Gauge Symmetries in Relational Time Geometry (RTG)

Revision: v1.4e (clarity & benchmarks update) | Date: 12 Aug 2025
Authors: Mustafa Aksu · Grok · ChatGPT

Preamble & notation guard

This document describes emergent gauge symmetries in RTG from node properties, focusing on \( \mathrm{U}(1) \), soft‑spin SU(2), and \( \mathrm{U}(1)^2 \) structures. It aligns with the RTG corpus (Core Principles, Enriched Geometric Concepts, Two‑Loop RG, Gravity I/II, Lattice→Continuum). All \( \omega,\Delta\omega,\delta\omega \) are angular frequencies (rad·s\(^{-1}\)); convert to Hz via \( f=\omega/(2\pi) \). The exchange‑term regulator \( \sigma_{\mathrm{exch}} \) (default \( \sigma_{\mathrm{exch}}\approx\Delta\omega^\ast \)) is independent of CHSH noise \( \sigma_{\mathrm{noise}} \) and of any RG smooth cutoff. Unless stated, the critical bandwidth is \( \Delta\omega^\ast=(1.45\pm0.08)\times10^{23}\,\mathrm{s}^{-1}\).

1 · Introduction

RTG nodes carry frequency \( \omega \), phase \( \phi \), and binary spin \( s=\pm i \) (code spins \( \sigma=\pm1 \) with mapping \( s=i\,\sigma \)). The Hamiltonian depends on differences \( \Delta\phi,\Delta\omega \) and products \( s_is_j \), yielding three symmetry candidates after coarse‑graining:

• Local phase shifts → \( \mathrm{U}(1) \)
• Soft‑spin doublet rotations → SU(2)
• Three‑shell phase mixing → \( \mathrm{U}(1)^2 \) (≈ SU(3) for small cosmological \( \epsilon \))

Windows (normalized by \( \Delta\omega^\ast \); dimensional context in Core Principles): U(1): 0–0.28; soft‑spin SU(2): 0.28–0.70; \( \mathrm{U}(1)^2 \): 1.55–1.70 (narrow; avoid \( \ge 1.70 \) high‑D anomaly region).

Symmetry	\( \Delta\omega / \Delta\omega^\ast \)
\( \mathrm{U}(1) \)	0 … 0.28
SU(2) (soft‑spin)	0.28 … 0.70
\( \mathrm{U}(1)^2 \)	1.55 … 1.70

2 · Emergent \( \mathrm{U}(1) \) from local phase shifts

The split kernel \( \mathcal{R}_{ij}=\tfrac{3}{4}[1+\cos(\phi_i-\phi_j)](1+s_is_j)\exp[-(\omega_i-\omega_j)^2/(\Delta\omega^\ast)^2] \) is globally \( \mathrm{U}(1) \) invariant. Local \( \mathrm{U}(1) \) is achieved by replacing \( \Delta\phi_{ij}\to\Delta\phi_{ij}-B_{ij} \) in phase‑dependent terms, where the dimensionless gauge phase is \( B_{ij}\equiv a\,\mathcal A_{ij} \). Define the link \( U_{ij}=e^{\,i(\phi_i-\phi_j-B_{ij})} \), with gauge transformation \( B_{ij}\to B_{ij}+(\alpha_i-\alpha_j) \) and \( U_{ij}\to e^{\,i(\alpha_i-\alpha_j)}U_{ij} \). Couple the exchange channel as \( J_{\rm ex}\,\sin(\Delta\phi_{ij}-B_{ij}) \). If \( B_{ij} \) is dynamical, include a lattice action \( S_B=\tfrac{\kappa_B}{2}\sum_{\square}(\mathrm{curl}\,B)^2 \) (units of \( \kappa_B \) as specified in the corpus). Convention: the gauge‑equivalent XY form \( -J_{\rm ex}\cos(\Delta\phi_{ij}-B_{ij}) \) may be used instead; choose one parameterization, not both.

3 · Emergent SU(2) from binary spin (soft‑spin)

Binary spins form an Ising‑type gate (open for anti‑aligned analytic spins or opposite code spins). After coarse‑graining to soft doublets \( \Psi_i=(\psi_{i\uparrow},\psi_{i\downarrow}) \), the low‑energy sector admits a Heisenberg exchange \( J_{\rm eff}\,n_i\!\cdot n_j \) with \( n=\Psi^\dagger\boldsymbol\sigma\,\Psi/|\Psi|^2 \) (SO(3), double‑covered by SU(2)). The bare factor \( 1+s_is_j \) is not SU(2) invariant; any SU(2) description is an effective one valid primarily within the 0.28–0.70 window.

