Conformal Transformations and Self-Sustaining Processes in Electric Circuits

This work establishes a theoretical framework connecting conformal symmetry in electromagnetism to self-sustaining processes in electrical circuits. Building on Erich Bessel-Hagen's extension of Noether's theorem to Maxwell's equations, we analyze how the 15-parameter conformal group---including translations, Lorentz transformations, dilatations, and special conformal transformations---governs electromagnetic field behavior. Through a Lagrangian formulation of circuit dynamics, we map these symmetries to component-level transformations and derive conformally extended versions of Kirchhoff's laws featuring: i) Geometry-dependent weighting factors (wᵢ ∝ λᵢ⁻¹); ii) A dilaton-like field interaction term (Φ_δ). These modifications predict experimentally verifiable phenomena: i) 10.2% deviations from classical current division in RF splitters; ii) 4.2% resonant frequency shifts and 2.67× quality factor enhancements in RLC circuits; iii) Power-law scaling (Jz ∝ a⁻²) for cylindrical conductor current densities. We propose a field equation for the dilaton-like field δ: \[ 1/r∂/∂ r(r∂δ/∂ r) + ⋯ = κ(E^2 - B^2) \] which mediates energy exchange via the dilaton potential Φ_δ in modified Kirchhoff's laws between the circuit and conformal background. This work bridges high-energy physics and electrical engineering, demonstrating how conformal symmetry can enable novel circuit behaviors---including self-sustaining oscillations in cylindrical geometries---that transcend lumped-element approximations.