Appendix AN — Derivation 30: Electromagnetism & Gauge Structure in PBG

AN.1 Motivation

Conventional EM uses fields, photons, and a primitive “charge” source.
Phase-Biased Geometry replaces all that with a single coherence medium and one variational cost.
Charge, force, and gauge symmetry emerge from phase-winding and anchoring-cost invariance.

AN.2 Core PBG Ingredients

A coherence mode

incurs an anchoring cost

where

• $\alpha$ (J m⁻¹) = spatial stiffness,
• $\beta$ (J m⁻³) = envelope penalty.

No additional gauge fields or charges are put in by hand.

AN.3 U(1) Gauge Invariance as Anchoring-Cost Freedom

AN.3.1 Global Phase

leaves $|\mathrm{\nabla }\phi |$ unchanged ⇒ $C[\psi ]$ unchanged.

AN.3.2 Local Phase

then $|\mathrm{\nabla }{\phi }^{\prime }{|}^2=|\mathrm{\nabla }\phi +\mathrm{\nabla }\theta {|}^2$ would raise the cost unless we introduce a compensator

and define the covariant derivative

so that

and $C[\psi ]$ remains form-invariant.
▶ Gauge potential $A_i$ is just the phase-derivative field.

AN.4 Emergent Electromagnetic Laws

From the one scalar potential $\mathrm{\Phi }(\mathbf{x},t)$ (Appendix E):

1. Static kernel (Gauss-like)

   $$\alpha {\mathrm{\nabla }}^2\mathrm{\Phi }-\beta \mathrm{\Phi }=-{\rho }_{\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}}⟹\mathbf{E}=-\mathrm{\nabla }\mathrm{\Phi },\mathrm{\nabla }\cdot \mathbf{E}=\frac{{\rho }_{\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{r}}}{\alpha }$$

2. Phase current (Ampère-like)

   $${\mathbf{J}}_{\varphi }={\rho }^2\mathrm{\nabla }\phi ,\mathrm{\nabla }\times \mathbf{B}=\frac{1}{\gamma }{\partial }_t\mathbf{E}+{\mathbf{J}}_{\varphi },c^2=\frac{\alpha }{\gamma }$$

3. No monopoles & Faraday

   $$\mathrm{\nabla }\cdot \mathbf{B}=0,\mathrm{\nabla }\times \mathbf{E}=-{\partial }_t\mathbf{B}.$$

All four Maxwell equations follow from phase continuity + anchoring gradients (see Appendix AM).

AN.5 Coulomb’s Law & Fine-Structure Constant

– Coulomb: static $\mathrm{\Phi }(r)\propto 1/r$ ⇒ $F\propto 1/r^2$, with

where $Q$ is the winding-charge of a phase mode.

– Fine-structure:

emerges as the ratio of phase-winding cost to free-propagation cost.

AN.6 Calibrated Constants

No other EM parameters are assumed.

AN.7 Why It Works

• Charge = topological winding in a coherence medium.
• Gauge invariance = freedom to rephase without cost.
• Fields and currents = derived from $\phi$ and $A_i={\partial }_i\theta$.
• Constants $c$, $k_e$, ${\alpha }_{\mathrm{f}\mathrm{s}}$ all follow from $\{\alpha ,\beta ,\gamma \}$.

PBG thereby replaces classical EM ontologies with coherence geometry and one scalar action.