Asymptotics of the Wigner 9j symbol

Abstract

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.

17 pages, 7 figures

References in corpus (4)

• 3nj Morphogenesis and Semiclassical Disentangling
• Pushing Further the Asymptotics of the 6j-symbol
• Uniform Semiclassical Approximation for the Wigner $6j$ Symbol in Terms of Rotation Matrices
• The Ponzano-Regge asymptotic of the 6j symbol: an elementary proof

Cited by in corpus (22)

• Discreteness of the volume of space from Bohr-Sommerfeld quantization
• SU(2) graph invariants, Regge actions and polytopes
• Semiclassical Mechanics of the Wigner 6j-Symbol
• Gluing polyhedra with entanglement in loop quantum gravity
• Discrete gravity dynamics from effective spin foams
• Searching for classical geometries in spin foam amplitudes: a numerical method
• Infrared divergences in the EPRL-FK Spin Foam model
• Semiclassical Analysis of the Wigner $9J$-Symbol with Small and Large Angular Momenta
• Pentahedral volume, chaos, and quantum gravity
• Renormalization of group field theories for quantum gravity: new scaling results and some suggestions
• Geometry of the shears mechanism in nuclei
• Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors
• Remarks on a melonic field theory with cubic interaction
• Symplectic and Semiclassical Aspects of the Schläfli Identity
• The Generic Superintegrable System on the 3-Sphere and the $9j$ Symbols of $\mathfrak{su}(1,1)$
• Maslov indices, Poisson brackets, and singular differential forms
• Introduction to SU(2) recoupling theory and graphical methods for loop quantum gravity
• Semiclassical Analysis of the Wigner $12j$ Symbol with One Small Angular Momentum
• The large j limit for certain 9-j symbols-power law behaviour
• Spinfoam tunneling of quantum geometries in angle variables
• Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, $3j$ Symbols, and Character Localization
• A Mapping between the Spin and Fermion Algebra