Electron acceleration due to photon absorption: A possible origin of the   infinity problems in relativistic quantum fields
  Based on the concept of extended particles recently introduced we perform a Gedankenexperiment accelerating single electrons with photons of suitably low frequency. Accounting for relativistic time dilation due to the acquired velocity and in infinite repetition of single absorption processes it can be shown that the kinetic energy in the infinite limit is equal to m_{e} c^{2}/2. However, the inertial mass of the electron seems enhanced, and it can be established that this enhancement is described by the relativistic mass effect. It appears, therefore, that although there exists a singularity in interactions - the frequency required to accelerate the particle near the limit of c becomes infinite - the energy of the particle itself approaches a finite limit. Comparing with calculations of the Lamb-shift by Bethe this result seems to provide the ultimate justification for the renormalization procedures employed. 
W. A. Hofer (TU-Wien)