Quantized meson fields in and out of equilibrium. II: Chiral condensate   and collective meson excitations
  We develop a quantum kinetic theory of the chiral condensate and meson quasi-particle excitations using the O(N) linear sigma model which describe the chiral phase transition both in and out of equilibrium in a unified way. A mean field approximation is formulated in the presence of mesonic quasi-particle excitations which are described by generalized Wigner functions. It is shown that in equilibrium our kinetic equations reduce to the gap equations which determine the equilibrium condensate amplitude and the effective masses of the quasi-particle excitations, while linearization of transport equations, near such equilibrium, determine the dispersion relations of the collective mesonic excitations at finite temperatures. Although all mass parameters for the meson excitations become at finite temperature, apparently violating the Goldstone theorem, the missing Nambu-Goldstone modes are retrieved in the collective excitations of the system as three degenerate phonon-like modes in the symmetry-broken phase. We show that the temperature dependence of the pole masses of the collective pion excitations has non-analytic kink behavior at the threshold of the quasi-particle excitations in the presence of explicit symmetry breaking interaction. 
M. Matsuo, T. Matsui