  Instabilities of equilibrium quantum mechanics are common and
well-understood. They are manifested for example in phase transitions, where a
quantum system becomes so sensitive to perturbations that a symmetry can be
spontaneously broken. Here, we consider the possibility that the time evolution
governing quantum dynamics may be similarly subject to an instability, at which
its unitarity spontaneously breaks down owing to an extreme sensitivity towards
perturbations. We find that indeed such an instability exists, and we explore
its immediate consequences. Interpretations of the results both in terms of
extreme sensitivity to the influence of environmental degrees of freedom, and
in terms of a possible fundamental violation of unitarity are discussed.
