Scattering theory for quantum electrodynamics. II. Reduction and cross-section formulas

Abstract

The quantum-electrodynamical $S$ matrix is obtained as the set of on-mass-shell values of the renormalized momentum-space Green's functions multiplied by $C_i{({m_i}^2-{p_i}^2)}^{1+{\zeta }_i}$ for each particle $i$, where ${\zeta }_i$ is proportional to the fine-structure constant and $C_i$ is a constant. A photon mass is not needed to eliminate virtual infrared divergences. Instead the parameters ${\delta }_i={m_i}^2-{p_i}^2$ regularize Feynman integrals in the infrared region, and the dependence on the ${\delta }_i$ is canceled against the expansion of ${({m_i}^2-{p_i}^2)}^{{\zeta }_i}$ multiplying lower-order Green's functions. Exact cross-section formulas are developed which express transition rates in terms of this $S$ matrix. They account for radiation damping nonperturbatively, whereas the $S$ matrix must be calculated perturbatively as a power series in $\alpha$. It is seen that in processes with very small energy loss to unobserved photons individual elements of the quantum-electrodynamical $S$ matrix are directly observable. Rules for practical calculations are summarized.