For each generic $(3\mbox{-}j)$ the column parities, $2(j \pm m)$, define $3$ intrinsic parities: $α, β,γ$.
In algebra $so(3)$ only $(3\mbox{-}j)\_α$ exists whereas super-algebra $osp(1|2)$ admits $3$ kinds of super-symbols
$(3\mbox{-}j)^{S}\_α, (3\mbox{-}j)^{S}\_β, (3\mbox{-}j)^{S}\_γ$.
Instead of 4 for $\{6\mbox{-}j\}$ symbols, Regge symmetry this time produces 5 partitions
${\tt S}\_{\boldsymbol{\wr}}(0),
{\tt S}\_{\boldsymbol{\wr}}(1), {\tt S}\_{\boldsymbol{\wr}}(2),
{\tt S}\_{\boldsymbol{\wr}}(4), {\tt S}\_{\boldsymbol{\wr}}(5)$,
with ${\tt S}\_{\boldsymbol{\wr}}(3) = \emptyset$. Valid for $(3\mbox{-}j)\_α,
(3\mbox{-}j)^{S}\_{α,γ}$ they reduce to 2 for $(3\mbox{-}j)^{S}\_β$ with ${\tt S}\_{\boldsymbol{\wr}}(0),
{\tt S}\_{\boldsymbol{\wr}}(1)$. Unexpectedly a symbol $(3\mbox{-}j)^{S}\_β$ and its 'Regge-transformed' may be opposite in sign.
In terms of integer parts and super-triangle $\triangle^{S}$ a formula similar to that of a $(3\mbox{-}j)$ is obtained for the $(3\mbox{-}j)^{S}$. Some forbidden $(3\mbox{-}j)^{S}\_β$ require an analytic prolongation, consistent with Regge $β$-partitions.
It shown that the five Regge transformations act as a spectrometric splitter on any 6-j symbol. Four unknown partitions are brought out: S4(0), S4(1), S(2) and S4(5). They are stable subsets, with well defined parameters depending only on triangles and quadrangles. These findings are easily generalized to super 6-jS symbols, properly labelled by their own parity alpha, beta, gamma. Super Regge symmetry is reduced only for beta where S4(2), S4(5) vanish. In addition, all tools for computing exact values of any 6-jS are provided.
We adapt the Gurau's proof (2008) about the asymptotic limit of Ponzano-Regge formula to supersymmetric 6jS symbols according to their intrinsic parities alpha, beta, gamma. The behaviour at a large scaling shows significant differences depending on these parities. The decay is slowed and the angles of the oscillating parts in cosine are generally shifted or more altered.Our results should be relevant in 3-D Quantum Supergravity and Spin Foam models. PACS: 03.65.Fd Algebraic methods PACS: 04.60.-m Quantum gravity PACS: 11.30.Pb Supersymmetry MSC: 81Q60; 83D05.
The Weyl quantization of classical observables on the torus (as phase space) without regularity assumptions is explicitly computed. The equivalence class of symbols yielding the same Weyl operator is characterized. The Heisenberg equation for the dynamics of general quantum observables is written through the Moyal brackets on the torus and the support of the Wigner transform is characterized. Finally, a dequantization procedure is introduced that applies, for instance, to the Pauli matrices. As a result we obtain the corresponding classical symbols.
The Wigner $3j$ 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 interpre…
We present new sum rules for $3jm$ coefficients, which involve, in addition to the usual weighting factor $(2j + 1)$ where $j$ is an angular momentum, the quantity $[j(j + 1)]^k$ with $k \ge 1$. The sum rules appear for instance in the statistical modeling of rotational spectra within the theory of moments, and enable one to deduce the expectation values of $r^k$ (used in the theory of Stark effect for hydrogenic ions) in parabolic coordinates from the expectation values of $r^k$ in spherical coordinates.
We present a novel hierarchical construction of projective spin networks of the Ponzano-Regge type from an assembling of five quadrangles up to the combinatorial 4-simplex compatible with a geometrical realization in Euclidean 4-space. The key ingrendients are the projective Desargues configuration and the incidence structure given by its space-dual, on the one hand, and the Biedenharn--Elliott identity for the 6j symbol of SU(2), on the other. The interplay between projective-combinatorial and algebraic features relies on the recoupling theory of angular momenta, an approach to discrete quant…
The contribution of Jacques Raynal to angular-momentum theory is highly valuable. In the present article, I intend to recall the main aspects of his work related to Wigner $3j$ symbols. It is well known that the latter can be expressed with a hypergeometric series. The polynomial zeros of the $3j$ coefficients were initially characterized by the number of terms of the series minus one, which is the degree of the coefficient. A detailed study of the zeros of the $3j$ coefficient with respect to the degree $n$ for $J = a + b + c \leq 240$ ($a$, $b$ and $c$ being the angular momenta in the first …
We examine the affine Wigner distribution from a quantization perspective with an emphasis on the underlying group structure. One of our main results expresses the scalogram as (affine) convolution of affine Wigner distributions. We strive to unite the literature on affine Wigner distributions and we provide the connection to the Mellin transform in a rigorous manner. Moreover, we present an affine ambiguity function and show how this can be used to illuminate properties of the affine Wigner distribution. In contrast with the usual Wigner distribution, we demonstrate that the affine Wigner dis…
The Ponzano-Regge state-sum model provides a quantization of 3d gravity as a spin foam, providing a quantum amplitude to each 3d triangulation defined in terms of the 6j-symbol (from the spin-recoupling theory of SU(2) representations). In this context, the invariance of the 6j-symbol under 4-1 Pachner moves, mathematically defined by the Biedenharn-Elliot identity, can be understood as the invariance of the Ponzano-Regge model under coarse-graining or equivalently as the invariance of the amplitudes under the Hamiltonian constraints. Here we look at length and volume insertions in the Biedenh…
We study the SL(2,C) Clebsch-Gordan coefficients appearing in the lorentzian EPRL spin foam amplitudes for loop quantum gravity. We show how the amplitudes decompose into SU(2) nj-symbols at the vertices and integrals over boosts at the edges. The integrals define edge amplitudes that can be evaluated analytically using and adapting results in the literature, leading to a pure state sum model formulation. This procedure introduces virtual representations which, in a manner reminiscent to virtual momenta in Feynman amplitudes, are off-shell of the simplicity constraints present in the theory, b…
