Variational Multi-Time Green's Functions for Nonequilibrium Quantum Fields

The time-dependent variational principle proposed by Balian and Veneroni is used to provide the best approximation to the generating functional for multi-time Green's functions of a set of (bosonic) observables Q_{\mu). By suitably restricting the trial spaces, the computation of the two-time Green's function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature ${\bf\Phi^4}$ theory in curved space and coupled to gravity.