Non-Abelian Electromagnetic Field and Its Particle Solutions

This work clarifies a long-standing conflation of the global symmetry of electromagnetic waves and the local symmetry of the electromagnetic field. Here we show that the electromagnetic field exhibits exactly Abelian global symmetry when treated as waves or photons, and yet its local properties at the field point can be non-Abelian. Typically, the crossing of two linearly propagating light beams manifests the global U(1) character of the waves, which is described by the direct sum of Lie algebras. Distortions at the intersection point, including self-bending phenomena, reflect the local symmetric properties of the field and ought to be treated via dynamic vectors superposition within a more fundamental vector space, which is described by the Cartan decomposition, and lead to two coupled generators at the field point with a nonvanishing Lie bracket, and result to be non-Abelian, and the field exhibits self-interaction. Meanwhile, both the electromagnetic potential A and the field strength F take values in the same Lie algebra and are therefore equivalent in characterizing the symmetry of the field. This work establishes a new research paradigm for analyzing F in 3-dimensional real space to demonstrate the symmetry of the electromagnetic field. This study shows that the field possesses a fundamental SO(3) symmetric structure, and can be reduced to U(1), or promoted to SU(2) depending on dynamical conditions. Thereby, exactly two steady-state curvature solutions are obtained: the zero-curvature solution corresponds to U(1), while the characteristic-frequency of the nonvanishing-curvature solution coincides precisely with the γ-ray associated with the electron. There exist exactly two steady-state particle solutions whose characteristic parameters are in close agreement with those of the electron and proton. Within this theoretical framework, the crossing of two γ rays can directly generate self-sustaining steady electromagnetic particle directly. Known experiments of “\gamma \gamma \rightarrow e^{+}e^{-}” indicate that this particle is the electron. This study extends the understanding of electromagnetic field symmetry and demonstrates that elementary particles can be constructed directly from electromagnetic fields, providing fundamental theoretical references for relevant areas.