ROOT-FQEDMathematics & Physics

Fractal Quantum Electrodynamics (FQED)

Extends gauge invariance and electromagnetic coupling to fractal network geometries, enforcing discrete boundary cycle preservation matching U(1) gauge symmetry.

ClassGauge Topology Fabric

Evidence TierTier 1 (Formal Root)

Model ProfileFORMAL_SPECIFICATION

AttributionIvan Pasev

Procedural PlateFormal procedural plate defining the bounded theoretical geometry and invariant obligations for this research fabric.

EPISTEMIC DEMARCATION // Tier 1 (Formal Root)

Inspect Results Ledger →

Constitutional Demarcation: A Research Fabric is a bounded epistemological architecture. Structural formulation, digital execution, and proof-assistant replay declare local invariants; they do not constitute unverified physical validation.

Architectural Overview

Fractal Quantum Electrodynamics (FQED) extends standard abelian gauge theory and photon-matter interaction dynamics to self-similar, multi-scale network geometries. Instead of integrating continuous gauge potentials over smooth 4-manifolds, FQED formulates electromagnetic phase factors and holonomies over discrete fractal boundary cycles preserving exact discrete $U(1)$ gauge invariance.

The discrete action functional on the self-similar lattice ${\mathcal{K}}_{\mathrm{\infty }}$ is given by:

$$S_{FQED}[A,\psi ]=\sum _{\mathrm{\Delta }\in {\mathcal{C}}_2(\mathcal{K})}\text{Tr}[1-\exp (ie{\oint }_{\partial \mathrm{\Delta }}A)]+\sum _{x\in {\mathcal{C}}_0(\mathcal{K})}\overline{\psi }(x){\mathcal{D}}_F\psi (x)$$

Where ${\mathcal{D}}_F$ is the scale-covariant fractal Dirac operator preserving local charge conservation across recursive scale transformations.

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Core Invariants

1. Exact Discrete $U(1)$ Gauge Invariance: Invariance under local phase transformations $\psi (x)\to e^{i\theta (x)}\psi (x)$ on arbitrary fractal lattice levels.
2. Ward–Takahashi Identity Preservation: Multi-scale current conservation guarantees anomaly-free radiative corrections.
3. Continuum Asymptotic Recovery Limit: In the smooth homogeneous limit (fractal dimension $d_f\to 4$), the action converges rigorously to standard QED perturbation theory.
4. Renormalization Group Monotonicity: Beta-function trajectories remain non-divergent under hierarchical scale decimation.

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Epistemic Boundary Notice

Theoretical Formal Specification

FQED represents an analytical and mathematical field-theoretic investigation into scale-covariant electrodynamics. It provides mathematical tools for studying photon-electron behavior in complex topological media. It does not replace standard QED as verified by precision anomalous magnetic moment ($g-2$) or Lamb shift measurements.

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Reciprocal Bridges & Related Research

• Fractal Quantum Field Theory (RF-007) — The general non-abelian field-theoretic framework.
• LPFR Strong-Field Metrology (RF-040) — Experimental laser-plasma interaction and strong-field physics.
• Extended TQFT (RF-010) — Topological quantum field theories across discrete manifolds.
• FQED Whitepaper Reader — Full mathematical derivations and formal proofs.