On the Foundation and the Applicability of Williams-Weizsäcker Method

Abstract

The energy spectrum of the virtual photon field accompanied by a high energy charged particle which has been derived classically by Williams and W eizsiiclcer independently is derived here by the Feynman-Dyson method, and the equivalence between the Williams-Weizsiicker and the Feynman­ Dyson methods is verified in a general way when the El).ergy transferred from an incident charged particle is small. It is also qualitatively shown that the cross section derived by the Williams-Weizsiidcer method gives a correct order of magnitude. Let us consider a fast charged particle moving with velocity almost equal to light velocity. The field of this particle is almost identical with that of a set of photons with various frequencies. The electromagnetic action of this particle on another charged particle, say at rest, is equivalent to that of these virtual photons. This simplified method (W-W method) is often of great help and yields the same results as the Feynman-Dyson method (F-D method) in several cases, for example Bremsstrahlung3>, the pair production by a charged particle4> and the 1C production by p.-meson5>, etc. This agreement is based on the fact that the longitudinal and scalar photon fields accompanied by the incident charged particle is neglected as compared with the transverse one as was shown classically by Williams and also implicitly on the fact that the energy spectrum of the virtual trans­ verse photon derived by him is essentially the same as the one to be derived &om the F-D method. In our previous. paper6l on the electron pair creation by charged particles, we showed that the contribution to the cross section arises mainly from the process that the spin of the incident particle does not flip and the direction of polarization of the virtual photon is transverse, and the energy spectrum of the virtual transverse photon becomes the same as the classical one when the energy transferred &om the incident particle to the created pair is very small. This conclusion was derived essentially by separating .. the contributions to the cross section according to the directions of the polariza­ tion of the virtual photon. Therefore this separation procedure suggests the possibility of extending the above conclusion to more general cases. To generalize this conclusion &om the quantum electrodynamical point of view is the content of this paper.