Wigner functional approach to quantum field dynamics

We introduce the Wigner functional representing a quantum field in terms of the field amplitudes and their conjugate momenta. The equation of motion for the functional of a scalar field points out the relevance of solutions of the classical field equations to the time evolution of the quantum field. We discuss the field in thermodynamical equilibrium and find the explicit solution of the equations of motion for the so-called ''rollover'' phase transition. Finally, we briefly discuss the approximate methods for the evaluation of the Wigner functional that may be used to numerically simulate the initial value problem