Contents
1. Introduction
This workbook outlines a systematic derivation from the lattice Hamiltonian of Relational Time Geometry (RTG) to its continuum action, incorporating \(\Delta\omega^* = (1.45 \pm 0.08) \times 10^{23} \, \text{s}^{-1}\) from the two-loop RG analysis.
2. Lattice Hamiltonian
The lattice Hamiltonian, including resonance effects, is:
\[ S_{\text{lattice}} = \sum_{\langle ij \rangle} \left[ K_{\text{lattice}} |\omega_i – \omega_j| \cdot \mathcal{R}_{ij} + J_{\text{lattice}} \frac{3}{4} \left( \psi_i^\dagger \psi_i + \psi_j^\dagger \psi_j + 2 \Re[\psi_i^\dagger \psi_j] \right) \cdot \mathcal{R}_{ij} \right] \]
Where \(\psi_i = s_i e^{i\phi_i}\) (\(s_i = \pm i\), \(|\psi_i|^2 = 1\)), and the resonance kernel is:
\[ \mathcal{R}_{ij} = \frac{3}{4} [1 + \cos(\phi_i – \phi_j)] (1 + s_i s_j) e^{-(\omega_i – \omega_j)^2 / (\Delta\omega^*)^2} \]
(Note: Excluding the exponential falloff simplifies the model; inclusion reduces \(J_{\text{eff}}\) by ~10% due to frequency damping.)
3. Continuum Action
The continuum action, reflecting relational interactions, is:
\[ S_{\text{cont}} = \int d^3x \left[ \frac{1}{2} K_{\text{cont}} (\nabla \omega)^2 \cdot \langle \mathcal{R} \rangle + J_{\text{cont}} \frac{3}{2} (\psi^\dagger \psi)^2 \cdot \langle \mathcal{R} \rangle + \frac{1}{2} m^2 \psi^\dagger \psi \right] \]
Where \(\psi(x)\) encodes spin and phase, with \((\psi^\dagger \psi)^2\) modeling phase-coherent energy density, scaled by \(\langle \mathcal{R} \rangle \approx 1 \pm 0.1\) (reflecting phase misalignment).
4. Frequency Term Mapping
For \(|\omega_i – \omega_j| \ll \omega_{\text{avg}} \approx 10\%\), the lattice term approximates: \[ K_{\text{lattice}} |\omega_i – \omega_j| \cdot \mathcal{R}_{ij} \approx \frac{1}{2} K_{\text{eff}} (\omega_i – \omega_j)^2 \cdot \mathcal{R}_{ij} \]
With \(K_{\text{eff}} = \frac{2 K_{\text{lattice}}}{a}\) (from \(\frac{d}{d\omega} |\omega| \approx 2\omega\) at small differences), and a 3D cubic lattice (6 neighbors), the continuum limit is: \[ \sum_{\langle ij \rangle} \frac{1}{2} K_{\text{eff}} (\omega_i – \omega_j)^2 \cdot \mathcal{R}_{ij} \approx \int d^3x \frac{1}{2} K_{\text{eff}} \cdot 6 a^2 (\nabla \omega)^2 \cdot \langle \mathcal{R} \rangle \]
Thus, \(K_{\text{cont}} = 6 K_{\text{lattice}} a \cdot \langle \mathcal{R} \rangle\). With \(a = c / \Delta\omega^* \approx 2.066 \times 10^{-16} \, \text{m}\), \(K_{\text{cont}} \approx 1.24 \times 10^{-15} K_{\text{lattice}} \cdot \langle \mathcal{R} \rangle\).
Footnote: The quadratic approximation holds reliably when \(|\omega_i – \omega_j| \lesssim 10\% \omega_{\text{avg}}\). Outside that range, higher-order terms renormalize \(K_{\text{cont}}\) nontrivially.
5. Interaction Term Mapping
The lattice interaction term: \[ \sum_{\langle ij \rangle} J_{\text{lattice}} \frac{3}{4} (2 + 2 \cos(\phi_i – \phi_j)) \cdot \mathcal{R}_{ij} \approx \int d^3x 3 J_{\text{lattice}} (\psi^\dagger \psi)^2 \cdot \langle \mathcal{R} \rangle \]
With \(\cos(\phi_i – \phi_j) \approx 1\) and \(\mathcal{R}_{ij} \approx \langle \mathcal{R} \rangle\), \(J_{\text{cont}} = J_{\text{lattice}} \cdot \langle \mathcal{R} \rangle\), with \(\langle \mathcal{R} \rangle\) varying ±10% based on node alignment.
In practical simulations, \(\langle \mathcal{R} \rangle\) may vary ±10% based on node alignment; include this as an error bound when estimating \(J_{\text{cont}}\).
6. Effective Coupling in Continuum
The effective coupling is: \[ g = \frac{\frac{3}{2} J_{\text{cont}}}{K_{\text{cont}}} \]
Substituting: \[ g = \frac{\frac{3}{2} J_{\text{lattice}} \cdot \langle \mathcal{R} \rangle}{6 K_{\text{lattice}} a \cdot \langle \mathcal{R} \rangle} = \frac{1}{4} \frac{J_{\text{lattice}}}{K_{\text{lattice}} a} \]
Clarification: Setting lattice spacing \(a = 1\) removes dimensional scaling from \(g\), yielding \(g \approx 0.2145\) as a dimensionless coupling. The large value (\(\approx 1 \times 10^{14}\)) in SI units is for dimensional consistency only.
Links to the two-loop \(\beta_g\) from “Two-Loop RG Derivation” (2025), where \(g^* \approx 1.14 \pm 0.02\).
7. Numerical Example
With \(K_{\text{lattice}} = 0.0904291\), \(J_{\text{lattice}} = 0.07760878\), and \(a \approx 2.066 \times 10^{-16} \, \text{m}\): \[ g = \frac{1}{4} \frac{0.07760878}{0.0904291 \cdot 2.066 \times 10^{-16}} \approx 1.04 \times 10^{14} \]
Rescaling \(J_{\text{lattice}} / a\) yields \(g \approx 0.2145\). The positive \(\beta_g(0.2145) \approx 0.72 \cdot 0.2145 – 0.63 \cdot (0.2145)^2 – 0.011 \cdot (0.2145)^3 \approx 0.154 > 0\) confirms UV flow toward stronger IR coupling. To access perturbative IR dynamics (\(g \approx 1.0–1.2\)), raise \(J_{\text{lattice}}\) from 0.0776 to \(\approx 0.36\) (for \(K_{\text{lattice}} \approx 0.09\)), placing simulations within the RG critical regime.
8. Conclusion
This workbook maps RTG’s lattice to continuum, integrating \(\Delta\omega^*\) and \(\mathcal{R}_{ij}\). The derivation enables precise predictions for dimensional transitions and resonance coherence.
9. Uncertainty & Parameter Sensitivity
- Residual \(\langle \mathcal{R} \rangle\) variability: ±10% ⇒ ±10% shifts in \(K_{\text{cont}}\), \(J_{\text{cont}}\).
- \(\Delta\omega^*\) uncertainty (±5%) impacts \(a\): influences \(g\) by a factor \((1 \pm 0.05)\).
- Recommendations: Perform sensitivity scans in simulation across \(\Delta\omega^*\), \(\langle \mathcal{R} \rangle\), and \(J_{\text{lattice}}\) to quantify effects.