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.

📖 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 spin configurations (j=0.5 to 3.0).
• Identifies trivial zero-volume states (J₁₂ ∩ J₃₄ = ∅).
• Confirms no non-trivial zero-volume states in the tested range.
• 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: 1,296 (j_i ∈ {0.5, 1.0, …, 3.0})
• Trivial zero-volume states: 60 (4.6%)
• Non-trivial zero-volume states: 0 (0.0%)
• Full-rank matrices (non-zero volume): 674 (52.0%)
• Other kernel matrices (not zero-volume): 562 (43.4%)

All trivial zero-volume cases satisfy the Diophantine condition:

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

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