  We investigate the possibility of Many-Body Localization in translation
invariant Hamiltonian systems, which was recently brought up by several
authors. A key feature of Many-Body Localized disordered systems is recovered,
namely the fact that resonant spots are rare and far-between. However, we point
out that resonant spots are mobile, unlike in models with strong quenched
disorder, and that these mobile spots constitute a possible mechanism for
delocalization, albeit possibly only on very long timescales. In some models,
this argument for delocalization can be made very explicit in first order of
perturbation theory in the hopping. For models where this does not work, we
present instead a non-perturbative argument that relies solely on ergodicity
inside the resonant spots.
