Renormalization of the Electromagnetic Energy-Momentum Tensor

Abstract

Close to the position $\mathbf {y}(t)$ of a charged particle, the electromagnetic field it creates diverges as $F^{\mu \nu }\sim 1/ r^2$, where $r=|\mathbf {x}-\mathbf {y}(t)|$ is the distance from the particle. Correspondingly, the electromagnetic energy-momentum tensor

$$\begin{aligned} T^{\mu \nu }_\mathrm{em}=F^{\mu \alpha }F_{\alpha }{}^\nu +{1\over 4}\,\eta ^{\mu \nu }F^{\alpha \beta }F_{\alpha \beta } \end{aligned}$$

diverges as

$$\begin{aligned} T^{\mu \nu }_\mathrm{em}\sim {1\over r^4}. \end{aligned}$$