  We show that particles can split only when their group velocity exceeds their
phase velocity. In this sense the splitting process is the quantum analog of
the modulational instability in anomalous dispersive media. In the case of a
neutrino whose dispersion relation contains a subdominant Lorentz-violating
correction of the form aP^k, the neutrino will decay into two neutrinos and an
antineutrino at a rate proportional to a^3G_F^2E^{2+3k}. Unlike the
Cohen-Glashow instability, the splitting instability exists even if all
particles involved in the interaction have the same dispersion relations at the
relevant energy scales. We show that this instability leads to strong
constraints even if the energy E is a function of both the momentum P and also
of the background density rho, for example we show that it alone would have
been sufficient to eliminate any model of the MINOS/OPERA velocity anomaly
which modifies the neutrino dispersion relation while leaving those of other
particles intact.
