A Method of Renormalization for Unstable Particles

Abstract

We investigate the Lee's model which is modified in such a way as the V state is in the unstable mass relation. A renormalization recipe is proposed, which reduces to the conventional one in the stable case but is essentially different from the latter in some respects. The result obtained by this method represents certainly the unstability of the V state. However, its decay character is more complicated than the ordinarily accepted exponential time·dependence. Some new features introduced .by our method are discussed. § I. Introduction Quantum electrodynamics has made so great progress that it is not only nearly complete in its form but also it gives satisfactory results in comparison with experiments. In its course of development; it has been the guiding principle that the theory must be formulated in a relativistically invariant form and that it is sufficient to treat only observable quantities. Further, quantum electrodyn .mics plays an important role as model in analysis of the phenomena in which so-called elementary particles take parts. The present field theory thus constructed, however, still involves many unsolved problems, and some of them seem to compel us to introduce some new concept or to reconstruct the field theory from somewhat different standpoint. It has so closed a form that any partial modification destroy the theorys at all. Therefore, it might be necessary to re­ formulate the theory from the beginning by introducing a new idea. Let us now take out the following two questions from tho~e which we bear in mind. i) Although the quantit2.tive agreement between theoretical calculations and experi­ ments is very surprising in quantum electrodynamics, meson theory is still nothing but a phenomenology in spite of its qualitative success. Why is this the case ? ii) The field theory has succeeded in qu...ntum electrodynamics to a certain extent, but at the preoent [t2.ge it owes the finite results to the temporal method of renormali­ zation. How should we treat this problem in future step ? Here we may remember the fact that the present field theory (especially the second quantization and renormalization) has been constructed on the ground of the stable particle image. If we could reconstruct field theory in such a way that it would contain the conventiortal field theory as the limit where all concerned particles are stable, the interpretation of the first problem would become clea.r. That ie, nucleons and mesons are more or less unstable in contrast to electrons and photons, and, in addition, meson