78 research outputs found
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 Symbol with One Small Angular Momentum
We derive an asymptotic formula for the Wigner symbol, in the limit of
one small and 11 large angular momenta. There are two kinds of asymptotic
formulas for the 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 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 symbol with small and large angular momenta. When applying the same
technique to the 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
symbol is expressed in terms of the vector diagram for a symbol. This
illustrates a general geometric connection between asymptotic limits of the
various symbols. This work contributes the first known asymptotic formula
for the symbol to the quantum theory of angular momentum, and serves as a
basis for finding asymptotic formulas for the Wigner 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 -Symbol with Small and Large Angular Momenta
We derive a new asymptotic formula for the Wigner -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 -symbols.
We display without proof some new asymptotic formulas for the -symbol and
the -symbol in the appendices. This work contributes a new asymptotic
formula of the Wigner -symbol to the quantum theory of angular momentum,
and serves as an example of a new general method for deriving asymptotic
formulas for -symbols.Comment: 18 pages, 16 figures. To appear in Phys. Rev.
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
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
Resonant Inelastic X-ray Scattering Studies of Elementary Excitations
In the past decade, Resonant Inelastic X-ray Scattering (RIXS) has made
remarkable progress as a spectroscopic technique. This is a direct result of
the availability of high-brilliance synchrotron X-ray radiation sources and of
advanced photon detection instrumentation. The technique's unique capability to
probe elementary excitations in complex materials by measuring their energy-,
momentum-, and polarization-dependence has brought RIXS to the forefront of
experimental photon science. We review both the experimental and theoretical
RIXS investigations of the past decade, focusing on those determining the
low-energy charge, spin, orbital and lattice excitations of solids. We present
the fundamentals of RIXS as an experimental method and then review the
theoretical state of affairs, its recent developments and discuss the different
(approximate) methods to compute the dynamical RIXS response. The last decade's
body of experimental RIXS data and its interpretation is surveyed, with an
emphasis on RIXS studies of correlated electron systems, especially transition
metal compounds. Finally, we discuss the promise that RIXS holds for the near
future, particularly in view of the advent of x-ray laser photon sources.Comment: Review, 67 pages, 44 figure
An efficient approach for spin-angular integrations in atomic structure calculations
A general method is described for finding algebraic expressions for matrix
elements of any one- and two-particle operator for an arbitrary number of
subshells in an atomic configuration, requiring neither coefficients of
fractional parentage nor unit tensors. It is based on the combination of second
quantization in the coupled tensorial form, angular momentum theory in three
spaces (orbital, spin and quasispin), and a generalized graphical technique.
The latter allows us to calculate graphically the irreducible tensorial
products of the second quantization operators and their commutators, and to
formulate additional rules for operations with diagrams. The additional rules
allow us to find graphically the normal form of the complicated tensorial
products of the operators. All matrix elements (diagonal and non-diagonal with
respect to configurations) differ only by the values of the projections of the
quasispin momenta of separate shells and are expressed in terms of completely
reduced matrix elements (in all three spaces) of the second quantization
operators. As a result, it allows us to use standard quantities uniformly for
both diagona and off-diagonal matrix elements
Asymptotics of the Wigner 9j symbol
We present the asymptotic formula for the Wigner 9j-symbol, valid when all
quantum numbers are large, in the classically allowed region. As in the
Ponzano-Regge formula for the 6j-symbol, the action is expressed in terms of
lengths of edges and dihedral angles of a geometrical figure, but the angles
require care in definition. Rules are presented for converting spin networks
into the associated geometrical figures. The amplitude is expressed as the
determinant of a 2x2 matrix of Poisson brackets. The 9j-symbol possesses
caustics associated with the fold and elliptic and hyperbolic umbilic
catastrophes. The asymptotic formula obeys the exact symmetries of the
9j-symbol.Comment: 17 pages, 7 figure
- …