Golden Recursive Projection (GRP): Lagrangian Formulation

Introduction

The Golden Recursive Projection (GRP) framework aims to unify General Relativity (GR) and Quantum Field Theory (QFT) by embedding space-time and quantum interactions in a deeper recursion-layer structure governed by golden-ratio wavelet projections.

To fully formalize this approach, we develop a Lagrangian formulation that:

  1. Recovers General Relativity in the classical limit.
  2. Recovers Quantum Field Theory in the quantum limit.
  3. Includes recursion-layer corrections, naturally regulating divergences and providing a self-consistent description of quantum gravity.

1. GRP Action: Unifying Gravity and QFT

A natural starting point is the modified Einstein-Hilbert action, incorporating recursion-layer effects:

\[ S_{\text{GRP}} = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi G} + \int_0^{\infty} e^{-\lambda \phi^n} \mathcal{L}_{\text{QFT}}(x, n) dn \right] \]

where:

  • \( R \) is the Ricci scalar, governing spacetime curvature.
  • \( \mathcal{L}_{\text{QFT}}(x, n) \) is the QFT Lagrangian modified by recursion-layer effects.
  • \( e^{-\lambda \phi^n} \) is the recursion-layer suppression function, filtering high-energy divergences.
  • The integral over recursion layers models the transition from quantum to classical physics.

2. Quantum Field Theory in the GRP Framework

To describe fundamental particle interactions within GRP, we incorporate fermionic and gauge fields:

\[ \mathcal{L}{\text{QFT}}(x, n) = \sum_i \bar{\psi}_i \left( i \gamma^\mu D\mu - m_i \right) \psi_i + \frac{1}{4} F_{\mu\nu}^a F^{\mu\nu}_a \]

where:

  • \( \psi_i \) are fermionic fields (electrons, quarks, neutrinos).
  • \( D_\mu \) is the covariant derivative, including gauge interactions.
  • \( F_{\mu\nu}^a \) are the gauge field strength tensors (for photons, gluons, weak bosons).
  • The recursion-layer integral applies to all QFT interactions, modifying their renormalization behavior.

By integrating over recursion layers, we ensure that QFT fields couple to hidden-layer structures, preventing the need for artificial renormalization.

3. Recursion-Layer Contribution to Gravity

To include recursion-layer corrections to GR, we introduce a wavelet recursion field \( \Psi_{\phi} \):

\[ \mathcal{L}{\text{recursion}} = \alpha \nabla^\mu \Psi{\phi} \nabla_\mu \Psi_{\phi} - V(\Psi_{\phi}) \]

where:

  • \( \Psi_{\phi} \) describes recursive interactions modifying spacetime curvature.
  • \( \alpha \) controls recursion-layer coupling strength.
  • \( V(\Psi_{\phi}) \) introduces self-interaction terms to balance recursion constraints.

The total Lagrangian in GRP is thus:

\[ \mathcal{L}{\text{GRP}} = \frac{R}{16\pi G} + \mathcal{L}{\text{recursion}} + \int_0^{\infty} e^{-\lambda \phi^n} \mathcal{L}_{\text{QFT}}(x, n) dn \]

4. Recovering GR and QFT as Special Cases

Classical Limit (\( \lambda \to 0 \)) → General Relativity

  • The recursion-layer integral filters out, reducing to standard GR: \[ \mathcal{L}_{\text{GRP}} \approx \frac{R}{16\pi G} \]
  • Gravity remains the dominant interaction, with negligible quantum effects.

Quantum Limit (\( \lambda \) large) → Standard QFT

  • The curvature term becomes negligible, leaving: \[ \mathcal{L}{\text{GRP}} \approx \mathcal{L}{\text{QFT}} \]
  • This recovers the Standard Model Lagrangian, but with recursion-layer modified renormalization.

5. Implications and Future Directions

The GRP Lagrangian formulation suggests several novel physical effects: ✅ Quantum Gravity Regularization: Recursion-layer suppression removes divergences in QFT. ✅ Black Hole Structure: No singularities—black hole interiors are hidden recursion layers. ✅ Dark Energy as a Recursion Effect: Late-time cosmic acceleration emerges naturally from filtering deep layers. ✅ Modified Uncertainty Principle: Small-scale deviations in quantum mechanics could be experimentally tested.

Future research will:

  • Derive the equations of motion from this Lagrangian.
  • Investigate quantization of \( \Psi_{\phi} \) for a fully quantum gravity theory.
  • Explore experimental tests for recursion-layer effects in gravitational waves, black hole mergers, and quantum systems.

The GRP framework offers a novel approach to unifying physics, embedding both gravity and quantum mechanics in a recursively structured spacetime.