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Sample records for 3j-symbols

  1. 3j Symbols: To Normalize or Not to Normalize?

    van Veenendaal, Michel

    2011-01-01

    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

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

    2014-01-01

    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. 3j-symbol for the modular double SLq(2, R) revisited

    Modular double of quantum group SLq(2, R) with |q| = 1 has a series of selfadjoint irreducible representations πs parameterized by s element of R+. Ponsot and Teschner in [Comm. Math. Phys. 224 (2001) 613] considered a decomposition of the tensor product πs1 πs2 into irreducibles. In our paper we give more detailed derivation and some new proofs.

  4. Coherent 3j-symbol representation for the loop quantum gravity intertwiner space

    Alesci, Emanuele; Mäkinen, Ilkka

    2016-01-01

    We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.

  5. New information and entropic inequalities for Clebsch-Gordan coefficients

    Chernega, V. N.; Manko, O. V.; Manko, V. I.; Seilov, Z.

    2016-01-01

    The Clebsch-Gordan coefficients of the group SU(2) are shown to satisfy new inequalities. They are obtained using the properties of Shannon and Tsallis entropies. The inequalities associated with the Wigner 3-j symbols are obtained using the relation of Clebsch-Gordan coefficients with probability distributions interpreted either as distributions for composite systems or distributions for noncomposite systems. The new inequalities were found for Hahn polynomials and hypergeometric functions

  6. Calculations of angular momentum coupling coefficients on a computer code

    In this study, Clebsch-Gordan coefficients, 3j symbols, Racah coefficients, Wigner's 6j and 9j symbols were calculated by a prepared computer code of COEFF. The computer program COEFF is described which calculates angular momentum coupling coefficients and expresses them as quotient of two integers multiplied by the square root of the quotient of two integers. The program includes subroutines to encode an integer into its prime factors, to decode of prime factors back into an integer , to perform basic arithmetic operations on prime-coded numbers, as well as subroutines which calculate the coupling coefficients themselves. The computer code COEFF had been prepared to run on a VAX. In this study we rearranged the code to run on PC and tested it successfully. The obtained values in this study, were compared with the values of other computer programmes. A pretty good agreement is obtained between our prepared computer code and other computer programmes

  7. Exact computation of the 3-j and 6-j symbols

    A simple FORTRAN program for the exaxt computation of 3-j and 6-j symbols has been written for the VAX with VMS version v5.1 in our university's computing center. It goes beyond and contains all of the 3-j and 6-j symbols evaluated in the book by M. Rotenberg, R. Bivins, N. Metropolis and J.K. Wooten Jr. The 3-j symbols up to (30/m1 30/m2 30/m3) and 6-j symbols up to {20/20 20/20 20/20} can be computed exactly by this program. Approximate values for larger j's up to (200/m1 200/m2 200/m3) and {200/200 200/200 200/220} can also be computed by this program. (orig.)

  8. A linear operator method to compute the rotational modes of asymmetric 3D Earth by vector spherical harmonics

    Zhang, Mian; Huang, Cheng-li

    2012-08-01

    Generalized spherical harmonics (GSH) are usually applied on the problems where the Earth model is elliptical and elastic stress tensor is involved in, as stress tensor can’t be represented in vector spherical harmonics. However, the divergence of the te ns or and a vector dot - product with the tensor are only needed on computation rotation modes of the Earth which can be written in the vector spherical harmonics. We extend the equations on the spherical Earth to asymmetric 3D model by means of linear operator method. This method doesn’t use the complicated generalized spherical harmonics nor Wigner 3 - j symbol. As a validation of this method, the practical calculation of rotational modes of 3D Earth will be made and discussed.

  9. Matrix model version of AGT conjecture and generalized Selberg integrals

    Mironov, A; Morozov, And

    2010-01-01

    Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $\\beta$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of...

  10. Generating function method and its applications to Quantum, Nuclear and the Classical Groups

    Hage-Hassan, Mehdi

    2012-01-01

    The generating function method that we had developing has various applications in physics and not only interress undergraduate students but also physicists. We solve simply difficult problems or unsolved commonly used in quantum, nuclear and group theory textbooks. We find simply: the generating function of the harmonic oscillator, the Feynman propagators of the oscillator and the oscillator in uniform magnetic field. We derive the invariants of SU(2) and the expressions of 3-j,6-j symbols. We find also the octonions or Hurwitz quadratic transformations. We show that the cross-product exist only in E3 and E7. We determine the {p} representation of hydrogen atom in three and n-dimensions. We generalize the Cramer's rule for the calculation of the rotational spectrum of the nucleus. We find the expression of the Hamiltonian in terms of quasi-bosons for study the collective vibration. We determine the basis and the expressions of 3-j symbols of SU (3) and SU(n).We find the Schr\\"odinger equation from Hamilton-Ja...

