  In this paper, the real-time dynamics of the $O(4)$ scalar theory is studied
within the functional renormalization group formulated on the Schwinger-Keldysh
closed time path. The flow equations for the effective action and its $n$-point
correlation functions are derived in terms of the "classical" and "quantum"
fields, and a concise diagrammatic representation is presented. An analytic
expression for the flow of the four-point vertex is obtained. Spectral
functions with different values of temperature and momentum are obtained.
Moreover, we calculate the dynamical critical exponent for the phase transition
near the critical temperature in the $O(4)$ scalar theory in $3+1$ dimensions,
and the value is found to be $z\simeq 2.023$.
