Gauge covariant formulation of Wigner representation through   deformational quantization --Application to Keldysh formalism with   electromagnetic field--
  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. 
N. Sugimoto, S. Onoda and N. Nagaosa