A Homothetic Gauge Theory and the Regularization of the Point Charge
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912989477601280 |
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| author | Sabetghadam, Fereidoun |
| author_facet | Sabetghadam, Fereidoun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06153 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Homothetic Gauge Theory and the Regularization of the Point Charge Sabetghadam, Fereidoun Mathematical Physics 02.40.-k, 11.15.-q, 03.50.De 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. |
| title | A Homothetic Gauge Theory and the Regularization of the Point Charge |
| topic | Mathematical Physics 02.40.-k, 11.15.-q, 03.50.De |
| url | https://arxiv.org/abs/2507.06153 |