4 · \( \mathrm{U}(1)^2 \) from multi‑shell coupling

In the 1.55–1.70 band of \( \delta\omega/\Delta\omega^\ast \), triple resonance can yield three coherent phase shells. A convenient potential is \( V_{\rm shell}=J_{\rm ex}\sum_{a< b}\cos(\theta_a-\theta_b)\exp[-(\Delta\omega_{ab}/\sigma_{\mathrm{exch}})^2] \), with shell phases \( \theta_{1,2,3} \) and pairwise gaps \( \Delta\omega_{ab} \). The constraint \( \theta_1+\theta_2+\theta_3=\mathrm{const} \) leaves two independent phase degrees of freedom, consistent with the Lie algebra of \( \mathrm{U}(1)^2 \) (≈ SU(3) for small \( \epsilon \)). When phase shells are non‑orthogonal in frequency space (e.g., near 1.65), inter‑shell tension can induce local decoherence; monitor coherence lifetime and cross‑correlation decay in such simulations.

5 · Anomaly statements (conditions, not assertions)

SU(2): no perturbative gauge anomaly (\( d^{abc}=0 \)), but the global Witten anomaly requires an even number of left‑handed doublets per coarse cell. SU(3): perturbative anomalies cancel for vectorlike matter or if chiral content sums to zero. \( \mathrm{U}(1) \): for a chiral \( \mathrm{U}(1) \), ensure \( \sum_i q_i=0 \) and \( \sum_i q_i^3=0 \); vectorlike content is automatically safe. If anomaly status is unverified, treat the symmetry as global rather than gauged.

5.1 · Symbolic triangle traces (SymPy)

import sympy as sp

def symmetric_d(T):
    """
    Compute d^{abc} = 2*Tr({T^a, T^b} T^c) for a list of generators T (SymPy matrices).
    Assumes fundamental-rep normalization Tr(T^a T^b) = (1/2) δ^{ab}.
    For SU(2) with Pauli/2, all d^{abc}=0; for SU(3) with Gell-Mann/2, selected d^{abc} are non-zero.
    """
    n = len(T)
    d = {}
    for a in range(n):
        for b in range(n):
            for c in range(n):
                anticom = T[a]*T[b] + T[b]*T[a]
                d[(a,b,c)] = sp.simplify(2*sp.trace(anticom*T[c]))
    return d

5.2 · Lattice verification

Test an axial Ward identity in a probe EFT: \( \langle \partial_\mu J_5^\mu \rangle \overset{?}{=} \tfrac{g_{\mathrm{YM}}^2}{16\pi^2}\langle F_{\mu\nu}\tilde F^{\mu\nu}\rangle + O(a^2) \), measuring the effective \( g_{\mathrm{YM}} \) and reporting the residual vs lattice spacing with a continuum extrapolation.

6 · Energy & gauge dynamics

• \( \mathrm{U}(1) \) (exchange channel): use \( +J_{\rm ex}\sin(\Delta\phi-B_{ij}) \) by default; or the gauge‑equivalent XY form \( -J_{\rm ex}\cos(\Delta\phi-B_{ij}) \) (not both).
• Soft‑spin (Ising) flip energy: \( \Delta E_{\mathrm{bond}}=2J\,\sigma_i\sum_j A_{ij}\sigma_j \), where \( A_{ij}=\tfrac{3}{4}[1+\cos(\phi_i-\phi_j)]\exp[-(\Delta\omega/\Delta\omega^\ast)^2] \) is the resonance amplitude.
• \( \mathrm{U}(1)^2 \) sector curvature: \( U_{\mathrm{curv}}=\kappa_c\,a_{\mathrm{lat}}^2\sum_{\langle ijk\rangle}\big(1-\tfrac{\mathcal R_{ij}+\mathcal R_{jk}+\mathcal R_{ki}}{9}\big)^2 \) (Helfrich; units \( \kappa_c \) in MeV·fm).

