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A Homothetic Gauge Theory and the Regularization of the Point Charge

Fereidoun Sabetghadam

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.

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 , 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 and then specialize to 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.

Paper Structure

This paper contains 16 sections, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weyl integrable Spacetime and Homothetic Dilaton Dressing
  3. Classical Weyl-integrable geometry
  4. Homothetic dilaton dressing
  5. Linearization
  6. A Homothetic Hodge Decomposition
  7. Homothetic de Rham complex and Hodge theory
  8. Homothetic Electromagnetism
  9. The Offset fields
  10. The Homothetic Lorenz gauge and Potential
  11. Homothetic Lorenz Gauge
  12. Homothetic Harmonic Gauge Fields and Potentials
  13. Application: A Non-Singular Point Electric Charge
  14. Conclusions and Outlook
  15. Principal H-bundle formulation and curvature
  16. ...and 1 more sections

Figures (1)

  • Figure 1: There is a chain isomorphism $\mathcal{H}\circ\iota^\bullet$ between the ordinary de Rham complex and the homothetically dilaton dressed complex.

Theorems & Definitions (3)

  • Remark 2.1
  • Remark 2.2
  • Remark 4.1: Conformal covariance