  We treat the probability distributions for quadratic quantum fields, averaged
with a Lorentzian test function, in four-dimensional Minkowski vacuum. These
distributions share some properties with previous results in two-dimensional
spacetime. Specifically, there is a lower bound at a finite negative value, but
no upper bound. Thus arbitrarily large positive energy density fluctuations are
possible. 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 bounds, the asymptotic forms for large positive argument,
and possible fits to the intermediate region. The first 65 moments are used for
these purposes. All of our results are subject to the caveat that these
distributions are not uniquely determined by the moments. However, we also give
bounds on the cumulative distribution function that are valid for any
distribution fitting these moments.We apply the asymptotic form of the
electromagnetic energy density distribution to estimate the nucleation rates of
black holes and of Boltzmann brains.
