Evolution of Inhomogeneous Condensates: Self-consistent Variational   Approach
  We establish a self-consistent variational framework that allows us to study numerically the non-equilibrium evolution of non-perturbative inhomogeneous field configurations including quantum backreaction effects. After discussing the practical merits and disadvantages of different approaches we provide a closed set of local and renormalizable update equations that determine the dynamical evolution of inhomogeneous condensates and can be implemented numerically. These incorporate self-consistently the backreaction of quantum fluctuations and particle production. This program requires the solution of a self-consistent inhomogeneous problem to provide initial Cauchy data for the inhomogeneous condensates and Green's functions. We provide a simple solvable ansatz for such an initial value problem for the Sine-Gordon and phi^4 quantum field theories in one spatial dimension. We compare exact known results of the Sine Gordon model to this simple ansatz. We also study the linear sigma model in the large N limit in three spatial dimensions as a microscopic model for pion production in ultrarelativistic collisions. We provide a solvable self- consistent ansatz for the initial value problem with cylindrical symmetry. For this case we also obtain a closed set of local and renormalized update equations that can be numerically implemented. A novel phenomenon of spinodal instabilities and pion production arises as a result of a Klein paradox for large amplitude inhomogeneous condensate configurations. 
D. Boyanovsky, F. Cooper, H. J. de Vega and P. Sodano