Wigner description of massless Dirac plasmas.

We derive a quantum kinetic model describing the dynamics of graphene electrons in phase space based on the Wigner-Moyal procedure. In order to take into account the quantum nature of the carriers, we start from the low-energy Schr\"{o}dinger equation describing the mean-field wavefunction for both the conduction and valence electrons. The equation of motion for the Wigner matrix is established, with the Coulomb interaction being introduced self-consistently (in the Hartree approximation). The long wavelength limit for the plasmon dispersion relation is obtained, for both ungated and gated situations. As an application, we derive the corresponding hydrodynamical equations and add discuss the correct value of the effective hydrodynamic mass of the carriers from first principles, an issue that is crucial in the establishment of the correct hydrodynamics of Dirac electrons, thus paving the stage towards a more comprehensive description of graphene plasmonics.