  We develop a general theory of non-equilibrium states based on the Keldysh
formalism, in particular, for charged-particle systems under static uniform
electromagnetic fields. The Dyson equation for the uniform stationary state is
rewritten in a compact gauge-invariant form by using the Moyal product in the
phase space of energy-momentum variables, which originally do not commute in
the case of the conventional operator algebra. Expanding the Dyson equation in
electromagnetic fields, a systematic method for the order-by-order calculation
of linear and non-linear responses from the zeroth-order Green's functions is
obtained. In particular, we find that for impurity problems, up to linear order
in the electric field, the present approach provides a diagrammatic method for
the St\u{r}eda formula. This approach also generalizes the semi-classical
Boltzmann transport theory to fully quantum-mechanical and/or multi-component
systems. In multi-component systems and/or for Hall transport phenomena,
however, this quantum Boltzmann transport theory, constructed from the
anti-symmetric combination of two different representations for the Dyson
equation, does not uniquely specify the non-equilibrium state, but the
symmetric combination is required. We demonstate the formalism to calculate
longitudinal and Hall electric conductivities in an isotropic single-band
electron system in the clean limit. It is found that the results are fully
consistent with those obtained by Mott and Ziman in terms of the semi-classical
Boltzmann transport theory.
