Probability distributions of smeared quantum stress tensors

TL;DR

This paper derives the probability distributions of smeared quantum stress tensors in various field theories, revealing that these distributions are often shifted Gamma distributions and highlighting the high likelihood of negative energy measurements in the vacuum state.

Contribution

It provides the first closed-form probability distributions for smeared quantum stress tensors in 2D conformal field theory and extends analysis to 4D scalar fields, revealing new distribution characteristics.

Findings

In 2D CFT, the distribution is a shifted Gamma distribution.

For small central charge, negative energy measurements are highly probable.

In 4D scalar fields, the energy density distribution differs from the square field distribution.

Abstract

We obtain in closed form the probability distribution for individual measurements of the stress-energy tensor of two-dimensional conformal field theory in the vacuum state, smeared in time against a Gaussian test function. The result is a shifted Gamma distribution with the shift given by the previously known optimal quantum inequality bound. For small values of the central charge it is overwhelmingly likely that individual measurements of the sampled energy density in the vacuum give negative results. For the case of a single massless scalar field, the probability of finding a negative value is 84%. We also report on computations for four-dimensional massless scalar fields showing that the probability distribution of the smeared square field is also a shifted Gamma distribution, but that the distribution of the energy density is not.