  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.
