  We developed a gauge-covariant formulation of the non-equilibrium Green
function method for the dynamical and/or non-uniform electromagnetic field by
means of the deformational quantization method. Such a formulation is realized
by replacing the Moyal product in the so-called Wigner space by the star
product, and facilitates the order-by-order calculation of a gauge-invariant
observable in terms of the electromagnetic field. An application of this
formalism to the linear response theory is discussed.
