A more sophisticated approach to quantum transport theory is supplied by the Green's-function formulation of many-body theory. This approach had its origins in the development of the theory of quantum electrodynamics in the late 1940's and early 1950's, and inherits this field's emphasis on perturbation expansions described in diagrammatic form. The non-equilibrium Green's-function theory was formulated by Kadanoff and Baym [59] and by Keldysh [60], was elaborated by Langreth [61], and is described in the text by Mahan [62]. The problems and promise of applying this approach to electron devices has been discussed by Jauho [63]. In particular, most of the development of the Green's function approach has assumed uniform electric fields, which is not adequate for the description of quantum devices. Among the more recent work in this area which addresses problems beyond the uniform field are those of Sols [64] and Rammer [65].
The non-equilibrium Green's functions are defined as expectation values of
single-particle creation and annihilation operators, and they describe the state and
time evolution of the system. There are four independent functions which appear in
the formalism. In the conventional notation 
gives the distribution of electrons
(and reduces to the density matrix or Wigner function in certain limits), 
gives
the distribution of holes, 
describes time evolution into the future and 
describes time evolution into the past. These Green's functions are determined by
solving a set of Dyson equations (an integral form of Schrödinger 's equation) which
form a convenient starting point for the development of a perturbation expansion.
Each of the Gs has two position and two time arguments, which can be transformed via the Wigner-Weyl procedure into one each of a position, momentum, time, and energy (or frequency) argument. The presence of the energy dependence (or the two time arguments) distinguishes the Green's function approach from the Wigner function scheme described above. Because the Wigner function measures the state of the device at a particular time, and its evolution is described by a first-order differential equation, it can only comprehend external interactions which occur instantaneously in time. Such behavior is termed ``Markovian.'' The energy dependence of the Green's functions permits a description of processes which are not local in time, or ``non-Markovian'' processes, because the energy argument provides a way to include convolution integrals over the past history of the system. An example of a process which is non-Markovian is the resonant absorption or emission of a phonon. In order for the energy of the phonon to be well-defined, the interaction must occur over a time greater than the oscillation period of the phonon. A non-Markovian Green's function approach can accurately describe such processes, the Markovian Wigner function approach cannot.
It is fair to say that work on the Green's function approach has produced a great many mathematical formulations and very few explicit calculations of realistic systems. Among the latter is the work of Lake and Datta [66]. They model the resonant-tunneling diode, and include interactions between the electrons and localized phonons. The locality of the interaction removes the momentum argument from the Green's function, and also removes any notion of momentum conservation from the model.
One can identify a general principle here, to the effect that if any of the position, momentum, or energy arguments are missing from the distribution function in a given theory, then a corresponding conservation law is not enforced within that theory. If the position argument is not present, as in the case of the Pauli master equation, then the continuity equation is not enforced. If the momentum argument is absent, as in the approach of Lake and Datta, then conservation of momentum is not enforced. And finally, if the energy argument is absent, as in the Markovian Wigner function theory, then conservation of energy is not enforced. Thus it appears that the only completely satisfactory theory will be a complete Green's function theory which includes all four arguments. Such an approach has not yet been developed into a numerically tractable form.