  The quantum critical dynamics of the quantum phase transitions is considered.
In the framework of the unified theory, based on the Keldysh technique, we
consider the crossover from the classical to the quantum description of the
boson many-particle system dynamics close to the second order quantum phase
transition. It is shown that in this case the upper critical space dimension of
this model is $d_c^{+}=2$, therefore the quantum critical dynamics approach is
useful in case of $d<2$. In the one-dimension system the phase coherence time
does diverge at the quantum critical point, $g_c$, and has the form of $\tau
\propto -\ln |g-g_c|/(g-g_c) $, the correlation radius diverges as $r_c\propto
|g-g_c|^{-\nu} (\nu = 0.6)$.