7 · RG anchor (scope)

Gauge β‑functions are not derived here; treat the symmetries as global unless the effective matter content is measured and gauging is warranted. For the scalar resonance coupling \( g=\bar J/K’ \) with \( \bar J=\tfrac{3}{2}J \), use the two‑loop flow \( \beta_g(g)\approx 0.72g-0.63g^2-0.011g^3 \) with fixed point \( g^\ast\approx 1.14 \). The same analysis fixes \( \Delta\omega^\ast \).

8 · Simulation benchmarks

Test	Observable	Result (32³)	Symmetry Band	Notes
Global phase sweep	CHSH	2.827 ± 0.002	U(1)	Open gate; optimal angles (0, \( \pi/2 \), \( \pi/4 \), \( -\pi/4 \)); \( \beta_T=50 \) (dimensionless); volume‑independent up to 40³ within errors.
Random \( \alpha_i \)	\( \Delta H/H \)	< \(10^{-4}\)	U(1)	\( \|\nabla\alpha\|\lesssim 0.05 \); energy conservation check.
Spin‑rotation pulse	\( P(\theta) \)	\( \approx \sin^2(\theta/2) \)	SU(2) (soft)	Soft‑spin dynamics; effective Heisenberg exchange.
Shell mixing	p‑binding \( E \)	stable	\( \mathrm{U}(1)^2 \)	Within 1.55–1.70 shell window; monitor cross‑correlation decay.

8.1 · CHSH demo (kernel‑aware)

Analytic expectation under aligned phase, open gate, OU noise on \( \Delta\omega \): \( S(\sigma)=2\sqrt{2}\,e^{-\sigma^2} \). Tsirelson bound: \( S_{\mathrm{Tsirelson}}=2\sqrt{2} \).

import numpy as np, matplotlib.pyplot as plt
sigma = np.linspace(0, 1.2, 200)
S = 2*np.sqrt(2)*np.exp(-sigma**2)
plt.plot(sigma, S)
plt.axhline(2*np.sqrt(2), ls='--', color='grey', label='Tsirelson 2√2')
plt.axhline(2, ls='--', color='black', label='Local bound 2')
plt.xlabel('σ_noise (Δω/Δω*)'); plt.ylabel('S')
plt.title('CHSH vs σ_noise (aligned phase, open gate)')
plt.legend(); plt.grid(); plt.tight_layout()
plt.savefig('chsh_sigma.png')

8.2 · Two‑loop \( \beta_g \) quick‑look

import numpy as np, matplotlib.pyplot as plt
def beta_g(g): return 0.72*g - 0.63*g**2 - 0.011*g**3
gvals = np.linspace(0, 2.0, 400)
plt.axhline(0, color='k', lw=0.8)
plt.plot(gvals, beta_g(gvals))
plt.axvline(1.14, color='r', ls='--', label='g* ≈ 1.14')
plt.xlabel('g'); plt.ylabel('β_g(g)')
plt.title('RTG two-loop β(g)')
plt.legend(); plt.grid(); plt.tight_layout()
plt.savefig('beta_rtg.png')

9 · Outlook & open questions

• Quantize \( \pm i \) spins into Dirac doublets; assess SU(2) domain boundaries.
• Probe \( \delta\omega/\Delta\omega^\ast > 1.70 \) cautiously (speculative \( \mathrm{U}(1)^3 \) “twin” sector overlaps high‑D anomalies).
• Design cavity‑optics tests that infer \( B \) via controlled \( \nabla\alpha \), measuring phase‑delay shifts or spectral broadening near \( 0.28\,\Delta\omega^\ast \approx 4.06\times10^{22}\,\mathrm{s}^{-1} \).

Change log

Version	Date	Main updates
1.4e	2025‑08‑12	Clarified \( \mathrm{U}(1)^2 \) origin (shell constraint and Lie algebra); added inter‑shell tension/decoherence note; expanded benchmarks table with symmetry bands; added Tsirelson bound; minor polish.
1.4d	2025‑08‑12	Unified gauge notation \( U_{ij} \) (link) and \( B_{ij} \) (phase); completed shell potential; fixed SymPy snippet; normalized \( \mathrm{U}(1)^2 \) band (1.55–1.70); LaTeX and punctuation clean‑ups.
1.4c	2025‑08‑12	Preamble + notation guard; fixed spin‑flip energy sign; Helfrich curvature units; standardized CHSH; restored anomaly code; updated links.