https://github.com/arcticoder/su2-3nj-uniform-closed-form
Universal closed-form hypergeometric representation of SU(2) 3nj symbols, featuring a LaTeX derivation, master generating functional, and Python scripts for symbolic expansion and numerical verification.
https://github.com/arcticoder/su2-3nj-uniform-closed-form
3nj computational-physics generating-function hypergeometric latex mathematics mathjax orthogonal-polynomials physics quantum-mechanics recoupling su2 symbolic-computation sympy
Last synced: 8 months ago
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Universal closed-form hypergeometric representation of SU(2) 3nj symbols, featuring a LaTeX derivation, master generating functional, and Python scripts for symbolic expansion and numerical verification.
- Host: GitHub
- URL: https://github.com/arcticoder/su2-3nj-uniform-closed-form
- Owner: arcticoder
- Created: 2025-05-25T21:01:38.000Z (about 1 year ago)
- Default Branch: main
- Last Pushed: 2025-06-21T14:13:56.000Z (12 months ago)
- Last Synced: 2025-06-21T14:38:32.000Z (12 months ago)
- Topics: 3nj, computational-physics, generating-function, hypergeometric, latex, mathematics, mathjax, orthogonal-polynomials, physics, quantum-mechanics, recoupling, su2, symbolic-computation, sympy
- Language: Python
- Homepage: https://arcticoder.github.io/su2-3nj-uniform-closed-form/
- Size: 170 KB
- Stars: 1
- Watchers: 0
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Universal Closed-Form Hypergeometric Representation of SU(2) 3nj Symbols
[](https://arcticoder.github.io/su2-3nj-uniform-closed-form/)
[](LICENSE)
[](https://python.org)
This repository presents a mathematical framework and illustrative computations for closed-form hypergeometric representations of SU(2) 3nj symbols, with particular focus on 12j symbols and a proposed generating functional. The materials emphasize derivations and representative computational checks rather than exhaustive numerical validation across all parameter regimes.
## Mathematical Framework
The project proposes a closed-form hypergeometric representation that aims to cover many SU(2) 3nj recoupling coefficients within a special-function framework. The repository documents derivations and provides representative computational checks. Key points:
- Symbolic derivations are provided for a range of topologies; some edge cases may require additional analysis.
- The framework can offer alternative evaluation approaches for certain symbol classes; users should benchmark against existing libraries for their specific parameter ranges.
- Derivations and proofs are included in the LaTeX source; consult the paper for assumptions and scope.
- Numerical verification is provided for selected cases but is not a comprehensive validation across all spins and couplings.
## 📖 Contents
- **LaTeX Source**: Mathematical derivations and supporting notes
- **GitHub Pages Website**: Interactive exposition with MathJax rendering
- **PDF Documentation**: Publication-ready mathematical exposition
- **Computational Scripts**: Python implementation and verification tools
- **Validation Data**: Numerical verification results and benchmarks for tested cases
## 🌐 Online Documentation
**📚 Read the paper online**: https://arcticoder.github.io/su2-3nj-uniform-closed-form/
The website features interactive exposition, downloadable PDF, example code, and links to related work in the SU(2) 3nj series.
## Computational Verification
The repository includes symbolic and numeric scripts intended as reproducibility artifacts for the included examples.
### 🔄 Symbolic Taylor Expansion
**Script**: `symbolic_taylor_expansion.py`
- Generates symbolic series for illustrative cases and inspects coefficients for internal consistency.
### Hypergeometric Correspondence
**Script**: `match_simplest_hypergeometric.py`
- Demonstrates correspondence with known 9j symbol representations for selected parameter choices.
### Numerical Validation
**Primary**: `verify_simple_9j_numeric.py`
- High-precision numeric checks for representative simple cases.
**Extended**: `verify_additional_9j_numeric.py`
- Additional numeric checks across a small set of cases; intended as a starting point for broader validation.
**Output**: Results and verification artifacts are stored in `data/` for the tested examples; these serve as reproducibility artifacts rather than proof of exhaustive correctness across all regimes.
## Installation & Usage
### Prerequisites
```bash
pip install sympy numpy scipy pandas matplotlib
```
### Running Verification Scripts
```bash
# 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.py
```
## 🔗 Related Work
This repository is part of an 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`: Generating functional approach
- `su2-node-matrix-elements`: Operator matrix elements for arbitrary-valence nodes
## Mathematical Background
### Core Theory
The universal representation is presented as a hypergeometric-based construction that relates angular-momentum coupling topologies to special-function expressions. See the paper for precise definitions and applicable assumptions.
### Key Considerations
- The generating functional is proposed as a compact formal expression covering many topologies under stated assumptions.
- Derived closed-form expressions are provided for several topologies; additional cases may require extended derivations or boundary data.
- Performance and numerical stability depend on parameter ranges and chosen numerical precision; validate against established implementations for production use.
## Applications
- Quantum mechanics: angular momentum coupling computations (research/analysis use)
- Computational physics: experimental evaluation of evaluation techniques
- Mathematical physics: special-function identities and illustrative examples
## License
This project is licensed under The Unlicense - see the `LICENSE` file for details.
## Contributing
Contributions are welcome; for major changes please open an issue first to discuss the proposal.
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## Scope / Validation & Limitations
- **Research-stage framework:** Materials document a theoretical framework and illustrative verifications. Claims about universality are intended as working hypotheses derived in the paper; maintainers and users should verify applicability for new topologies and large-spin regimes.
- **Numerical stability & validation:** Numerical checks included here cover selected examples; reproduce these checks in your environment and extend them for other parameter ranges. Consider using high-precision arithmetic for large spins and cross-validate against established libraries.
- **Uncertainty & reproducibility:** When publishing numeric comparisons, include environment details, numerical precision, random seeds (if any), and input parameter sets under `docs/` or `data/` to support reproducibility.
- **Limitations:** Derivations assume the conditions stated in the paper; edge cases (boundary conditions, degenerate couplings) may require additional analysis or boundary data.