Version 1.4 — 13 Aug 2025 (Added spin-gate table, wave equation derivation, cliff plot code, photon-photon units tie; aligned with Forces v1.1, Relativistic Effects v1.5)
Contents
1 Why a dedicated definition?
The forthcoming Photon Dynamics & Electromagnetism in RTG assumes a photon object already exists. Here we pin down that object from first principles so every simulation or derivation starts with a corpus-consistent definition.
Units note — All ω, Δω, δω in rad s⁻¹; Hz via /2π. σexch = independent UV regulator for exchange (typically O(Δω*)), σnoise = dimensionless CHSH/noise parameter (Quantum Behaviours v1.0). ħ, c explicit (ħc = 3.16×10⁻²⁶ J·m). Δω* = (1.45 ± 0.08)×10²³ s⁻¹ (Δω*/2π ≈ 2.31×10²² Hz) from two-loop RG v1.3.1.
2 Single-Photon Node Construction
- Frequency locking. All constituent nodes share carrier ωγ (visible photons: 3×10¹⁴ – 3×10¹⁵ rad s⁻¹ ≪ Δω*).
- Kernel maximisation. Gated kernel: \[ \mathcal R_{ij}=\frac34[1+\cos(\Delta\phi_{ij})]\,G_{ij}\,e^{-(\Delta\omega_{ij}/\Delta\omega^*)^2}, \quad G_{ij}=\frac{1-\sigma_i\sigma_j}{2} \] peaks at ℛij=3 for Δφ=0, G=1. In theory spins s=±i, so (+i, −i) ⇒ G=1.
| Convention | Pair | Gate |
|---|---|---|
| Analytic ±i | (+i, −i) | 1+s_i s_j=2 (open) |
| Analytic ±i | (+i, +i) or (−i, −i) | 0 (closed) |
| Code ±1 | (+1, −1) | 1-σ_iσ_j=2 (open) |
| Code ±1 | (+1, +1) or (−1, −1) | 0 (closed) |
Open for anti-aligned analytic spins, opposite code σ (Core Principles v1.3 §2).
- Bond-energy context. (Forces v1.1) \[ E_{ij} = K’ \frac{|\omega_i – \omega_j|}{\Delta\omega^*} + J\,\mathcal{R}_{ij} + J_{\mathrm{ex}}\sin(\Delta\phi_{ij})e^{-(\Delta\omega_{ij}/\sigma_{\mathrm{exch}})^2} \]
- Masslessness. Δω=0 ⇒ m=ħΔω/c²=0 ⇒ massless composite.
3 Null Propagation, Energy & Momentum
Centre-of-phase Φ obeys wave equation1:
\[ \partial_t^2\Phi – c^2\nabla^2\Phi = 0 \]
⇒ v= c, λγ=2πc/ωγ, E=ħωγ, p=E/c.
3.1 Collective-Wave Equivalence
For λ≫a (a≈c/Δω*≈2×10⁻¹⁵ m), a spin-pair is the quantum of a larger packet obeying the same equation. Pulse energy Epulse ⇒ Nγ=Epulse/ħωγ quanta; phases add coherently ⇒ classical field (Photon Dynamics v1.3).
4 Polarisation in RTG
Internal SU(2) rotation of (+i, −i): θ=0,π ⇒ linear X/Y; θ=±π/2 ⇒ circular ± (Gauge Symm v1.4 §3). Gauge-free U(1) polarisation group recovered.
5 Bandwidth & Decoherence Cliff
Bandwidth >0.70 Δω* ≈ 1.02×10²³ s⁻¹ (~1.62×10²² Hz) ⇒ ℛij<1 ⇒ pair unbinds (SU(2)→U(1)² corridor; RG v1.3.1, Forces v1.1). Experimental: high-Q ring-down cavity swept to δω≈1.0×10²³ s⁻¹ ⇒ (5±1)% Q-drop (Thermodynamics v1.6); lensing offsets (Relativistic Effects v1.5).
import numpy as np, matplotlib.pyplot as plt
δω = np.linspace(0, 1.5, 200)
R = np.exp(-δω**2)
plt.plot(δω, R)
plt.axvline(0.7, ls='--', color='r', label='Cliff 0.70 Δω*')
plt.xlabel('δω / Δω*'); plt.ylabel('R_ij (norm.)')
plt.title('Resonance kernel vs bandwidth')
plt.legend(); plt.grid(); plt.tight_layout()
plt.savefig('cliff_plot.png')
6 Leading Interactions
- Photon–photon. Elastic exchange ∼ J²/Δω* ≈ 10⁻⁵ × QED σγγ (HL-LHC testable; Forces v1.1 §6).
- Photon–gravity. Phase-collapse near massive bodies via residual-elastic kernel; +0.01″ lensing surplus (Relativistic Effects v1.5).
7 Key Take-aways & Corpus Links
- RTG photon = spin-anti-spin pair at ℛij max.
- Wave–field dual for λ≫a.
- Travels at c; E=ħω, p=E/c; bosonic under pair exchange.
- Cliff at 0.70 Δω* ⇒ (~5 %) Q-drop + lensing offsets.
- Links: Forces v1.1 | RG v1.3.1 | Gauge Symm v1.4 | Relativistic Effects v1.5 | Thermodynamics v1.6.
Appendix A — Mini-Glossary
| Symbol | Description |
|---|---|
| ωγ | photon angular frequency (rad s⁻¹) |
| Δω* | critical bandwidth = 1.45×10²³ s⁻¹ (2.31×10²² Hz) |
| Gij | spin gate = (1 − σiσj)/2 |
| σexch | exchange-term UV regulator ≈ O(Δω*) |
| σnoise | dimensionless CHSH/noise amplitude |
| a | lattice spacing ≈ c/Δω* ≈ 2×10⁻¹⁵ m |
Change Log
| Version | Date | Main updates |
|---|---|---|
| 1.0 | 2025-07-31 | Spin-kernel proof, units, E/p note, decoherence-cliff fig. |
| 1.1 | 2025-07-31 | Wave–field paragraph, footnote, cross-links. |
| 1.2 | 2025-07-31 | Kernel logic, refined field-mode text, Bose/commutator note, cavity-Q prediction. |
| 1.3 | 2025-08-13 | Explicit units & σ terms; kernel with gate; bond-energy context; 0.70 Δω* numeric; added glossary. |
| 1.4 | 2025-08-13 | Added spin-gate table, wave eq derivation, cliff plot code, photon-photon units tie; aligned with Forces v1.1, Relativistic Effects v1.5. |
1 Derivation: small-angle expand Eij ≈ (J/2)(Δφij)² for open gate ⇒ φ̈i ≈ J a² ∇² φi (lattice Laplacian). Second time derivative ⇒ continuum d’Alembertian (Forces v1.1 §4).