The effect of the interaction with the radiation field in changing the energy levels of an electron in an external field is calculated using the conventional form of perturbation theory. The infinite self-energy of the electron which occurs in the same approximation is removed by subtracting from the Hamiltonian a "mass operator" $M$. The criteria used in deriving $M$ are that it should correctly give the self-energy for a free electron in the absence of an external field and that the amended Hamiltonian should give a properly convariant form for the level shift in an external field. It is pointed out that $M$ is uniquely determined by these requirements.

The results give 1051 mc/sec. for the $2s_{\frac{1}{2}}-2p_{\frac{1}{2}}$ separation in hydrogen and also show the surplus magnetic moment of the electron as $\frac{\alpha }{2\pi }$ Bohr magnetons, as first found by Schwinger. The amended Hamiltonian can be used for determining the radiative corrections for other processes.

• Received 10 December 1948

DOI:https://doi.org/10.1103/PhysRev.75.1240