Covariant Theory of Radiation Damping

Schwinger's expression of the $S$ matrix in the Cayley form, in terms of a Hermitian operator $K$, is shown to be identical with the previous noncovariant expression used in Heitler's theory of radiation damping. The comparison of the two formalisms leads, furthermore, to a clear understanding of mass renormalization which is necessary for internal consistency, quite independently of the eventual removal of divergences. For the computation of $\overline{K}$ in a covariant way, new formulas generalizing and connecting Gupta's and Fukuda and Miyazima's results are presented. The $n$th order approximations of $\overline{K}$ and $S$ are closely related, and ${\overline{K}}_n$ may be expressed in terms of the $S_p$ of order $p\leqq n$ or in terms of their anti-Hermitian parts only.