Conformal Transformations and Self-Sustaining Processes in Electric Circuits

This work establishes the first derivation of geometry-dependent Kirchhoff’s laws via conformal symmetry, enabling new types of self-sustaining circuits unattainable in classical lumped-element theory. Building on Bessel-Hagen’s extension of Noether’s theorem to Maxwell’s equations, we develop a conformal circuit formalism that fundamentally extends traditional circuit theory through two key innovations: (1) Geometry-dependent weighting factors (wi∝ai−1) in Kirchhoff’s laws derived from scaling symmetry; (2) A dilaton-like field (δ) mediating energy exchange between circuits and conformal backgrounds. Unlike prior symmetry applications in electromagnetism, our approach directly maps the 15-parameter conformal group to component-level circuit transformations, predicting experimentally verifiable phenomena: (i) 10.2% deviations from classical current division in RF splitters; (ii) 4.2% resonant frequency shifts with 2.67× Q-factor enhancement; (iii) Power-law scaling (Jz∝a−2) in cylindrical conductors. This theoretical framework proposes how conformal symmetry could enable novel circuit behaviors, including potential self-sustaining oscillations, subject to experimental validation.