A Universal Generating Functional for SU(2) 3nj Symbols

Author: Arcticoder

About this Repository

This repository contains a paper on a universal generating functional for SU(2) 3nj symbols.

Mathematical Formulation

The master generating functional for Wigner 3nj recoupling coefficients is

$$ G({x_e}) =;\int \prod_{v=1}^n \frac{d^2w_v}{\pi},\exp\Bigl(-\sum_v\lVert w_v\rVert^2\Bigr) ;\prod_{e=\langle i,j\rangle}\exp\bigl(x_e,\epsilon(w_i,w_j)\bigr) =;\frac{1}{\sqrt{\det!\bigl(I - K({x_e})\bigr)}}, $$

where $K$ is the antisymmetric adjacency matrix of edge variables $x_e$.

6-j Symbol Example

For the 6-j case ($n=4$) with two edge variables $x,y$,

$$ G(x,y) =;\frac{1}{\sqrt{,(1 - x y - x - y),(1 + x y - x + y),(1 + x y + x - y),(1 - x y + x + y),}}. $$

9-j and 15-j Symbols

More generally,

$$ G(x,y,z) =;\frac{1}{\sqrt{\det!\bigl(I_6 - K(x,y,z)\bigr)}}, \qquad G(x_1,\dots,x_7) =;\frac{1}{\sqrt{\det!\bigl(I_8 - K(x_1,\dots,x_7)\bigr)}}. $$

Included Scripts

This repository includes Python scripts that calculate and verify the mathematical results, as well as uncertainty‐quantification routines:

• compute_G_xy_series_coefficients.py: Computes series coefficients of the generating function for 6-j symbols.
• compute_hilbert_series_coefficients_n2_6.py: Computes Hilbert series coefficients for ( n=2 ) to ( n=6 ).
• test_15j_generating_function.py: Tests specific terms of the 15-j generating function.
• uq_determinant_stability.py: Quantify numerical error in determinant evaluation of (G({x_e})).
• uq_3nj_sensitivity.py: Local sensitivity analysis of the 6-j symbol under small spin perturbations.
• uq_compare_rational_vs_numeric.py: Compare symbolic vs numeric evaluations of (G({x_e})) for benchmark inputs.
• CI workflow: The Biedenharn–Elliott (pentagon) identity is verified automatically via GitHub Actions, as defined in .github/workflows/pentagon-identity.yml.

Data and Results

Results of running these scripts:

• series_coefficients_G_xy_up_to2.csv
• hilbert_series_coeffs_n2_6.csv
• 15j_generating_function_tests.csv
• uq_determinant_stability.csv
• uq_3nj_sensitivity.csv
• uq_compare_rational_vs_numeric.csv