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

1975; American Physical Society; Volume: 11; Issue: 12 Linguagem: Inglês

10.1103/physrevd.11.3504

1538-4500

Daniel Zwanziger,

Quantum Mechanics and Applications

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 ${C}_{i}{({{m}_{i}}^{2}\ensuremath{-}{{p}_{i}}^{2})}^{1+{\ensuremath{\zeta}}_{i}}$ for each particle $i$, where ${\ensuremath{\zeta}}_{i}$ 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 ${\ensuremath{\delta}}_{i}={{m}_{i}}^{2}\ensuremath{-}{{p}_{i}}^{2}$ regularize Feynman integrals in the infrared region, and the dependence on the ${\ensuremath{\delta}}_{i}$ is canceled against the expansion of ${({{m}_{i}}^{2}\ensuremath{-}{{p}_{i}}^{2})}^{{\ensuremath{\zeta}}_{i}}$ 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 $\ensuremath{\alpha}$. 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.