  Radiation spectrum of extreme relativistic electrons and a probability of
electron-positron pair production by energetic photons in a strong plasma
wakefield are derived in the framework of a semiclassical approach. It is shown
that that the radiation losses of the relativistic electron in the plasma
wakefield scale as $ \propto \epsilon ^{2 / 3}$ in the quantum limit when the
energy of the radiated photon becomes close to the electron energy, $\epsilon
$. The quantum effects will play a key role in future plasma-based accelerators
operating at ultrahigh energy of the electrons.
