A nonequilibrium Ward identity ͑NE WI͒ connecting the scalar transport vertex correction with one-particle self-energy is derived using the global U͑1͒ symmetry of the fermion nonequilibrium Green's functions ͑NGF͒. The nonperturbative derivation does not depend on the details of the many-body system. A renormalized multiplicative composition rule for the NGF, reflecting time coherence, is obtained and related to the NE WI. Applications involve ͑i͒ testing the consistency of approximations shown in the example of a self-consistent Born approximation for disorder scattering, and ͑ii͒ in the general quantum transport theory, the formalism permits one to assess routes to generalized master equations, in particular those based on the generalized Kadanoff-Baym ansatz.

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This review deals with the state of the art and perspectives of description of nonequilibrium many-body systems using the nonequilibrium Green's function (NGF) method. The basic aim is to describe time evolution of the many-body system from its initial state over its transient dynamics to its long time asymptotic evolution. First, we discuss basic aims of transport theories to motivate the introduction of the NGF techniques. Second, this article summarizes the present view on construction of the electron transport equations formulated within the NGF approach to nonequilibrium. We discuss incorporation of complex initial conditions to the NGF formalism, and the NGF reconstruction theorem, which serves as a tool to derive simplified kinetic equations. Three stages of evolution of the nonequilibrium, the first described by the full NGF description, the second by a non-Markovian generalized master equation and the third by a Markovian master equation will be related to each other.