In the previous chapters we have calculated a whole series of reactions in QED to lowest order. The results were always finite and agreed well with experiment. We know, however, that, in principle the theory includes higher order terms and that for a precise comparison with experiment, these too must be taken into account. After the development of QED at the end of the 1920s, it quickly became clear that in calculations of higher order terms, the so-called radiative corrections, one arrived at infinite results. It required much effort by many physicists in order to come to grips with these infinities and to find a systematic procedure for calculating higher order contributions with a finite result. This procedure is known as renormalization. Important original contributions on this subject are collected in Schwinger (1958). Here we will not go into the details of renormalization theory but will illustrate the problems and methods of solution by means of a few examples.
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