Dynamical behavior of spatially inhomogeneous relativistic λφ⁴ quantum field theory in the Hartree approximation

We study the dynamics of a spatially inhomogeneous quantum λφ⁴ field theory in 1+1 dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case and, unlike the latter, shows some universality at long times. Nevertheless, we show that, within a family of Gaussian spatial profiles for the mean field, the evolution fails to thermalize in a quantum canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasiclassical inhomogeneous configurations (e.g. topological defects) in a background of dynamical self-consistent quantum fluctuations.