lubon-visualization

"Interactive 3D visualization of the Lubón particle, linked to the Riemann zeta function zeros."

3D Visualization of the Lubón Particle

This repository provides an interactive 3D visualization of the hypothetical Lubón (Lub) particle, a scalar boson linked to the non-trivial zeros of the Riemann zeta function ( \zeta(s) ). The visualization is implemented in a Jupyter Notebook, optimized for Binder, and features animations, sliders, and detailed annotations.

Overview

The Lubón particle is a theoretical construct with the following properties:

• Type: Scalar boson (spin-0)
• Mass: ~10⁻³ eV/c² (lighter than neutrinos)
• Energy: ( E_n = \hbar \omega \gamma_n ), where ( \gamma_n ) are the imaginary parts of the zeta function's non-trivial zeros
• Nature: Resonance in a quantum system with chaotic dynamics (GUE statistics)
• Connection: Defined by the Riemann zeta function ( \zeta(s) )
• Potential Detection: Quantum dots, photonic crystals, or cosmological observations
• Dark Matter: Possible candidate
• Significance: A bridge between mathematics and physics

The visualization plots the probability density ( |\text{lub}n(t)|^2 ) and the effective potential ( V{\text{eff}}(t) = -k |\zeta(1/2 + it)|^2 ) in 3D space, corresponding to the first three zeros (( \gamma_1 = 14.1347 ), ( \gamma_2 = 21.0220 ), ( \gamma_3 = 25.0108 )).

Features

• Exact Eigenfunctions: Computes ( \text{lub}n(t) ) as eigenfunctions of the operator ( \hat{H} ) with kernel ( K{\text{sym}}(t, t') ).
• Animation: Visualizes the phase evolution ( e^{-i \omega \gamma_n t} ) with "Play"/"Pause" buttons.
• Sliders: Allows switching between ( \gamma_n ) for different ( |\text{lub}_n(t)|^2 ).
• Annotations: Marks ( \gamma_n ) positions on the graph.
• Surface Plot: Displays a semi-transparent surface for ( V_{\text{eff}}(t) ).
• Centered Visualization: Graph is centered with optimized camera angles.
• Export: Saves the plot as lubon_visualization.html for offline viewing.
• Colors:
  • Blue: ( |\text{lub}_1(t)|^2 ) (( \gamma_1 = 14.1347 ))
  • Green: ( |\text{lub}_2(t)|^2 ) (( \gamma_2 = 21.0220 ))
  • Purple: ( |\text{lub}_3(t)|^2 ) (( \gamma_3 = 25.0108 ))
  • Red (dashed): ( V_{\text{eff}}(t) )
  • Red (semi-transparent): Surface of ( V_{\text{eff}}(t) )

How to Use

1. Launch in Binder:
   • Click the Binder badge above or visit mybinder.org and enter the repository URL (https://github.com/YYurq/lubon-visualization).
   • Wait for the environment to load.
2. Run the Notebook:
   • Open binder-lubon.ipynb.
   • Execute the first cell to install dependencies.
   • Execute the second cell to generate the visualization.
3. Interact:
   • Use sliders to switch between ( \gamma_n ).
   • Click "Play"/"Pause" for animation.
   • Rotate and zoom the 3D plot.
   • Download lubon_visualization.html for offline use.
4. Local Setup (optional):
   • Clone the repository:

     git clone https://github.com/YYurq/lubon-visualization.git
     cd lubon-visualization

   • Create a virtual environment and install dependencies:

     python -m venv env
     source env/bin/activate  # On Windows: env\Scripts\activate
     pip install -r requirements.txt

   • Run the notebook:

     jupyter notebook binder-lubon.ipynb

Repository Structure

• binder-lubon.ipynb: Jupyter Notebook with the visualization code.
• requirements.txt: Python dependencies.
• environment.yml: Binder environment configuration.
• .github/workflows/binder.yml: GitHub Actions for Binder automation.
• README.md: This file.

Dependencies

• Python 3.11
• numpy==1.26.4
• matplotlib==3.9.2
• mpmath==1.3.0
• scipy==1.14.1
• plotly==5.24.1
• sympy==1.13.3

Results

The notebook generates an interactive 3D plot with:

• Animated probability densities ( |\text{lub}_n(t)|^2 ) for three zeta zeros.
• Effective potential ( V_{\text{eff}}(t) ) as a dashed line and semi-transparent surface.
• Annotations marking ( \gamma_n ).
• Sliders for selecting different ( \gamma_n ).
• Centered visualization with export to HTML.
• Console output describing the Lubón particle and color legend.

Notes

• The eigenfunctions ( \text{lub}_n(t) ) are computed numerically, which may take a few seconds in Binder.
• For faster performance, reduce the number of points (e.g., change t = np.linspace(-T, T, 300) to 200).
• The animation uses a short time interval (( 10^{-12} , \text{s} )) for clarity; adjust time in np.linspace for longer animations.

Contributing

Contributions are welcome! Please open an issue or submit a pull request for improvements, such as:

• Enhanced visualizations (e.g., additional animations).
• Optimization of eigenfunction calculations.
• Support for more zeta zeros.

License

MIT License. See LICENSE for details.

Contact

For questions, contact y.yurqa@gmail.com.