TFP Section 8 (v8)

SECTION 8 — UNIFIED CLASSICAL FIELDS (v8.3)

Geometry, Electromagnetism, and Scalar Dynamics from Coarse-Grained Alignment
By John Gavel

8.0 Overview

The coarse-grained alignment field \( A(x,t) \) (Section 2.3) is the fundamental classical observable of Temporal Flow Physics. All classical fields — scalar, vector, and metric — emerge as derived quantities:

• Scalar field: \( \phi(x,t) \equiv A(x,t) \)
• Vector potential: \( A_\mu \) from charge and current (Section 6)
• Metric: \( g_{\mu\nu} \) from correlations (Section 3)

No new fields are postulated. The unified classical action is the continuum limit of substrate tension minimization (Axiom 6).

8.1 Scalar Field as Alignment Order Parameter

From Section 2.3.1, the alignment field is:

\[ \phi(x,t) = A(x,t) = \lim_{R \to x} \frac{1}{|R|} \sum_{i \in R} F_i(t) \tag{8.1} \]

Its dynamics derive from the tension functional (Section 5.2.9.2):

\[ T[\phi] = \int d^d x \left[ \frac{1}{2} D (\nabla \phi)^2 - \frac{1}{2} \mu_{\text{eff}}^2 \phi^2 \right] \tag{8.2} \]

where:

• \( D = k_{\text{avg}} a^2 / (T_{\text{eff}} \tau_0) \) (Eq 2.12) penalizes spatial misalignment
• \( \mu_{\text{eff}}^2 = \kappa \tau_0 \) (Eq 5.21) reflects instability of mixed states (\( \phi = 0 \))

The equation of motion is \( \partial T / \partial \phi = 0 \):

\[ D \nabla^2 \phi - \mu_{\text{eff}}^2 \phi = 0 \quad \Rightarrow \quad \nabla^2 \phi = \frac{\mu_{\text{eff}}^2}{D} \phi \tag{8.3} \]

This is the static Klein-Gordon equation, with mass scale \( m_\phi^2 = \mu_{\text{eff}}^2 / D \) (Section 5.4).

8.2 Electromagnetic Coupling

From Section 4.5.1, charge density is net alignment:

\[ \rho_{\text{dim}} = \phi \tag{8.4} \]

From Section 4.10, motif velocity \( \mathbf{v}_C \) defines current:

\[ \mathbf{J}_{\text{dim}} = \phi \mathbf{v}_C \tag{8.5} \]

Charge conservation (Section 6.1.2) gives:

\[ \frac{\partial \phi}{\partial t} + \nabla \cdot (\phi \mathbf{v}_C) = 0 \tag{8.6} \]

The vector potential is the convolution (Section 6.3):

\[ A_\mu^{\text{dim}} = \left( \Phi_{\text{dim}}, \mathbf{A}_{\text{dim}} \right) = \int \frac{j_\mu^{\text{dim}}(\mathbf{x}')}{|\mathbf{x} - \mathbf{x}'|} d^3x' \tag{8.7} \]

where \( j_\mu^{\text{dim}} = (\rho_{\text{dim}}, \mathbf{J}_{\text{dim}}) \). In physical units (Section 5):

\[ A_\mu^{\text{phys}} = A_\mu^{\text{dim}} \cdot \frac{q_0}{L_c} \tag{8.8} \]

8.3 Gravitational Coupling

From Section 3.2, the metric \( g_{\mu\nu} \) is defined by operational distance \( d_{ij} \). In the continuum, minimal coupling replaces partial derivatives with covariant derivatives:

\[ \partial_\mu \phi \to \nabla_\mu \phi = \partial_\mu \phi - \Gamma^\lambda_{\mu\nu} \phi \tag{8.9} \]

The scalar field action becomes:

\[ S_\phi = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi - V(\phi) \right] \tag{8.10} \]

where \( V(\phi) = -\frac{1}{2} \mu_{\text{eff}}^2 \phi^2 \) (from Eq 8.2). This couples \( \phi \) to curvature via \( \sqrt{-g} \) and \( g^{\mu\nu} \) (Section 7).

8.4 Unified Classical Action

Combining scalar, vector, and metric sectors, the total action is:

\[ S = \int d^4x \sqrt{-g} \left[ \frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi - V(\phi) - \frac{1}{4} F_{\mu\nu} F^{\mu\nu} \right] \tag{8.11} \]

where \( F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \) (Section 6.4). All terms derive from:

• \( \phi \): coarse-grained alignment (Section 2.3)
• \( A_\mu \): charge/current convolution (Section 6.3)
• \( g_{\mu\nu} \): correlation geometry (Section 3.2)

No external coupling constants are introduced — all scales are fixed by \( D \), \( \mu_{\text{eff}} \), \( L_c \), \( T_c \) (Sections 2, 5).

8.5 Field Equations

Varying (8.11) yields the classical field equations:

8.5.1 Scalar (Klein-Gordon)

\[ \nabla_\mu \nabla^\mu \phi + V'(\phi) = 0 \quad \Rightarrow \quad \Box \phi - \mu_{\text{eff}}^2 \phi = 0 \tag{8.12} \]

8.5.2 Vector (Maxwell)

\[ \nabla_\mu F^{\mu\nu} = j^\nu \tag{8.13} \]

where \( j^\nu = q_0 (\phi, \phi \mathbf{v}_C) \) (Section 6.1).

8.5.3 Tensor (Einstein)

\[ G_{\mu\nu} = 8\pi G_{\text{obs}} T_{\mu\nu} \tag{8.14} \]

with stress-energy:

\[ T_{\mu\nu} = \nabla_\mu \phi \nabla_\nu \phi - g_{\mu\nu} \left[ \frac{1}{2} (\nabla \phi)^2 - V(\phi) + \frac{1}{4} F_{\alpha\beta} F^{\alpha\beta} \right] \tag{8.15} \]

and \( G_{\text{obs}} = L_c^3 / (M_c T_c^2) \) (Eq 7.14).

8.6 Axiomatic Closure

Field Concept	Substrate Origin	Axiom
Scalar \( \phi \)	Coarse-grained alignment \( A(x,t) \)	A2, A3
Vector \( A_\mu \)	Charge/current convolution	A2, A6
Metric \( g_{\mu\nu} \)	Correlation distance \( d_{ij} \)	A3
Unified Action	Tension functional + calibration	A6, Section 5

8.7 Bridge to Section 9 — Quantization

Section 8 provides:

• Classical scalar field \( \phi = A(x,t) \)
• Unified action coupling \( \phi \), \( A_\mu \), and \( g_{\mu\nu} \)
• Field equations with no free parameters

Section 9 quantizes these fields by identifying \( \phi \) with flow multiplets \( \Psi = [F^{(0)}, F^{(1)}, F^{(2)}, F^{(3)}] \) (Section 13.3), recovering uncertainty, superposition, and CPT symmetry from substrate dynamics.