A Homothetic Gauge Theory and the Regularization of the Point Charge

Abstract

We introduce a Homothetic Hodge de Rham (HHDR) theory that extends the de Rham complex and Hodge decomposition to homothetically dressed differential forms. The dressing, governed by a dilaton field and a Weyl weight $w$, defines the homothetic Hodge machinery. Imposing homothetic symmetry on physical laws yields scale covariant interaction terms that arise canonically from the geometry and can be interpreted as penalty-type couplings in the language of differential equations. On this geometric foundation, we construct a Homothetic Gauge Theory (HGT) for a general weight $w$ and then specialize to $w=1$ to formulate homothetic electromagnetism, obtaining homothetic Maxwell equations for a coupled system of the physical gauge field and a homothetic offset field. As a central application, we revisit the divergence of the self-energy of a point charge: modeling the charge as a boundary condition and choosing an appropriate dilaton profile, we show that both the electric field and its total self-energy remain finite at the origin. The HHDR/HGT framework thus provides a mathematically controlled extension of gauge theory with potential implications for field theory, classical electrodynamics, and numerical penalty methods.