  We consider the scattering of probe particles on an ultra-boosted beam of
charge, in the case that the fields of the beam are strong and must be treated
non-perturbatively. We show that the fields of the ultra-boosted beam act as
stochastic plane waves - scattering amplitudes (of elastic scattering,
nonlinear Compton and nonlinear Breit-Wheeler) are obtained without
approximation by averaging plane wave scattering amplitudes over all possible
plane wave parameters. The relevant plane waves are ultra-short and, as such,
scattering on ultra-boosted beams does not exhibit the conjectured strong-field
behaviour of QED based on the locally constant field approximation.
