(Received October 25, 1948)
The recent achievements both in experiment and theory have confirmed that the reaction of the radiation field is a really observable phenomenon and that one can calculate this effect in our present formalism of quantum theory if one employs some subtraction prescription. Thus Bethe computed successfully the level shift of the hydrogen atom revealed by the experiment of Lamb and Retherford. Following Bethe's idea, Tomonaga has developed, independently of American authors, a so-called “self-consistent” subtraction method which aims at disposing of the infinities in a self-consistent manner and obtaining finite results for various processes. In fact he and collaborators could show that the radiative corrections for the elastic scattering of the electron and the Compton scattering were made finite by this method.
Although Tomonaga's theory has recourse to a relativistic canonical transformation which may be regarded as a generalization of the transformation used by Bloch and Nordsieck and by Pauli and Fierz, yet his method can equally be applied to the conventional perturbation formalism. We have here calculated the radiative correction of an electron moving in an external (electromagnetic) field along the line of the ordinary perturbation theory. To get definite numerical results we had to content ourselves with non-relativistic approximation for the initial and final states.
According to Tomonaga there are three kinds of infinities, that is, the electron's self energy, or the mass type infinity, the photon's self energy, or the vacuum polarization type infinity, and an infinity depending on the external field. We shall see that the subtraction of the first and the third type infinity is necessary in our case.
A similar calculation making use of the canonical transformation has been carried out by Fukuda, Miyamoto and Tomonaga. Comparison of the obtained results is not without interest in view of the provisional character of the present theory.
URL : http://ptp.ipap.jp/link?PTP/4/82/
DOI : 10.1143/PTP.4.82