>[!warning] >This content has not been peer reviewed. # Standard Model — RST application **Pillar:** The running coupling constants of the Standard Model forces are governed by the same fidelity function $\mu(\eta, n)$ that drives galaxy rotation and steel fracture. The transition sharpness $n$ is derived from the **number of gauge generators** of each symmetry group — the morphism connectivity of the quantum channel. This is the seventh sector and the first **quantum** application of the fidelity function, extending the Classical Desktop into the Standard Model. --- ## 1. Standard view (running couplings) - **QCD (SU(3)):** The strong coupling $\alpha_s(Q)$ runs from $\sim 0.12$ at the Z mass (91.2 GeV) to $\sim 0.5$ at 1 GeV. At high Q: asymptotic freedom ($\alpha_s \to 0$). At low Q: confinement ($\alpha_s \to \infty$). The 1-loop running is $\alpha_s(Q) = 12\pi / [(33 - 2n_f)\ln(Q^2/\Lambda_{QCD}^2)]$. - **QED (U(1)):** The fine-structure constant $\alpha$ runs from $1/137.036$ at low energy to $\sim 1/128$ at the Z pole. Logarithmic vacuum polarization: $\alpha(Q) = \alpha(0)/[1 - (\alpha/3\pi)\ln(Q^2/m_e^2)]$. - **Weak (SU(2)):** The weak mixing angle $\sin^2\theta_W$ runs from $\sim 0.238$ at low energy to $0.2312$ at the Z pole. --- ## 2. RST mapping: gauge generators as morphism connectivity The [[expanded theory/RST Four-Force Bridge|Four-Force Bridge]] identifies *what* each force is (which projection of the Resource Triangle). The n-derivation identifies *how sharp* the transition is (the topology of the quantum channel). | Gauge group | Generators | n (derived) | Physical backbone | Channel | |:---|:---|:---|:---|:---| | **U(1)** | 1 (photon) | **1** | 1D phase rotation | EM coupling | | **SU(2)** | 3 (W$^+$, W$^-$, Z) | **3** | 3D isospin rotation | Weak mixing | | **SU(3)** | 8 (8 gluons) | **8** | 8D color space | Strong coupling | **The derivation:** In the classical sectors, $n$ is the Hausdorff dimension of the relational pathway (Axiom A5). For gauge forces, the "relational pathway" is the gauge orbit — the set of transformations that maintain the symmetry. The dimension of this orbit is the number of independent generators: $n = \dim(\mathfrak{g})$. - **U(1) → n = 1:** A single phase rotation (circle). The "slowest" channel — matches the extremely gradual running of $\alpha$. - **SU(2) → n = 3:** A 3D rotation manifold ($S^3$). Moderate sharpness — matches the electroweak transition. - **SU(3) → n = 8:** An 8D color manifold. High sharpness — matches the catastrophic confinement transition, analogous to turbulence ($n = 20$) on the classical ladder. **Graph-theoretic origin:** The generator counts 1, 3, 8 arise from the information possibilities of 1-, 2-, and 3-node cliques: $K_2 \to 3$ Pauli ($SU(2)$), $K_3 \to 8$ Gell-Mann ($SU(3)$). See [[N=2 — Weak Force and Pauli (RST)]] and [[N=3 — Strong Force and Gell-Mann (RST)]]. --- ## 3. Field equation (running coupling from fidelity) The RST model for running couplings uses the same additive form as the [[../Electronic Transport/Electronic Transport (RST)|electronic transport sector]]: $\alpha(Q) = \alpha_0 + A \cdot \left[\frac{1}{\mu(Q/\Lambda,\, n)} - 1\right]$ where $\mu$ is the universal fidelity function: $\mu(\eta, n) = \frac{\eta}{(1 + \eta^n)^{1/n}}, \qquad \eta = \frac{Q}{\Lambda}$ **Parameters:** - **$\alpha_0$** — asymptotic coupling (Q → ∞): the "residual" coupling at infinite energy - **A** — coupling scale: magnitude of the fidelity-deficit contribution - **$\Lambda$** — confinement / transition scale: the noise floor of the gauge field - **n** — transition sharpness: fixed by gauge group ($n = \dim(\mathfrak{g})$) **Asymptotic behavior:** - High Q ($\eta \gg 1$): $\mu \to 1$, $\alpha \to \alpha_0$ (asymptotic freedom for QCD) - $Q \sim \Lambda$ ($\eta \sim 1$): transition, coupling rises - Low Q ($\eta \to 0$): $\mu \to \eta = Q/\Lambda$, $\alpha \to \alpha_0 + A \cdot (\Lambda/Q - 1) \to \infty$ (confinement) **The parallel:** $\Theta_D$ (Debye temperature) is to electrical resistivity what $\Lambda_{QCD}$ is to the strong coupling. The phonon noise floor in solids maps to the confinement scale in QCD. The same fidelity function governs both. --- ## 4. Key prediction: algebraic vs logarithmic running Standard perturbative QFT