The Fourth-Order Contribution to the Self-Energy of the Electron

The self-energy of a free electron is calculated to order $e^4$. The modification of quantum electrodynamics suggested by Feynman is used to make the matrix elements finite. After mass renormalization, using the self-energy to order $e^2$, there remain four Feynman diagrams whose matrix elements must be calculated. Two of these represented the two ways in which two virtual quanta may be emitted and absorbed by the electron, both quanta being present at the same time. The third diagram represents the emission of a virtual quantum which creates a virtual pair, the quantum produced by the pair annihilation being absorbed by the electron. The last diagram corresponds to the emission and absorption of a virtual quantum accompanied by a mass renormalization.