Probability distributions for quantum stress tensors in two and four dimensions

TL;DR

This paper investigates the probability distributions of quantum stress tensor fluctuations in two and four dimensions, providing exact results in two dimensions and estimations in four dimensions, with implications for black hole and Boltzmann brain nucleation.

Contribution

It presents the first exact analytic form for quantum stress tensor distributions in two dimensions and estimates for four dimensions using moment calculations.

Findings

Exact distribution in 2D for quantum stress tensors.

Estimated distribution bounds and tail behavior in 4D.

Implications for black hole and Boltzmann brain nucleation rates.

Abstract

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.