Loop Quantum Gravity Fractal
Creators: name: Scott Monroe, Scott
Publisher: name: Zenodo
Year: 2015
Subjects: subject: Quantum physics
subject: loop quantum gravity
subject: Physical cosmology
subject: astrophys

description: Using My Expanding Zoom-Out Recursive Fractal Algorithm to Support Loop Quantum Gravity and the Quantum Bounce

Abstract

In this paper, I present my Expanding Zoom-Out Recursive Fractal Algorithm as a potential framework to visualize and analyze space-time quantization in Loop Quantum Gravity (LQG). My fractal algorithm introduces a self-similar, discrete structure governed by recursive expansion rules dependent on the Hubble parameter. I propose that this model can illustrate key LQG concepts, including spin network evolution, space-time discreteness, and the quantum bounce. Mathematical comparisons between my fractal's equations and known LQG formulations support the idea that recursive fractal geometry can model the quantum fabric of space-time.

Introduction

Loop Quantum Gravity (LQG) predicts that space-time is fundamentally discrete, composed of quantized units connected by spin networks. Unlike classical General Relativity, which assumes a smooth continuum, LQG describes the universe as evolving through discrete steps. My fractal algorithm follows a similar recursive structure, making it a suitable candidate for modeling LQG principles. In this work, I explore how my fractal's recursive expansion laws can represent discrete space-time, mimic spin network dynamics, and provide insight into the quantum bounce that replaces the traditional Big Bang singularity.

Fractal Model and Recursive Equations

My Expanding Zoom-Out Recursive Fractal Algorithm is defined by a set of recursive equations that govern its expansion and rotation. These equations include a scaling function, an escape radius condition, and recursive tiling transformations.

1. **Scaling Function:**  R_n = R_0 * (H_0 / H(t))^n  - Controls fractal expansion in relation to the Hubble parameter.

2. **Escape Radius Condition:**  R_esc(t) = c / H(t)  - Defines a natural boundary for structure formation.

3. **Recursive Growth Rule:**  Z_{n+1} = Z_n + (a_n * Z_n) + (b_n * R_n) + g_n  - Governs hierarchical fractal expansion with time-dependent coefficients.

4. **Rotation Function:**  B(t) = B_0 * exp(-t / tau)  - Introduces dynamic rotational transformations.

Connections to Loop Quantum Gravity

LQG describes space-time using spin networks, which evolve in discrete steps. My fractal model exhibits a similar discrete evolution, where each recursion step represents a quantum transition in space-time. The mathematical parallels between my fractal and LQG suggest that recursive fractal geometry could serve as a powerful tool for understanding space-time quantization.

Mathematical Comparisons to LQG Concepts

1. **Discrete Space-Time Quantization:**

  - LQG Area Quantization:  A_j = 8πγ l_p^2 Σ_i sqrt(j_i(j_i+1))

  - My Fractal Scaling:  R_n = R_0 * (H_0 / H(t))^n

2. **Evolution via the Hamiltonian Constraint:**

  - LQG Evolution:  Ψ_{n+1} = U_n * Ψ_n

  - My Fractal Recursion:  Z_{n+1} = Z_n + (a_n * Z_n) + (b_n * R_n) + g_n

3. **Avoidance of Singularities and the Quantum Bounce:**

  - LQG Bounce Equation:  H^2 = (8πG / 3) ρ (1 - ρ / ρ_c)

  - My Fractal Escape Condition:  R_esc(t) = c / H(t)

Implications and Future Research

If my fractal model accurately captures quantum gravity effects, it could provide a new visualization framework for LQG and its implications for early universe physics. Potential research directions include numerical simulations to test whether fractal recursion can recover known LQG predictions, such as discrete area spectra and quantum gravity corrections to cosmology.

Conclusion

My Expanding Zoom-Out Recursive Fractal Algorithm demonstrates strong parallels with Loop Quantum Gravity, particularly in its discrete, self-similar structure and recursive expansion laws. By mathematically comparing my fractal to LQG’s known formulations, I have shown that fractal geometry may provide an intuitive and computationally powerful tool for studying quantum space-time. Future work will focus on refining this approach and exploring its implications for quantum gravity research.