  We study the local group velocity defined as the weak value of the velocity
operator in the (1+1) dimensional Klein-Gordon as well as Dirac theory. It was
shown by Berry [ J. Phys. A 45, 185308 (2012)] that when the pre- and
post-selected states for evaluating the weak value are chosen at random from an
ensemble of available states, the local group velocity has a universal
probability distribution which can have both subluminal and superluminal
components. In this work, we numerically explore the role of Lorentz boost and
its impact on the superluminal fraction of the total probability distribution.
We show that the dependence (enhancement) of the superluminal fraction on
Lorentz boost of the total probability distribution differs both qualitatively
and quantitatively for the Klein-Gordon waves and Dirac waves. For the
Klein-Gordan waves, the asymmetry in the distribution of group velocities
around the zero velocity point in the laboratory frame is entirely responsible
for the observation of relative enhancement in the boosted frame. On the other
hand, for the Dirac waves, we observe an enhancement irrespective of whether
the laboratory frame velocity distribution is symmetric or not.
