  The real time evolution and relaxation of expectation values of quantum
fields and of quantum states are computed as initial value problems by
implementing the dynamical renormalization group (DRG).Linear response is
invoked to set up the renormalized initial value problem to study the dynamics
of the expectation value of quantum fields. The perturbative solution of the
equations of motion for the field expectation values of quantum fields as well
as the evolution of quantum states features secular terms, namely terms that
grow in time and invalidate the perturbative expansion for late times. The DRG
provides a consistent framework to resum these secular terms and yields a
uniform asymptotic expansion at long times. Several relevant cases are studied
in detail, including those of threshold infrared divergences which appear in
gauge theories at finite temperature and lead to anomalous relaxation. In these
cases the DRG is shown to provide a resummation akin to Bloch-Nordsieck but
directly in real time and that goes beyond the scope of Bloch-Nordsieck and
Dyson resummations. The nature of the resummation program is discussed in
several examples. The DRG provides a framework that is consistent, systematic
and easy to implement to study the non-equilibrium relaxational dynamics
directly in real time that does not rely on the concept of quasiparticle
widths.
