  In the present contribution, by studying a fractional version of Dirac's
equation for the electron, we show that the phenomenon of Zitterbewegung in a
coarse-grained medium exhibits a transient oscillatory behavior, rather than a
purely oscillatory regime, as it occurs in the integer case, $\alpha=1$. Our
result suggests that, in such systems, the Zitterbewegung-type term related to
a trembling motion of a quasiparticle is tamed by its complex interactions with
other particles and the medium. This can justify the difficulties in the
observation of this interesting phenomenon. The possibility that the
Zitterbewegung be accompanied by a damping factor supports the viewpoint of
particle substructures in Quantum Mechanics.
