Nonequilibrium evolution of ${\Phi }^4$ theory in $1+1$ dimensions in the two-particle point-irreducible formalism

We consider the out-of-equilibrium evolution of a classical condensate field and its quantum fluctuations for a ${\Phi }^4$ model in $1+1$ dimensions with a symmetric and a double well potential. We use the two-particle point-irreducible (2PPI) formalism and go beyond the Hartree approximation by including the sunset term. In addition to the mean field $\phi \mathrm{}\left(t\right)=〈\Phi 〉$ the 2PPI formalism uses as a variational parameter a time dependent mass ${\mathcal{M}}^2\mathrm{}\left(t\right)\mathrm{}$ which contains all local insertions into the Green’s function. We compare our results to those obtained in the Hartree approximation. In the symmetric ${\Phi }^4$ theory we observe that the mean field shows a stronger dissipation than the one found in the Hartree approximation. The dissipation is roughly exponential in an intermediate time region. In the theory with spontaneous symmetry breaking, i.e., with a double well potential, the field amplitude tends to zero, i.e., to the symmetric configuration. This is expected on general grounds: in $\mathrm{}(1+1)\mathrm{}$-dimensional quantum field theory there is no spontaneous symmetry breaking for $T>\mathrm{}0,$ and so there should be none at finite energy density (microcanonical ensemble), either. Within the time range of our simulations the momentum spectra do not thermalize and display parametric resonance bands.