Abstract
The Wick-Hori theorem is used to write the $S$ matrix in the Feynman and Schwinger formulations in which virtual-photon effects and virtual-electron-positron effects are respectively taken into account to all orders in $\frac{{e}^{2}}{\mathbf{\ensuremath{\hbar}} c}$. Physical amplitudes in the Feynman formulation are represented by a finite number of graphs containing the exact off-mass-shell particle-to-particle amplitudes of electron-positron processes. In the Schwinger formulation physical amplitudes are represented by an infinite number of graphs containing the exact off-mass-shell particle-to-particle amplitudes of photon processes. Functional representation of the Compton scattering is given, and its asymptotic behavior is discussed. The differential equations which interrelate the off-mass-shell particle-to-particle amplitudes are given in the Appendix. Problems related to divergences are not considered.
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