Quantized meson fields in and out of equilibrium. I : Kinetics of meson   condensate and quasi-particle excitations
  We formulate a kinetic theory of self-interacting meson fields with an aim to describe the freezeout stage of the space-time evolution of matter in ultrarelativistic nuclear collisions. Kinetic equations are obtained from the Heisenberg equation of motion for a single component real scalar quantum field taking the mean field approximation for the non-linear interaction. The mesonic mean field obeys the classical non-linear Klein-Gordon equation with a modification due to the coupling to mesonic quasi-particle excitations which are expressed in terms of the Wigner functions of the quantum fluctuations of the meson field, namely the statistical average of the bilinear forms of the meson creation and annihilation operators. In the long wavelength limit, the equations of motion of the diagonal components of the Wigner functions take a form of Vlasov equation with a particle source and sink which arises due to the non-vanishing off-diagonal components of the Wigner function expressing coherent pair-creation and pair-annihilation process in the presence of non-uniform condensate. We show that in the static homogeneous system, these kinetic equations reduce to the well-known gap equation in the Hartree approximation, and hence they may be considered as a generalization of the Hartree approximation method to non-equilibrium systems. As an application of these kinetic equations, we compute the dispersion relations of the collective mesonic excitations in the system near equilibrium. 
T. Matsui, M. Matsuo