  We quantize non-commutative Maxwell theory canonically in the background
field gauge for weak and slowly varying background fields. We determine the
complete basis for expansion under such an approximation. As an application, we
derive the Wigner function which determines the leading order high temperature
behavior of the perturbative amplitudes of non-commutative Maxwell theory. To
leading order, we also give a closed form expression for the distribution
function for the non-commutative $U (1)$ gauge theory at high temperature.
