  We review recent achievements in the solution of the initial-value problem
for quantum back-reaction in scalar and spinor QED. The problem is formulated
and solved in the semiclassical mean-field approximation for a homogeneous,
time-dependent electric field. Our primary motivation in examining
back-reaction has to do with applications to theoretical models of production
of the quark-gluon plasma, though we here address practicable solutions for
back-reaction in general. We review the application of the method of adiabatic
regularization to the Klein-Gordon and Dirac fields in order to renormalize the
expectation value of the current and derive a finite coupled set of ordinary
differential equations for the time evolution of the system. Three time scales
are involved in the problem and therefore caution is needed to achieve
numerical stability for this system. Several physical features, like plasma
oscillations and plateaus in the current, appear in the solution. From the
plateau of the electric current one can estimate the number of pairs before the
onset of plasma oscillations, while the plasma oscillations themselves yield
the number of particles from the plasma frequency.
  We compare the field-theory solution to a simple model based on a
relativistic Boltzmann-Vlasov equation, with a particle production source term
inferred from the Schwinger particle creation rate and a Pauli-blocking (or
Bose-enhancement) factor. This model reproduces very well the time behavior of
the electric field and the creation rate of charged pairs of the semiclassical
calculation. It therefore provides a simple intuitive understanding of the
nature of the solution since nearly all the physical features can be expressed
in terms of the classical distribution function.
