📝 SummarXiv

Abstract

We introduce a Homothetic Hodge-de Rham (HHDR) theory that extends the de Rham complex to homothetically transformed differential forms and develops the associated homothetic Hodge machinery. Enforcing homothetic symmetry on physical laws naturally yields scale-covariant couplings, interpretable as canonical penalty terms in the framework of differential equations. Building on this foundation, we construct a Homothetic Gauge Theory (HGT) and apply it to classical electromagnetism, leading to generalized homothetic Maxwell equations. The approach establishes a link between homothetic scaling and conformal field theory concepts such as the dilaton field and vacuum expectation values (VEVs). As a concrete physical application, we address the classical divergence of point charge self-energy. By modeling the source as a penalized Dirichlet boundary condition, we obtain a non-singular electrostatic potential. This framework highlights a mathematically rigorous and physically meaningful extension of gauge theory with potential relevance across field theory and electrodynamics.