  11. The many faces of Ocneanu cells

    We define generalised chiral vertex operators covariant under the Ocneanu 'double triangle algebra' A, a novel quantum symmetry intrinsic to a given rational 2d 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 A, reappear in several guises. With A and its dual algebra A-circumflex one associates a pair of graphs, G and G-tilde. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs G-tilde classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining A-circumflex. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra

  12. The many faces of Ocneanu cells

    Petkova, V.B. E-mail: valentina.petkova@unn.ac.uk; Zuber, J.-B. E-mail: zuber@spht.saclay.cea.fr

    2001-06-11

    We define generalised chiral vertex operators covariant under the Ocneanu 'double triangle algebra' A, a novel quantum symmetry intrinsic to a given rational 2d 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 A, reappear in several guises. With A and its dual algebra A-circumflex one associates a pair of graphs, G and G-tilde. While G are known to encode complete sets of conformal boundary states, the Ocneanu graphs G-tilde classify twisted torus partition functions. The fusion algebra of the twist operators provides the data determining A-circumflex. The study of bulk field correlators in the presence of twists reveals that the Ocneanu graph quantum symmetry gives also an information on the field operator algebra.

  13. Quantum stereodynamics of the F + H2 → HF + H reaction by the stereodirected S-matrix approach

    Reaction stereodynamics can be studied in quantum mechanics using alternative representations of the S matrix. In this paper we employ the equations for the orthogonal transformations (expressed in terms of Wigner 3j symbols) that convert the S matrix from the body fixed (vertical bar jΩ>) representation into the stereodirected one (vertical bar νΩ>). This representation is characterized by the introduction of the steric quantum number ν, which in the vector model of quantum mechanics is put into correspondence with given precession cones of attack of the incoming atom on the diatomic molecule for the reactants' channels, and of cones of escape for the departing atom away from the diatomic molecule for the products' channels. The angles of aperture of such cones are determined from the uncertainty principle. As the ν quantum number increases (semiclassical limit), the grid of discrete values of the precession cones more finely scans the angle between the Jacobi vectors. Using a time-independent hyperspherical coordinate method we have generated the full S matrix including all open reactive and inelastic channels for two potential energy surfaces corresponding to the F + H2 → HF + H reaction and they have been used to calculate, via vertical bar jΩ>→ vertical bar νΩ> matrix transformations, the attack and exit cumulative reaction probabilities. During the calculations, we have distinguished between ortho-H2 and para-H2. Clear stereodynamical effects have being identified, in particular, regarding the reaction entrance channel, that F-atom attacks are preferred at the transition state (bent) geometry, while for the exit channel the H-atom departs in a collinear geometry by the H-end side of HF

  14. Eugene Wigner and Symmetries In Physics

    Moshinsky, Marcos

    2002-04-01

    Concepts of symmetry in physics have had a long history, particularly if they are of a geometric or crystallographic origin, yet in classical physics they had a somewhat esoteric position. This situation changed radically when in the XX Century we passed from classical to quantum mechanics. In the former a state for a system of particles was given by a number of points in phase space and the transformation groups related with symmetries mainly gave the invariance of concepts such as energy or angular momentum. In the latter the state is characterized by a vector in Hilbert space in which the transformations had a representation. Eugene Wigner was the right man (for his mathematical ability and physical intuition) at the right place and time (Germany, in the twenties) to take full advantage of this new situation. His first interest was atomic spectroscopy (then a very active field) and the fact that its basic states were related with irreducible representation of the orthogonal group in three dimensions O(3). The German version of his book on ``Group theory and Application" published in 1931 established, as he quotes ``that almost all rules of spectroscopy follow from the symmetry of the problem". His later extension to the direct product of two or more representations led to his development of the 3-j symbol, that he explicitly derived, and his interest in the properties of 6-j, 9-j, etc. His awareness of the time inversion as an antiunitary operator, and the analysis of its combination with the unitary representations of other symmetries, proved fundamental for deriving the features of time reversed reactions from their direct behavior. His interest in space reflection and the concept of parity led to important selection rules, and was of relevance even in weak interactions where parity is not a good symmetry. His later interest in nuclear physics, solid state, elementary particles etc., was almost never without a component of the role of symmetry in these