Fourth-Order Radiative Corrections to Atomic Energy Levels

In this paper the fourth-order radiative corrections to the elastic scattering of an electron in the field of a fixed potential are examined, using the Dyson $S$-matrix formulation of quantum electrodynamics. The result can be represented as an addition to the interaction energy density of the electron with the external potential: $-\frac{\mathrm{ie}\overline{\psi }\mathrm{}\left(x\right)\mathrm{}{\gamma }_{\mu }\psi \mathrm{}\left(x\right)\mathrm{}{\square }^2}{{\kappa }^2{A_{\mu }}^e\mathrm{}\left(x\right)\left[\right(\frac{{\alpha }^2}{4{\pi }^2}\left)\right(0.52\mathrm{}\pm 0.21\left)\right]},$ plus the anomalous magnetic moment already known. The contributions that arise from the vacuum-polarization currents are omitted. This term calculated contributes to the level shift in a hydrogenic atom an energy of $\left(\frac{{\alpha }^4{Z}^4}{n^3}\right)\mathrm{Ry}{\delta }_{l,0\mathrm{}}\left[\right(\frac{4}{{\pi }^2}\left)\right(0.52\mathrm{}\pm 0.21\left)\right].$ For the $2S$ level of hydrogen this is 0.24±0.10 Mc/sec.