Appendix AN — Electromagnetism and Gauge Symmetry

L.1 Introduction

In PBG, classical electromagnetism emerges not from external vector fields, but from phase-coherent modal structures and anchoring bias dynamics. This appendix reconstructs the electromagnetic sector from coherence principles, including gauge symmetry, Maxwell equations, and photon structure.

L.2 Modal Structure and Coherence Anchoring

Each coherence-bearing mode ( \psi(\vec{x}) ) is composed of:

• An amplitude envelope ( \rho(\vec{x}) ),
• A phase structure ( \theta(\vec{x}) ),
• A modal anchoring profile determined by bias gradients.

The modal coherence field ( B(\vec{x}) ) generated by a mode evolves under:

where ( \gamma_0 ) is the anchoring strength constant, and ( k ) defines coherence decay scale.

L.3 Charge as Phase Topology

Charge arises from modal phase winding. Let ( \theta(\vec{x}) ) be the phase field of a mode:

• Topological charge ( q ) is associated with phase winding number around a closed loop:

  $$q\propto {\oint }_{\partial S}\mathrm{\nabla }\theta \cdot d\overrightarrow{l}$$

• No separate charge field is required. Phase topology determines bias behaviour that mimics electric field interactions.

L.4 Electromagnetic Analogues in Coherence Fields

We define effective fields:

• Effective electric field:

  $$\overrightarrow{E}\equiv -\mathrm{\nabla }\mathrm{\Phi }$$

• Effective magnetic field:

  $$\overrightarrow{B}\equiv \mathrm{\nabla }\times \overrightarrow{A}$$

where ( \Phi ) is the modal anchoring potential, and ( \vec{A} ) is the coherence phase vector derived from modal flow.

L.5 Modal Anchoring Dynamics and Maxwell Analogues

Starting from the coherence flow ( \vec{j}_c = \rho_c \vec{v}_c ) where ( \rho_c = |\psi|^2 ), and ( \vec{v}_c = -\nabla \theta ), we recover:

• Continuity equation (charge conservation):

  $$\frac{\partial {\rho }_c}{\partial t}+\mathrm{\nabla }\cdot ({\rho }_c{\overrightarrow{v}}_c)=0$$

• Bias evolution analogous to Faraday's law:

  $$\mathrm{\nabla }\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$$

• Modal anchoring balance analogous to Gauss's law:

  $$\mathrm{\nabla }\cdot \overrightarrow{E}\propto {\rho }_c$$

L.6 Gauge Freedom and Coherence Redefinition

The overall phase ( \theta(\vec{x}) ) of a mode is physically meaningful only through its gradients. Thus:

• Gauge-like freedom naturally exists:

  $$\theta (\overrightarrow{x})\to \theta (\overrightarrow{x})+f(t)$$

Local phase redefinitions correspond to effective gauge transformations.

L.7 Photons as Modal Coherence Structures

A photon is not a particle or excitation of a field, but:

• A self-sustaining coherence-bearing wavepacket,
• A mode ( \psi_\gamma(\vec{x}) ) with minimal internal anchoring cost,
• Phase topology ensuring preservation during propagation.

L.8 Polarisation and Coherence Phase

Polarisation states correspond to structured phase-coherent cross-sections of the photon mode:

• Linear polarisation: Phase gradients aligned across wavefront.
• Circular polarisation: Helical phase structure along propagation axis.

L.9 Electromagnetic Radiation as Modal Turnover

Radiation emission and absorption arise from:

• Turnover events causing coherence reconfiguration,
• Phase-slippage creating new free-propagating modes (photons),
• Anchoring field adjustment driving re-emission.

L.10 Summary

The electromagnetic sector emerges from:

• Modal phase gradients,
• Coherence anchoring bias,
• Topological phase structure.

Maxwellian behaviour, gauge freedom, radiation, and photon properties all arise naturally from coherence-based modal dynamics without external fields or classical forces.