  A result from particle tracking states that, after a microbunched electron
beam is kicked, its trajectory changes while the orientation of the
microbunching wavefront remains as before. Experiments at the LCLS showed that
radiation in the kicked direction is produced practically without suppression.
This could be explained if the orientation of the microbunching wavefront is
readjusted along the kicked direction. In previous papers we showed that when
the evolution of the electron beam modulation is treated according to
relativistic kinematics, the orientation of the microbunching wavefront in the
ultrarelativistic asymptotic is always perpendicular to the electron beam
velocity. There we refrained from using advanced theoretical concepts to
explain or analyze the wavefront rotation. For example, we only hinted to the
relation of this phenomenon with the concept of Wigner rotation. This more
abstract view of wavefront rotation underlines its elementary nature. The
Wigner rotation is known as a fundamental effect in elementary particle
physics. The composition of non collinear boosts does not result in a simple
boost but, rather, in a Lorentz transformation involving a boost and a
rotation, the Wigner rotation. Here we show that during the LCLS experiments, a
Wigner rotation was actually directly recorded for the first time with a
ultrarelativistic, macroscopic object: an ultrarelativistic electron bunch in
an XFEL modulated at nm-scale of the size of about 10 microns. Here we point
out the role of Wigner rotation in the analysis and interpretation of
experiments with ultrarelativistic, microbunched electron beams in FELs. After
the beam splitting experiment at the LCLS it became clear that, in the
ultrarelativistic asymptotic, the projection of the microbunching wave vector
onto the beam velocity is a Lorentz invariant, similar to the helicity in
particle physics.
