Electron self-energy in a magnetic field

In the lowest order in the fine-structure constant, the electron self-energy in an external magnetic field can be written in the form of a double integral representation containing the exact information about the radiative shift and width of the energy levels, without approximation in the field strength. In the low-field expansion of the radiative shift, the leading term is conveniently interpreted in terms of the electron's anomalous magnetic moment, whilst in very strong fields the enhancement of the cyclotron motion makes the shift a positive, slowly increasing function of the field intensity. It follows that, even in superstrong magnetic fields, the electromagnetic interaction cannot give rise to an instability of the electron-positron vacuum.