  It is suggested that the Minkowski vacuum of quantum field theories of a
large number of fields N would be gravitationally unstable due to strong vacuum
energy fluctuations unless an N dependent sub-Planckian ultraviolet momentum
cutoff is introduced. We estimate this implied cutoff using an effective
quantum theory of massless fields that couple to semi-classical gravity and
find it (assuming that the cosmological constant vanishes) to be bounded by
$M_Planck/N^1/4$. Our bound can be made consistent with entropy bounds and
holography, but does not seem to be equivalent to either, and it relaxes but
does not eliminate the implied bound on N inherent in entropy bounds.
