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 ($w_i \propto \lambda_i^{-1}$); ii) A dilaton-like field interaction term ($\Phi_\delta$). 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$\times$ quality factor enhancements in RLC circuits; iii) Power-law scaling ($J_z \propto a^{-2}$) for cylindrical conductor current densities. We propose a field equation for the \textbf{dilaton-like field} $\delta$: \[ \frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\delta}{\partial r}\right) + \cdots = \kappa(E^2 - B^2) \] which mediates energy exchange via the \textbf{dilaton potential} $\Phi_\delta$ 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.