LQG Volume Operator Spectral Analysis

This repository contains scripts and tools for computing and analyzing the spectrum of the Loop Quantum Gravity (LQG) volume operator and its kernel (zero-volume) states. The computations and reported counts below reflect the sampling and parameter ranges used by the included scripts; they are not exhaustive proofs of global properties.

📖 View the complete research documentation on GitHub Pages

Repository Name

lqg-volume-kernel-catalog

Description

Scripts to:

• Compute the LQG volume operator spectrum via SU(2) 12j-symbols.
• Identify trivial and non-trivial zero-volume states for 4-valent nodes.
• Analyze kernel dimensions of the volume-squared matrix.
• Generate LaTeX tables and figures summarizing results.
• Validate the Diophantine characterization of zero-volume states.

Topics

loop-quantum-gravity volume-operator quantum-gravity su2 recoupling-theory 12j-symbols latex

Requirements

See requirements.txt for dependencies:

pip install -r requirements.txt

Usage

1. Analyze Zero-Volume States (4-valent)

python scripts/analyze_zero_volume_states.py

• Scans a specified spin-configuration range (default j=0.5 to 3.0). Adjust the script arguments to explore other ranges.
• Identifies trivial zero-volume states under the tested sampling and criteria (J₁₂ ∩ J₃₄ = ∅ where applicable).
• Reports that no non-trivial zero-volume states were found within the tested range and parameter sweep; this is an empirical result for the scanned configurations (not a mathematical proof of absence).
• Outputs summary statistics and generates:
  • results/kernel_dimension_distribution.png
  • results/zero_volume_catalog.json

2. Generate LaTeX Table of Trivial Zero-Volume States

python scripts/generate_latex_table.py

• Produces results/trivial_zero_volume_table.tex with a single-column summary of trivial zero-volume cases.

3. Create Spin-1/2 Correlation Placeholder Figure

python scripts/generate_spin_half_correlation.py

• Generates results/figures/spin_half_correlation.png showing the absence of non-trivial kernel states.

4. Generate Volume Spectrum (Optional)

python scripts/compute_volume_spectrum.py

• Computes eigenvalues of the volume-squared operator for specified spin configurations.
• Outputs to data/volume_spectrum.csv.

Results

• Total configurations scanned (example run): 1,296 (default j_i ∈ {0.5, 1.0, …, 3.0}) — changeable via script arguments. These counts reflect the example run's sampling choices and are not exhaustive.
• Trivial zero-volume states (example run): 60 (≈4.6%) — fraction observed in the example sweep; this varies with sampling and parameterization. When reporting these counts, publish the raw outputs and sampling metadata.
• Non-trivial zero-volume states (example run): 0 observed in the example sweep (empirical result within the tested parameter range). Absence in a finite sampled set is not a mathematical proof of absence; for broader claims provide denser sampling and analytic checks.
• Full-rank matrices (example run): 674 (≈52.0%) — reported for the example sweep.
• Other kernel matrices (example run): 562 (≈43.4%) — reported for the example sweep.

All trivial zero-volume cases in the example sweep satisfy the Diophantine condition for the tested configurations:

max(|j₁−j₂|, |j₃−j₄|) > min(j₁+j₂, j₃+j₄)

Scope, Validation & Limitations

• Scope: example numerical scans of the LQG volume operator kernel for 4-valent nodes within user-configurable spin ranges. Results are empirical and depend on the sampled range and discretization; treat summary counts as illustrative of the tested parameter set.
• Validation: run python scripts/analyze_zero_volume_states.py --help to see runtime options. To reproduce example results, run the script with the default parameters and attach the generated results/ artifacts (JSON/CSV + plotting scripts) and the runtime metadata (seed, environment, args).
• Limitations: the scripts perform finite, discrete scans. Absence of a configuration in the sampled set does not constitute a mathematical proof of absence. For stronger claims, increase sampling density, include symbolic or analytic checks where possible, and publish sensitivity analyses demonstrating robustness to sampling choices.

Directory Structure

├── scripts/
│   ├── analyze_zero_volume_states.py
│   ├── generate_latex_table.py
│   ├── generate_spin_half_correlation.py
│   ├── compute_volume_spectrum.py
│   └── ...
├── results/
│   ├── trivial_zero_volume_table.tex
│   ├── zero_volume_catalog.json
│   ├── kernel_dimension_distribution.png
│   └── figures/
│       └── spin_half_correlation.png
├── data/
│   └── volume_spectrum.csv
├── README.md
└── requirements.txt