7 research outputs found
The Screen representation of spin networks: 2D recurrence, eigenvalue equation for 6j symbols, geometric interpretation and Hamiltonian dynamics
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
Schr\"odinger-like equations and an asymptotic analysis illustrates
semiclassical and classical limits in terms of Hamiltonian evolution.Comment: 14 pages,9 figures, presented at ICCSA 2013 13th International
Conference on Computational Science and Applicatio
The Screen representation of spin networks. Images of 6j symbols and semiclassical features
This article presents and discusses in detail the results of extensive exact
calculations of the most basic ingredients of spin networks, the Racah
coefficients (or Wigner 6j symbols), exhibiting their salient features when
considered as a function of two variables - a natural choice due to their
origin as elements of a square orthogonal matrix - and illustrated by use of a
projection on a square "screen" introduced recently. On these screens, shown
are images which provide a systematic classification of features previously
introduced to represent the caustic and ridge curves (which delimit the
boundaries between oscillatory and evanescent behaviour according to the
asymptotic analysis of semiclassical approaches). Particular relevance is given
to the surprising role of the intriguing symmetries discovered long ago by
Regge and recently revisited; from their use, together with other newly
discovered properties and in conjunction with the traditional combinatorial
ones, a picture emerges of the amplitudes and phases of these discrete
wavefunctions, of interest in wide areas as building blocks of basic and
applied quantum mechanics.Comment: 16 pages, 13 figures, presented at ICCSA 2013 13th International
Conference on Computational Science and Applicatio
Symmetric angular momentum coupling, the quantum volume operator and the 7-spin network: a computational perspective
A unified vision of the symmetric coupling of angular momenta and of the
quantum mechanical volume operator is illustrated. The focus is on the quantum
mechanical angular momentum theory of Wigner's 6j symbols and on the volume
operator of the symmetric coupling in spin network approaches: here, crucial to
our presentation are an appreciation of the role of the Racah sum rule and the
simplification arising from the use of Regge symmetry. The projective geometry
approach permits the introduction of a symmetric representation of a network of
seven spins or angular momenta. Results of extensive computational
investigations are summarized, presented and briefly discussed.Comment: 15 pages, 10 figures, presented at ICCSA 2014, 14th International
Conference on Computational Science and Application
The screen representation of vector coupling coefficients or Wigner 3j symbols: exact computation and illustration of the asymptotic behavior
The Wigner 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.Comment: 14 pages, 8 figures, presented at ICCSA 2014, 14th International
Conference on Computational Science and Application