Compton Scattering. I. Spectral Forms for the Invariant Amplitudes to Ordere4

The six invariant amplitudes for Compton scattering are explicitly calculated to order $e^4$. Gauge-invariant amplitudes which are free of kinematic singularities and zeros are derived in double-spectral form. They are calculated in the framework of Schwinger's source theory which makes crucial use of causality and space-time extrapolation. The accompanying single-spectral forms (contact terms) are determined by analysis of forward scattering. The resulting amplitudes contain all the information necessary to calculate radiative corrections to polarized electron-photon scattering experiments.