(Received June 9, 1949)
Among the divergence difficulties appearing in quantum electrodynamics, there are two types, one the self-energy of charged particles, and the other the type hitherto called vacuum polarization. In calculations for various collision processes, too, many divergences appear when higher approximations of perturbation theory are taken. Detailed examinations of various cases of these divergences have recently been performed by many authors. As a result, the divergences appearing in various processes have respectively been traced down to certain operators in the Hamiltonian function. For instance, the self-energy of the electron, and a part of the divergences appearing in the elastic scattering of an electron have been avccounted for by the electronic mass-term of the form δm \intψ*βψdv. Discussions of these diverging terms have been made particularly by the method of contact transformation in the perfectly relativistic form proposed by Tomonaga and Schwinger.
These authors further proposed theories dissolving the divergence difficulties by subtracting beforehand in the Hamiltonian function various separated terms responsible for the divergences. This was based on the idea that the subtraction of diverging terms would merely cause changes in the mass and charge of the charged particle. As a result experimental facts such as the energy level shift of the normal state of the hydrogen atom from levels predicted by the Dirac equation for a bound electron, and the hyperfine structures of various atomic spectra were satisfactorily accounted for.
In being interpreted in terms of the difference of the self-energies of electrons in a bound and in a free state, these experimental facts show the reality of the self-energy appearing in higher order perturbation calculations. Therefore, it is necessary simultaneously to find other means of dissolving the divergence difficulties besides making discussions based on the phenomenological subtraction of diverging terms. For this purpose, we must search into the physical nature of the divergences, and find substantialistic solutions appropriate at each stage of theory.
Hitherto, the C-meson theory has been proposed as an attempt to dissolve the difficulty of the self-energy of the electron, and to provide a footing of the future theory. This was put forward by Prof. Sakata with the idea of analysing the divergence difficulties through considering synthetically the existence of various fields. Prof. Sakata has explained the relation between his theory and that of Tomonaga and Schwinger which appeared subsequently as a relation between concrete and abstract.
We desired to extend the C-meson theory and find substansialistic solutions possible at the present stage by making a synthetic investigation covering the whole range of quantum electrodynamics with various fields considered. For this purpose, we first of all examined many divergences by simultaneously considering various charged particle fields in the domain of vacuum polarization. As a result, it was found that, among the electromagnetic divergences appearing in our processes, those of the self-energies of photons and charged particles could be provisionally dissolved by simultaneously considering the C-meson and charged particles.
However, in our calculations up to now, detailed discussions inclusive of finite terms could not be made, owing to the imperfectness of perturbation calculation from the relativistic standpoint. We now intend to make detailed discussions by means of the improved perturbation calculation put forward in a separate paper. For this purpose, it is convenient first of all to obtain some general formulae for various quantities appearing in vacuum polarization regardless of the type of charged particles. This is most readily done by perturbation calculation, so we shall, in the following, derive the formulae by this method.
URL : http://ptp.ipap.jp/link?PTP/4/423/
DOI : 10.1143/PTP.4.423