A quantum kinetic equation, valid for relativistic unmagnetized plasmas, is derived here. This equation describes the evolution of a quantum quasi-distribution, which is the Wigner function for relativistic spinless charged particles in a plasma, and it is exactly equivalent to a Klein-Gordon equation. Our quantum kinetic equation reduces to the Vlasov equation in the classical limit, where the Wigner function is replaced by a classical distribution function. An approximate form of the quantum kinetic equation is also derived, which includes first order quantum corrections. This is applied to electron plasma waves, for which a new dispersion relation is obtained. It is shown that quantum recoil effects contribute to the electron Landau damping with a third order derivative term. The case of high frequency electromagnetic waves is also considered. Its dispersion relation is shown to be insensitive to quantum recoil effects for equilibrium plasma distributions.
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J. T. Mendonça (2011) studied this question.