On the Invariant Regularization in Relativistic Quantum Theory

Abstract

The formal method of regularization of mathematical expressions of sums of products of different types of $\ensuremath{\delta}$-functions is first applied to the example of vacuum polarization. It is emphasized that only a regularization of the whole expression without factorization leads to gauge invariant results. It is further shown, that for the regularization of the expression for the magnetic moment of the electron, a single auxiliary mass is sufficient, provided that different functions of the same particle (e.g., the photon functions $\overline{D}$ and ${D}^{(1)}$) are regularized in the same way and that the regularization of products of two electron functions is never factorized. The result is then the same as that of using Schwinger's method of introducing suitable parameters as new integration variables in the argument of $\ensuremath{\delta}$-functions, without using any auxiliary masses.