Recurrence relation for the 6j-symbol of su_q(2) as a symmetric eigenvalue problem

A well known recurrence relation for the 6j-symbol of the quantum group su_q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in the appendix. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q=1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley-Lieb recoupling theory to simplify intermediate calculations.Comment: v3: 13 pages, ws-ijgmmp; minor corrections, slight update to presentation; close to published versio

Semiclassical Analysis of the Wigner 12j12j Symbol with One Small Angular Momentum

We derive an asymptotic formula for the Wigner $12j$ symbol, in the limit of one small and 11 large angular momenta. There are two kinds of asymptotic formulas for the $12j$ symbol with one small angular momentum. We present the first kind of formula in this paper. Our derivation relies on the techniques developed in the semiclassical analysis of the Wigner $9j$ symbol [L. Yu and R. G. Littlejohn, Phys. Rev. A 83, 052114 (2011)], where we used a gauge-invariant form of the multicomponent WKB wave-functions to derive asymptotic formulas for the $9j$ symbol with small and large angular momenta. When applying the same technique to the $12j$ symbol in this paper, we find that the spinor is diagonalized in the direction of an intermediate angular momentum. In addition, we find that the geometry of the derived asymptotic formula for the $12j$ symbol is expressed in terms of the vector diagram for a $9j$ symbol. This illustrates a general geometric connection between asymptotic limits of the various $3nj$ symbols. This work contributes the first known asymptotic formula for the $12j$ symbol to the quantum theory of angular momentum, and serves as a basis for finding asymptotic formulas for the Wigner $15j$ symbol with two small angular momenta.Comment: 15 pages, 14 figure

Bohr-Sommerfeld Quantization of Space

We introduce semiclassical methods into the study of the volume spectrum in loop gravity. The classical system behind a 4-valent spinnetwork node is a Euclidean tetrahedron. We investigate the tetrahedral volume dynamics on phase space and apply Bohr-Sommerfeld quantization to find the volume spectrum. The analysis shows a remarkable quantitative agreement with the volume spectrum computed in loop gravity. Moreover, it provides new geometrical insights into the degeneracy of this spectrum and the maximum and minimum eigenvalues of the volume on intertwiner space.Comment: 32 pages, 10 figure

Semiclassical Analysis of the Wigner 9J9J-Symbol with Small and Large Angular Momenta

We derive a new asymptotic formula for the Wigner $9j$-symbol, in the limit of one small and eight large angular momenta, using a novel gauge-invariant factorization for the asymptotic solution of a set of coupled wave equations. Our factorization eliminates the geometric phases completely, using gauge-invariant non-canonical coordinates, parallel transports of spinors, and quantum rotation matrices. Our derivation generalizes to higher $3nj$-symbols. We display without proof some new asymptotic formulas for the $12j$-symbol and the $15j$-symbol in the appendices. This work contributes a new asymptotic formula of the Wigner $9j$-symbol to the quantum theory of angular momentum, and serves as an example of a new general method for deriving asymptotic formulas for $3nj$-symbols.Comment: 18 pages, 16 figures. To appear in Phys. Rev.

Volcano Popocatepetl, Mexico: ULF geomagnetic anomalies observed at Tlamacas station during March?July, 2005

International audienceIn this paper the first results of ULF (Ultra Low Frequency) geomagnetic anomalies observed at Tlamacas station (Long. 261.37, Lat. 19.07) located at 4 km near the volcano Popocatepetl (active volcano, Long. 261.37, Lat. 19.02) for the period March?July, 2005 and their analysis are presented. The geomagnetic data were collected with a 3-axial fluxgate magnetometer designed at UCLA (University of California, Los Angeles, 1 Hz sampling rate frequency, GPS). Our analysis reveals some anomalies which are suspected to be generated by local volcanic origin: the EM background in the vicinity of the volcano is significantly noisier than in other reference stations; the sporadic strong noise-like geomagnetic activity observed in the H-component; locally generated geomagnetic pulsations (without preferred polarization) are detected only at Tlamacas station

From Poincare to affine invariance: How does the Dirac equation generalize?

A generalization of the Dirac equation to the case of affine symmetry, with SL(4,R) replacing SO(1,3), is considered. A detailed analysis of a Dirac-type Poincare-covariant equation for any spin j is carried out, and the related general interlocking scheme fulfilling all physical requirements is established. Embedding of the corresponding Lorentz fields into infinite-component SL(4,R) fermionic fields, the constraints on the SL(4,R) vector-operator generalizing Dirac's gamma matrices, as well as the minimal coupling to (Metric-)Affine gravity are studied. Finally, a symmetry breaking scenario for SA(4,R) is presented which preserves the Poincare symmetry.Comment: 34 pages, LaTeX2e, 8 figures, revised introduction, typos correcte

Geomagnetic anomalies observed at volcano Popocatepetl, Mexico

International audienceResults of the ULF geomagnetic monitoring of the volcano Popocatepetl (Mexico) and their analysis are summarized and presented for the period 2003?2006. Our analysis reveals some anomalies which are considered to be of local volcanic origin: the EM background in the vicinity of the volcano was found to be significantly noisier than at other reference stations; sporadic strong noise-like geomagnetic activity was observed in the H-component; some geomagnetic pulsations were observed only at the Tlamacas station (located at 4 km near the volcano). The results are discussed in terms of a physical mechanism involving the presence of a second magmatic chamber within the volcano and, finally, further perspective directions to study volcanic geodynamical processes besides the traditional ones are given

Surface embedding, topology and dualization for spin networks

Spin networks are graphs derived from 3nj symbols of angular momentum. The surface embedding, the topology and dualization of these networks are considered. Embeddings into compact surfaces include the orientable sphere S^2 and the torus T, and the not orientable projective space P^2 and Klein's bottle K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces.Comment: LaTeX 17 pages, 6 eps figures (late submission to arxiv.org

Exact and asymptotic computations of elementary spin networks: classification of the quantum-classical boundaries

Increasing interest is being dedicated in the last few years to the issues of exact computations and asymptotics of spin networks. The large-entries regimes (semiclassical limits) occur in many areas of physics and chemistry, and in particular in discretization algorithms of applied quantum mechanics. Here we extend recent work on the basic building block of spin networks, namely the Wigner 6j symbol or Racah coefficient, enlightening the insight gained by exploiting its self-dual properties and studying it as a function of two (discrete) variables. This arises from its original definition as an (orthogonal) angular momentum recoupling matrix. Progress also derives from recognizing its role in the foundation of the modern theory of classical orthogonal polynomials, as extended to include discrete variables. Features of the imaging of various regimes of these orthonormal matrices are made explicit by computational advances -based on traditional and new recurrence relations- which allow an interpretation of the observed behaviors in terms of an underlying Hamiltonian formulation as well. This paper provides a contribution to the understanding of the transition between two extreme modes of the 6j, corresponding to the nearly classical and the fully quantum regimes, by studying the boundary lines (caustics) in the plane of the two matrix labels. This analysis marks the evolution of the turning points of relevance for the semiclassical regimes and puts on stage an unexpected key role of the Regge symmetries of the 6j.Comment: 15 pages, 11 figures. Talk presented at ICCSA 2012 (12th International Conference on Computational Science and Applications, Salvador de Bahia (Brazil) June 18-21, 2012