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An effective action technique for the time evolution of a closed system consisting of one or more mean fields interacting with their quantum fluctuations is presented. By marrying large-N expansion methods to the Schwinger-Keldysh closed time path formulation of the quantum effective action, causality of the resulting equations of motion is ensured and a systematic, energy-conserving and gauge-invariant expansion about the quasiclassical mean field (s) in powers of 1/N developed. The general method is exposed in two specific examples, O (N) symmetric scalar ^4 theory and quantum electrodynamics (QED) with N fermion fields. The ^4 case is well suited to the numerical study of the real time dynamics of phase transitions characterized by a scalar order parameter. In QED the technique may be used to study the quantum nonequilibrium effects of pair creation in strong electric fields and the scattering and transport processes in a relativistic e^+e^- plasma. A simple renormalization scheme that makes practical the numerical solution of the equations of motion of these and other field theories is described.
Cooper et al. (Mon,) studied this question.