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

Probability distributions for quantum stress tensors in two and four dimensions