Probability distributions for quantum stress tensors in four dimensions

TL;DR

This paper investigates the probability distributions of quantum stress tensors in four-dimensional Minkowski space, revealing properties like a finite negative lower bound and unbounded positive fluctuations, with implications for black hole and Boltzmann brain nucleation.

Contribution

It extends the analysis of quantum stress tensor distributions from two to four dimensions using moment calculations and bounds, providing new insights into energy fluctuation probabilities.

Findings

Distributions have a finite negative lower bound and no upper bound.

Used 65 moments to estimate distribution bounds and asymptotic behavior.

Applied results to estimate black hole and Boltzmann brain nucleation rates.

Abstract

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.