Quantum Electrodynamics of Charged Particles without Spin

Feynman's formulation of quantum electrodynamics is shown to be equivalent to the Schwinger-Tomonaga theory also for spinless charged particles (mesons) as developed by Kanesawa and Tomonaga. The divergencies of the scattering matrix are analyzed to all orders in the fine-structure constant and it is found that mass and charge renormalizations do not remove all divergencies, unlike the electron case. The remaining divergence is associated with the meson-meson interaction and occurs in all orders of radiative corrections except the lowest (second) order in which the process can exist. In order to make the scattering matrix completely finite a direct interaction term $\lambda {\phi }^{*}\mathrm{}\left(x\right)\mathrm{}{\phi }^{*}\mathrm{}\left(x\right)\mathrm{}\phi \mathrm{}\left(x\right)\mathrm{}\phi \mathrm{}\left(x\right)\mathrm{}$ in the Hamiltonian must be postulated. The infinite coupling constant $\lambda$ is to be renormalized by an infinite renormalization. One obtains a finite amount of direct interation which must be determined from experiment. The identical cancellation of certain divergencies to all orders of the fine-structure constant and valid spin 0, ½, and 1 is proven in the Appendix.