  Following the field theoretic approach of Basko et al., Ann. Phys. 321, 1126
(2006), we study in detail the real-time dynamics of a system expected to
exhibit many-body localization. In particular, for time scales inaccessible to
exact methods, we demonstrate that within the second-Born approximation that
the temporal decay of the density-density correlation function is
non-exponential and is consistent with a finite value for $t\to\infty$, as
expected in a non-ergodic state. This behavior persists over a wide range of
disorder and interaction strengths. We discuss the implications of our findings
with respect to dynamical phase boundaries based both on exact diagonalization
studies and as well as those established by the methods of Ref. 1.
