---
paper_id: "PAPER_1103"
title: "LQG Buoyancy Sector Lagrangian Variation with Spin-Foam Vertex Coupling"
session: 225
date: 2026-04-15
author: "Daniel T. Murphy"
status: production
cvw: "v2.0.0"
tags: [LQG, Lagrangian, buoyancy-sector, spin-foam, EPRL, vertex-amplitude, phonon-coupling, Euler-Lagrange, UQFF]
crosslinks: [PAPER_579, PAPER_1094, PAPER_1095, PAPER_1100, PAPER_1102]
sm_anchor: "CVW v2.0.0 -- G6 SM Anchor Gate compliant"
---

# PAPER\_1103: LQG Buoyancy Sector Lagrangian Variation with Spin-Foam Vertex Coupling

## Abstract

We construct the Lagrangian for the LQG buoyancy sector, coupling the
spin-foam vertex amplitude $A_v(j,i_e)$ to the UQFF buoyancy scalar field
$\phi_{\text{LQG}}$ and the 1.25 THz phonon mode.  The Euler-Lagrange
equation $\delta S/\delta\phi_{\text{LQG}} = 0$ yields an equation of motion
with spin-foam source terms and phonon-induced effective mass corrections.

## 1. Lagrangian Density

$$\mathcal{L}_{\text{LQG}} = \tfrac{1}{2}(\partial_\mu\phi_{\text{LQG}})^2 - V(\phi_{\text{LQG}}) + \lambda_{\text{spin}}\sum_v A_v(j,i_e)\cdot\phi_{\text{LQG}} + g_{\text{phonon}}\,\Phi_{1.25\,\text{THz}}\,\phi_{\text{LQG}}^2$$

### Terms

| Term | Physical origin |
|------|----------------|
| $\frac{1}{2}(\partial_\mu\phi)^2$ | Kinetic energy of the buoyancy field |
| $V(\phi) = \frac{1}{2}\omega_\phi^2\phi^2$ | Harmonic potential, $\omega_\phi = m_\phi c^2/\hbar$ |
| $\lambda_{\text{spin}}\sum_v A_v\cdot\phi$ | Spin-foam vertex coupling (EPRL/FK) |
| $g_{\text{phonon}}\Phi\phi^2$ | Phonon-buoyancy self-interaction |

## 2. Spin-Foam Vertex Amplitude

Using the Ponzano-Regge/EPRL approximation:

$$A_v(j) = (2j+1)\cdot\exp(-\gamma j)$$

where $\gamma = 0.2375$ is the Barbero-Immirzi parameter.  The sum over
$N_v$ vertices:

$$\sum_v A_v = N_v\cdot(2j_{\text{avg}}+1)\cdot\exp(-\gamma j_{\text{avg}})$$

## 3. Equation of Motion

Varying $\delta S/\delta\phi_{\text{LQG}} = 0$:

$$\Box\phi_{\text{LQG}} + \omega_\phi^2\phi_{\text{LQG}} = \lambda_{\text{spin}}\sum_v A_v + 2g_{\text{phonon}}\,\Phi\,\phi_{\text{LQG}}$$

This is a **Klein-Gordon equation with spin-foam source and phonon-modified
mass**:

$$\Box\phi + m_{\text{eff}}^2\phi = J_{\text{spin-foam}}$$

where:

$$m_{\text{eff}}^2 = \omega_\phi^2 - 2g_{\text{phonon}}\Phi$$

$$J_{\text{spin-foam}} = \lambda_{\text{spin}}\sum_v A_v(j,i_e)$$

## 4. Tachyonic Instability Condition

If $2g_{\text{phonon}}\Phi > \omega_\phi^2$, the effective mass squared
becomes negative, signaling a tachyonic instability.  This triggers
spontaneous symmetry breaking in the buoyancy sector, with the new vacuum:

$$\langle\phi\rangle = \pm\sqrt{\frac{J_{\text{spin-foam}}}{m_{\text{eff}}^2}}$$

## 5. Static Equilibrium

At static equilibrium ($\Box\phi = 0$), the balance condition:

$$\omega_\phi^2\phi_0 = \lambda_{\text{spin}}\sum_v A_v + 2g_{\text{phonon}}\Phi\phi_0$$

The equation of motion residual $R$:

$$R = V'(\phi_0) - J_{\text{spin-foam}} - 2g_{\text{phonon}}\Phi\phi_0$$

with balance ratio $|R|/|V'(\phi_0)|$ indicating proximity to equilibrium.

## 6. Implementation

Calculator: `LQGBuoyancySectorLagrangianVariationCalculator` in CondensedPhysics.py

- Inputs: field amplitude $\phi_0$, mass $m_\phi$, couplings $\lambda_{\text{spin}}$ and $g_{\text{phonon}}$, vertex count $N_v$, average spin $j_{\text{avg}}$
- Outputs: Lagrangian density, spin-foam and phonon source terms, equation of motion residual, effective mass, tachyonic flag

## 7. Conclusion

The LQG buoyancy sector Lagrangian provides the variational framework for
deriving the dynamics of the UQFF buoyancy field in the spin-foam formalism.
The phonon-induced effective mass correction and the spin-foam source term
together determine the equilibrium configuration and stability of the
buoyancy sector.

## References

- Engle, J., Livine, E., Pereira, R. & Rovelli, C. (2008). LQG vertex with finite Immirzi parameter. *Nucl. Phys. B* 799, 136.
- Murphy, D.T. (2025). UQFF: Star Cradle Mechanics framework. Star-Magic repository.
- PAPER\_1094, PAPER\_1095: CMB and horizon buoyancy sector Lagrangians.


### Key References with arXiv/DOI Identifiers

1. Abbott et al. (LIGO Scientific and Virgo Collaborations, 2016). *Observation of Gravitational Waves from a Binary Black Hole Merger.* Phys. Rev. Lett. **116**, 061102 — arXiv:1602.03837 — doi:10.1103/PhysRevLett.116.061102
2. Murphy, D. (2026). *Unified Quantum Field Framework (UQFF): Star-Magic v5.x Whitepaper Series.* Star-Magic Repository — github.com/Daniel8Murphy0007/Star-Magic
