Unified Field Theory: A Nonlinear Electromagnetic Unified Framework with Topological Solitons

Abstract

Abstract General relativity and the Standard Model are the two foundational frameworks of modern physics, each supported by extensive high-precision experimental data. They govern, respectively, macroscopic gravitation and the microscopic electromagnetic, weak, and strong interactions. Yet at their foundations the two theories rest on very different mathematical structures and field assumptions — differences sufficiently fundamental that a series of core problems in fundamental physics have so far defied any unified explanation. We do not question that existing theories work extremely well at the scales where they apply. Instead, we construct a research framework of nonlinear topological electromagnetic fields, taking the four-dimensional electromagnetic potential Aμ as the fundamental dynamical variable. The key advance is that we circumvent the constraint that Derrick's theorem places on three-dimensional soliton solutions. We add a minimal Skyrme-type gradient term (γ/2)(∂ρFμν)² to the Lagrangian and then prove rigorously that the modified Lagrangian admits spherically symmetric, static soliton solutions with finite energy. This is the first rigorous construction of stable soliton solutions within a purely electromagnetic unification scheme. We lay out five internally consistent fundamental postulates. Each one starts with a guiding physical statement and then gives the corresponding Lorentz-covariant mathematical definitions and the equations that govern the dynamics. The framework draws a distinction between topologically trivial electromagnetic waves — the photons — and solitons that are stabilized by Noether charge — the matter particles. Using background-field perturbation theory, we derive unified expressions for the equivalent interaction potentials of the electromagnetic, strong, weak, and gravitational interactions. High-energy phenomena like quark confinement, asymptotic freedom, and the confining field energy of hadrons are discussed analytically; and using the intrinsic negative pressure of the vacuum field, we attempt to explain why the universe expands at an accelerating rate — providing a new perspective on the ΛCDM vacuum-energy fine-tuning problem. The central quantitative result of this work reads: by precisely numerically solving the single-soliton equation f''' + f' - f³ = 0, a dimensionless integral I = 1.1932847 is obtained, and the total unbound mass of three solitons is 0.9294 GeV, deviating from the experimental proton mass of 0.938272 GeV by less than 1%. This result is in excellent agreement with the well-known lattice QCD finding that roughly 99% of the proton mass comes from strong-interaction field energy. Moreover, it emerges naturally from first principles — without invoking the Higgs field or requiring large-scale numerical simulations.