SO_0(1,d+1) Racah coefficients: Type I representations

We use AdS/CFT inspired methods to study the Racah coefficients for type I representations of the Lorentz group SO_0(1,d+1) with d>1. For such representations (a multiple of) the Racah coefficient can be represented as an integral of a product of 6 bulk-to-bulk propagators over 4 copies of the hyperbolic space H_{d+1}. To compute the integrals we represent the bulk-to-bulk propagators in terms of bulk-to-boundary ones. The bulk integrals can be computed explicitly, and the boundary integrations are carried out by introducing Feynman parameters. The final result is an integral representation of the Racah coefficient given by 4 Barnes-Mellin type integrals.Comment: 20 pages, 1 figure. v2: Case d=1 corrected, case d>1 clarifie

Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis

We consider the Lie group $\mathbb{R}^D_\kappa$ generated by the Lie algebra of $\kappa$-Minkowski space. Imposing the invariance of the metric under the pull-back of diffeomorphisms induced by right translations in the group, we show that a unique right invariant metric is associated with $\mathbb{R}^D_\kappa$. This metric coincides with the metric of de Sitter space-time. We analyze the structure of unitary representations of the group $\mathbb{R}^D_\kappa$ relevant for the realization of the non-commutative $\kappa$-Minkowski space by embedding into $(2D-1)$-dimensional Heisenberg algebra. Using a suitable set of generalized coherent states, we select the particular Hilbert space and realize the non-commutative $\kappa$-Minkowski space as an algebra of the Hilbert-Schmidt operators. We define dequantization map and fuzzy variant of the Laplace-Beltrami operator such that dequantization map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami operator on the half of de Sitter space-time.Comment: 21 pages, v3 differs from version published in Fortschritte der Physik by a note and references added and adjuste

Relating Spin Foams and Canonical Quantum Gravity: A Discrete Step Evolution Formulation of Spin Foams

This article has been replaced by gr-qc/0412011Comment: This article has been replaced by gr-qc/041201

Contracted Representation of Yang's Space-Time Algebra and Buniy-Hsu-Zee's Discrete Space-Time

Motivated by the recent proposition by Buniy, Hsu and Zee with respect to discrete space-time and finite spatial degrees of freedom of our physical world with a short- and a long-distance scales, $l_P$ and $L,$ we reconsider the Lorentz-covariant Yang's quantized space-time algebra (YSTA), which is intrinsically equipped with such two kinds of scale parameters, $\lambda$ and $R$. In accordance with their proposition, we find the so-called contracted representation of YSTA with finite spatial degrees of freedom associated with the ratio $R/\lambda$, which gives a possibility of the divergence-free noncommutative field theory on YSTA. The canonical commutation relations familiar in the ordinary quantum mechanics appear as the cooperative Inonu-Wigner's contraction limit of YSTA, $\lambda \to 0$ and $R \to \infty.

Two Dimensional Fractional Supersymmetry from the Quantum Poincare Group at Roots of Unity

A group theoretical understanding of the two dimensional fractional supersymmetry is given in terms of the quantum Poincare group at roots of unity. The fractional supersymmetry algebra and the quantum group dual to it are presented and the pseudo-unitary, irreducible representations of them are obtained. The matrix elements of these representations are explicitly constructed.Comment: 10 pages. Some misprints are corrected. To appear in J. Phys.

A minimal approach for the local statistical properties of a one-dimensional disordered wire

We consider a one-dimensional wire in gaussian random potential. By treating the spatial direction as imaginary time, we construct a `minimal' zero-dimensional quantum system such that the local statistical properties of the wire are given as products of statistically independent matrix elements of the evolution operator of the system. The space of states of this quantum system is found to be a particular non-unitary, infinite dimensional representation of the pseudo-unitary group, U(1,1). We show that our construction is minimal in a well defined sense, and compare it to the supersymmetry and Berezinskii techniques.Comment: 10 pages, 0 figure

Quantum particle on hyperboloid

We present quantization of particle dynamics on one-sheet hyperboloid embedded in three dimensional Minkowski space. Taking account of all global symmetries enables unique quantization. Making use of topology of canonical variables not only simplifies calculations but also gives proper framework for analysis.Comment: 7 pages, no figures, revtex

Group-theoretical approach to a non-central extension of the Kepler-Coulomb problem

Bound and scattering states of a non-central extension of the three-dimensional Kepler-Coulomb Hamiltonian are worked out analytically within the framework of the potential groups of the problem, SO(7) for bound states and SO(6,1) for scattering states. In the latter case, the S matrix is calculated by the method of intertwining operators.Comment: 12 pages, to appear in J. Phys. A : Math. Theo

Gravitational Wilson Loop and Large Scale Curvature

In a quantum theory of gravity the gravitational Wilson loop, defined as a suitable quantum average of a parallel transport operator around a large near-planar loop, provides important information about the large-scale curvature properties of the geometry. Here we shows that such properties can be systematically computed in the strong coupling limit of lattice regularized quantum gravity, by performing a local average over rotations, using an assumed near-uniform measure in group space. We then relate the resulting quantum averages to an expected semi-classical form valid for macroscopic observers, which leads to an identification of the gravitational correlation length appearing in the Wilson loop with an observed large-scale curvature. Our results suggest that strongly coupled gravity leads to a positively curved (De Sitter-like) quantum ground state, implying a positive effective cosmological constant at large distances.Comment: 22 pages, 6 figure

Evolution Equation for Generalized Parton Distributions

The extension of the method [arXiv:hep-ph/0503109] for solving the leading order evolution equation for Generalized Parton Distributions (GPDs) is presented. We obtain the solution of the evolution equation both for the flavor nonsinglet quark GPD and singlet quark and gluon GPDs. The properties of the solution and, in particular, the asymptotic form of GPDs in the small x and \xi region are discussed.Comment: REVTeX4, 34 pages, 3 figure