On the Interaction of Mesons with the Electromagnetic Field. I

At present, the meson theory is in such an unsatisfactory state that it has not proceeded much beyond its original hypothetical character and can hardly explain the experiments quantitatively. Perhaps this is partly due to the fact that various meson models we now consider do not fit the ``real'' meson well, and partly to the defect in the methods of calculation. The former is related to the fact that experimental evidences about mesons are very few and indirect though for electrons they are quite sufficient to construct the electron theory on them. As to the latter we have now at hand a new means of analysis, the Tomonaga-Schwinger theory, which has proved very useful in the case of electrons. Whether this method is also effective in its application to meson theories or not is, however, not well-known at present, especially in the case of nuclear interaction. In order to solve such problems in good approximation, it might to necessary to resort to other ways of approach, e. g., covariant analogy to the strong coupling treatment or one appropriate to the intermediate coupling. Meanwhile, it seems to be natural to consider that the Tomonaga-Schwinger theory in its present form can be applied also to mesons interacting with the electromagnetic field if some necessary modifications are made, because of the smallness of the coupling constant in this case. It is therefore expected that problems concerning mesons and electromagnetic field in interaction are free to a considerable extent from defects of calculation and that we can find out from this side an effective way of analyzing the meson models. The interaction of the meson fields with the electromagnetic field has been investigated by various authors especially in connection with the problems of polarization of mesonic vacuum, photon self-energy and the electromagnetic self-energy of a meson, most of these investigations having been confied to the examination of applicability of the Tomonaga-Schwinger theory to mesons. It is found that there appears no essential difference between the treatment of meson and electron in the e^2-approximation. We give here some results about the treatment of higher order processes, especially the dynamical reaction of the electro-magnetic field to the mesonic charge-current which involves quantities to be compared with experiments in principle. As is well-known, the Duffin-Kemmer form of equation for (scalar and vector) mesons is preferable when it is convenient to treat mesons as particles. For our purpose, it is necessary to generalize this formalism in a perfectly relativistic way which can be accomplished analogously to the Dirac equation. This form of the meson theory is better than the usual Proca and Klein Gordon form in the sense that in the Duffin-Kemmer case almost all formulas have the same form as those of the electron when γ_μ's, etc. in the latter are replaced by β_μ's etc. and thus the physical meanings of all these quantities can be better understood. Results obtained by these two alternative formalisms are, of course, identical. Our discussion will be given in two papers, since it is too long to be published together. In this paper we shall present the covariant formulation of the meson theory and the S-matrix method describing the scattering of mesons in general (S 2) and then apply it to find the second order dynamical reaction of electro-magnetic field to the mesonic charge-current (S 3). Effects involved in it are discussed in detail one by one, beginning with the vacuum polarization part (S 4). Corrections of mass renormalization type (S 5) and one of the Lamb-shift type (S 6) as well as the discussions about the problems involved in the meson theory will be dealt with in the second paper.