RenormalizedS-Matrix for Scalar Electrodynamics

By a consistent use of the concepts of mass and charge renormalization Dyson has demonstrated the possibility of constructing a divergence-free $S$-matrix for scattering problems in spinor-electrodynamics, valid to all orders in the fine-structure constant. In this paper a proof is given of the possibility of renormalizing theories of charged scalar (or pseudoscalar) particles in the presence of the electromagnetic field. It is found that in addition to the renormalization of mass and charge, an infinite constant of direct interaction $\delta \lambda$ has to be introduced in a term $\delta \lambda {\phi }^{*2\mathrm{}}{\phi }^2$ added to the Hamiltonian, in order to cancel consistently all divergences arising from the Møller scattering of one spinless particle by another. This, combined with the fact that theories of charged scalar and pseudoscalar mesons in scalar interaction with the nucleons can be renormalized with the same additional term in the Hamiltonian, seems to be of some significance.