of ours [i.e., see I.E.E.E. J. Ocean. Eng. 20, 285-293 (1995)] based on the method of images that uses the addition theorem for the spherical wave functions, to relate all the scattered sound fields to a common origin and thus obtain the (modified) SCS of the cloud now near the boundary. This formulation accounts for all orders of multiple scattering and yields an infinite set of coupled algebraic equations for the coupling coefficients. This set is then solved for the coupling coefficients in terms of infinite sums of products of pairs of Wigner 3-j symbols, which are then used to construct and evaluate the form function. We display numerical results in four cases that correspond to geographical sites in which the bubble concentrations within the cloud have been measured along a couple of oblique upward directions, or have been assumed to have increasing (and in a few instances, purposely unrealistically high) values. In all cases considered here the bubble clouds are only a few meters beneath the sea surface and consist of ideal bubbles. The results are also compared to those found in the absence of a boundary in all the cases considered.

  8. DIRAC: A new version of computer algebra tools for studying the properties and behavior of hydrogen-like ions (United States)

    McConnell, Sean; Fritzsche, Stephan; Surzhykov, Andrey


    restricted to the non-relativistic framework only. DiracGreensIntegralRadial[] - Evaluates the two-dimensional radial integrals with the wave- and Green's functions both in non-relativistic and relativistic frameworks. DiracAngularMatrixElement[] - Calculates the angular matrix elements for various irreducible tensor operators. The elimination of some redundant procedures. In particular, the previous version supported evaluation of the spherical Bessel functions, Wigner 3j symbols, Clebsch-Gordan coefficients and spherical harmonics functions. These tools are now superseded by in-built procedures of Mathematica. The development of a full featured interactive help system which follows the style of the Mathematica Help Pages. Extensive revision of the source code in order to correct a number of bugs and inconsistencies that have been identified during use of the previous version of Dirac. The DIRAC package is distributed as a compressed tar file from which the DIRAC root directory can be (re-)generated. The root directory contains the source code and help libraries, a "Readme" file, Dirac_Installation_Instructions, as well as the notebook DemonstrationNotebook.nb that includes a number of test cases to illustrate the use of the program. These test cases, which concern the theoretical analysis of wavefunctions and the fine-structure of hydrogen-like ions, has already been discussed in detail in Ref. [1] and are provided here in order to underline the continuity between the previous (Maple) and new (Mathematica) versions of the DIRAC program. Unusual features: Even though all basic features of the previous Maple version have been retained in as close to the original form as possible, some small syntax changes became necessary in the new version of DIRAC in order to follow Mathematica standards. First of all, these changes concern naming conventions for DIRAC's procedures. As was discussed in Ref. [1], previously rather long names were employed in which each word was separated by