  Observations and cosmological simulations show galaxy clusters as a family of
nearly self-similar objects with properties that can be described by scaling
relations as a function of e.g. mass and time. Here we study the scaling
relations between the galaxy velocity dispersion and X-ray quantities like
X-ray bolometric luminosity and temperature in galaxy clusters at high
redshifts (0.64 $\leq$ z $\leq$ 1.46). We also compare our results with the
similar study of the local HIFLUGCS sample. For the analysis, we use a set of
15 distant galaxy clusters extracted from the literature plus a sample of 10
newly discovered clusters selected in X-rays by the \XMM Distant Cluster
Project (XDCP) with more than 10 confirmed spectroscopic members per cluster.
We also study the evolution of this scaling relation by comparing the high
redshift results with the data from the local HIFLUGCS sample. We also
investigated the $L_X - T_X$ and the $\sigma_v - T_X$ relations for the 15
clusters in the literature sample. We report the results of the X-ray and
kinematic analysis of 10 newly detected high redshift clusters and provide
their spectroscopic and kinematic details. For the entire, distant sample we
find a slope fully consistent with the one typical of local clusters, albeit
with a large associated uncertainty. The study on the evolution of the
amplitude reveals a positive offset if the self-similar evolution is neglected,
hence possibly indicating the need for including evolutionary effects. However,
the $L_X - T_X$ relation is found to be in good agreement with the local
relation without any significant redshift evolution. Finally, the $\sigma_v -
T_X$ relation appears to slightly deviate from the theoretical expectation that
galaxies and gas particles have a similar specific kinetic energy. However, the
associated uncertainty is currently too large for making any conclusive
statement in this regard.