predicts **logarithmic** running: $\alpha_s^{(1\text{-loop})}(Q) \propto \frac{1}{\ln(Q/\Lambda_{QCD})}$ RST predicts **algebraic** (power-law) running at high Q: $\alpha_{RST}(Q) \approx \alpha_0 + A \cdot \frac{\Lambda^n}{n \cdot Q^n} \quad (Q \gg \Lambda)$ For QCD ($n = 8$): the RST correction decays as $1/Q^8$ — faster than the logarithmic 1-loop running. This means RST predicts that asymptotic freedom "turns off" more rapidly at high energies than perturbative QCD suggests. This is the **RST-logarithmic boundary** — analogous to the RST-Arrhenius boundary in fluids and the RST-Gaussian boundary in dynamical friction. The fidelity function is algebraic; perturbative QFT is logarithmic. They agree at intermediate energies but diverge at the extremes. --- ## 4b. Quantum fidelity extension (boundary extension, conjectural) **Boundary extension** — not removal. A conjectural extension that may improve fit across the RST-logarithmic boundary: $\mu_Q(\eta, n) = \mu(\ln(1 + \eta), n) = \frac{\ln(1+\eta)}{(1 + [\ln(1+\eta)]^n)^{1/n}}$ By mapping $\eta \to \ln(1+\eta)$, the argument becomes logarithmic while preserving the algebraic structure of $\mu$. At small $\eta$: $\ln(1+\eta) \approx \eta$, so $\mu_Q \approx \mu$. At large $\eta$: $\ln(1+\eta) \sim \ln\eta$, potentially capturing logarithmic running. **Success criterion:** Does $\mu_Q$ replicate $\alpha_s(Q)$ and $\alpha(Q)$ to R² > 0.98? Script: `rst_quantum_fidelity.py` — compare $\mu_Q$ vs perturbative QFT running. See [[Standard Model Results]] §6 for current constrained fit quality. **Status:** Conjectural. This is a boundary *extension* (adding a logarithmic map) not a claim to "close" the boundary. The algebraic $\mu$ remains the geometric foundation; $\mu_Q$ is a composite for the quantum sector. --- ## 5. Connection to existing RST theory The [[expanded theory/RST Super-Relational Mapping|Super-Relational Mapping]] derives the four forces as projections of the Resource Triangle: | Force | Projection | This sector's contribution | |:---|:---|:---| | **EM** | $\mu = \Omega/W$ | $n = 1$ (U(1) generator count) → shape of $\alpha(Q)$ | | **Gravity** | $\nu = N/W$ | $n \approx 1.24$ (derived); SPARC confirms | | **Strong** | $1 - \mu$ | $n = 8$ (SU(3) generators) → shape of $\alpha_s(Q)$ | | **Weak** | $\Omega^2/(W\cdot\tau)$ | $n = 3$ (SU(2) generators) → shape of $\sin^2\theta_W(Q)$ | The projections tell us *what* each force is. The gauge-derived $n$ tells us *how sharply* the transition occurs. These are complementary. --- ## 6. Limitations 1. **Algebraic vs logarithmic.** The $\mu$ function produces power-law running; QFT produces logarithmic running. The two agree qualitatively (coupling increases toward $\Lambda$) but differ quantitatively, especially at intermediate $Q$. 2. **Non-perturbative regime.** Below $Q \sim \Lambda_{QCD}$, neither perturbative QCD nor the current RST model is reliable. Lattice QCD provides the only first-principles reference in this regime. 3. **Electroweak mixing.** The weak sector uses $\sin^2\theta_W$ rather than a simple coupling constant. The mapping is less direct than for QCD/QED. 4. **No particle spectrum.** This module predicts coupling *running*, not particle masses or mixing angles. Mass derivation requires the Substrate Eigenvalues framework ([[expanded theory/Substrate Eigenvalues]]). --- ## 7. Results See **[[Standard Model Results]]** for: - QCD constrained fit ($n = 8$) and comparison to 1-loop - QED constrained fit ($n = 1$) and comparison to vacuum polarization - Weak constrained fit ($n = 3$) - Discovery mode (free $n$): does the fitted $n$ match the generator count? ## 8. Links - **Code:** [[Standard Model - Code]] - **Results:** [[Standard Model Results]] - **Topological spectrum:** [[expanded theory/The Spectrum of Relational Topologies]] - **Super-Relational Mapping:** [[expanded theory/RST Super-Relational Mapping]] - **Four-Force Bridge:** [[expanded theory/RST Four-Force Bridge]] - **Substrate Eigenvalues:** [[expanded theory/Substrate Eigenvalues]] - **Electronic transport (analogous classical sector):** [[../Electronic Transport/Electronic Transport (RST)]] - **Relational Dynamics (Lorentz identity):** [[../Relational Dynamics/Relational Dynamics (RST)]] - **Roadmap:** [[../../Applications Roadmap]]