  The probability distributions for the smeared energy densities of quantum
fields, in the two and four-dimensional Minkowski vacuum are discussed. These
distributions share the property that there is a lower bound at a finite
negative value, but no upper bound. Thus arbitrarily large positive energy
density fluctuations are possible. In two dimensions we are able to give an
exact unique analytic form for the distribution. However, in four dimensions,
we are not able to give closed form expressions for the probability
distribution, but rather use calculations of a finite number of moments to
estimate the lower bound, and the asymptotic form of the tail of the
distribution. The first 65 moments are used for these purposes. All of our
four-dimensional results are subject to the caveat that these distributions are
not uniquely determined by the moments. One can apply the asymptotic form of
the electromagnetic energy density distribution to estimate the nucleation
rates of black holes and of Boltzmann brains.
