Quantum Electrodynamics of Charged Particles without Spin

Abstract

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 $\ensuremath{\lambda}{\ensuremath{\varphi}}^{*}(x){\ensuremath{\varphi}}^{*}(x)\ensuremath{\varphi}(x)\ensuremath{\varphi}(x)$ in the Hamiltonian must be postulated. The infinite coupling constant $\ensuremath{\lambda}$ 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, \textonehalf{}, and 1 is proven in the Appendix.