Regularized Electromagnetic Force-Density Clouds from the Maxwell Stress Tensor: A Multi-Particle Formulation

The central idea is relatively straightforward: instead of treating charged particles as mathematical points (Dirac deltas), we replace the source distributions with 3D Gaussian mollifiers of width σ. This single substitution propagates through the entire electromagnetic description and produces a smooth, physically regularized force-density field f_σ(r) derived consistently from the Maxwell stress tensor. The framework spans four levels of description: 1. Classical point particle (σ → 0, flat spacetime): recovers the standard Lorentz force exactly. 2. Regularized flat-spacetime cloud (σ > 0): a smooth, integrable force-density field f_σ derived from the divergence of the Maxwell stress tensor, with a manifestly covariant relativistic extension via the four-force density f^μ = F^{μν}(J_ν)_σ. 3. Application to charged particle beams: the Gaussian cloud naturally yields the Kapchinskij-Vladimirskij envelope equation, the 1/γ² relativistic suppression of space charge, and the collider luminosity formula — all from the same object f_σ, without additional postulates. 4. General-relativistic extension: replacing η^{μν} with the curved metric g^{μν} couples the regularized stress tensor to the Einstein field equations. The Gaussian regularization removes the r⁻⁴ singularity of the point-charge energy density, renders the electromagnetic self-energy U_self = q²/(8π^{3/2} ε₀ σ) finite for all σ > 0, and regularizes the Reissner-Nordström metric at the origin. The Unruh temperature at the cloud boundary also provides a quantum lower bound on the effective thermal temperature of the system.