This repository contains a mathematical framework for the closed-form hypergeometric representations of SU(2) 3nj symbols, with particular focus on 12j symbols and their universal generating functional.
Our work presents a universal closed-form hypergeometric representation that unifies all SU(2) 3nj recoupling coefficients under a single special-function framework. This provides:
- Exact symbolic expressions for arbitrary 3nj symbols
- Computational efficiency through hypergeometric function evaluation
- Mathematical rigor with derivations and proofs
- Numerical validation across multiple test cases
- LaTeX Source: Complete mathematical derivation and proofs
- GitHub Pages Website: Interactive presentation with MathJax rendering
- PDF Documentation: Publication-ready mathematical exposition
- Computational Scripts: Python implementation and verification tools
- Validation Data: Numerical verification results and benchmarks
📚 Read the paper online: https://arcticoder.github.io/su2-3nj-uniform-closed-form/
The website features:
- Complete mathematical exposition with interactive equations
- Downloadable PDF version
- Source code examples and usage instructions
- Cross-references to related work in the SU(2) 3nj series
The theoretical framework is validated through computational verification:
Script: symbolic_taylor_expansion.py
- Constructs explicit symbolic Taylor expansion of the universal generating functional
- Generates series with 26 coefficients of the form
C_j12_j23_j34 - Covers angular momentum values: 0, 1/2, and 1
- Provides symbolic verification of convergence properties
Script: match_simplest_hypergeometric.py
- Demonstrates correspondence with known 9j symbol representations
- Validates 4F3 hypergeometric function equivalence
- Focuses on simplest case: (j12=0, j23=0, j34=1/2)
- Confirms theoretical predictions through symbolic computation
Primary: verify_simple_9j_numeric.py
- Numerical verification of simplest case (j12=0, j23=0, j34=1/2)
- High-precision arithmetic validation
- Error analysis and convergence testing
Extended: verify_additional_9j_numeric.py
- Additional test case: (j12=0, j23=1/2, j34=0)
- Robustness verification across parameter space
- Cross-validation with established numerical libraries
Output: All results stored in data/ directory as CSV files with complete numerical verification confirming theoretical accuracy.
pip install sympy numpy scipy pandas matplotlib# Symbolic Taylor expansion
python symbolic_taylor_expansion.py
# Hypergeometric matching
python match_simplest_hypergeometric.py
# Numerical validation
python verify_simple_9j_numeric.py
python verify_additional_9j_numeric.pyThis repository is part of a SU(2) 3nj symbol research series:
- su2-3nj-closedform: Closed-form hypergeometric product formula
- su2-3nj-recurrences: Finite closed-form recurrence relations
- su2-3nj-generating-functional: Universal generating functional approach
- su2-node-matrix-elements: Operator matrix elements for arbitrary-valence nodes
The universal representation unifies all 3nj symbols through:
3nj Symbol = Hypergeometric_Series(angular_momenta, coupling_structure)
- Universal generating functional: Single expression for all 3nj topologies
- Closed-form hypergeometric: Exact special function representation
- Computational efficiency: Direct evaluation without recursion
- Mathematical elegance: Unified framework for all recoupling coefficients
- Quantum Mechanics: Angular momentum coupling calculations
- Computational Physics: Efficient 3nj symbol evaluation
- Mathematical Physics: Special function theory and applications
- Numerical Libraries: High-performance recoupling coefficient computation
This project is licensed under The Unlicense - see the LICENSE file for details.
Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.