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With the goal in mind of deriving a method to compute quantum corrections forthe real-time evolution in quantum field theory, we analyze the problem fromthe perspective of the Wigner function. We argue that this provides the mostnatural way to justify and extend the classical approximation. A simpleproposal is presented that can allow to give systematic quantum corrections tothe evolution of expectation values and/or an estimate of the errors committedwhen using the classical approximation. The method is applied to the case of afew degrees of freedom and compared with other methods and with the exactquantum results. An analysis of the dependence of the numerical effort involvedas a function of the number of variables is given, which allow us to beoptimistic about its applicability in a quantum field theoretical context.

Real-time Quantum evolution in the Classical approximation and beyond