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We developed a gauge-covariant formulation of the non-equilibrium Greenfunction method for the dynamical and/or non-uniform electromagnetic field bymeans of the deformational quantization method. Such a formulation is realizedby replacing the Moyal product in the so-called Wigner space by the starproduct, and facilitates the order-by-order calculation of a gauge-invariantobservable in terms of the electromagnetic field. An application of thisformalism to the linear response theory is discussed

progress of theoretical physics

Gauge Covariant Formulation of the Wigner Representation through Deformation Quantization: Application to Keldysh Formalism with an Electromagnetic Field