  With the goal in mind of deriving a method to compute quantum corrections for
the real-time evolution in quantum field theory, we analyze the problem from
the perspective of the Wigner function. We argue that this provides the most
natural way to justify and extend the classical approximation. A simple
proposal is presented that can allow to give systematic quantum corrections to
the evolution of expectation values and/or an estimate of the errors committed
when using the classical approximation. The method is applied to the case of a
few degrees of freedom and compared with other methods and with the exact
quantum results. An analysis of the dependence of the numerical effort involved
as a function of the number of variables is given, which allow us to be
optimistic about its applicability in a quantum field theoretical context.
