Restrictions on Negative Energy Density in Flat Spacetime
L. H. Ford, Thomas A. Roman
TL;DR
Ford and Roman provide a streamlined derivation of quantum inequality bounds on negative energy density for quantum fields in flat spacetime, using a plane-wave mode expansion. They extend prior results for the massless scalar field to the massive scalar field in 4D and 2D, and derive a bound for the electromagnetic field, all in Minkowski space. The bounds depend on a sampling time and the field mass, with the massless limit recovered and the constraints tightening as mass increases; the results imply that sustained macroscopic negative energy is difficult to achieve and has strong implications for wormholes and related exotic phenomena. In the infinite sampling-time limit, the averaged weak energy condition is recovered, underscoring that these quantum inequalities act as a bridge between pointwise energy conditions and global averages in quantum field theory.
Abstract
In a previous paper, a bound on the negative energy density seen by an arbitrary inertial observer was derived for the free massless, quantized scalar field in four-dimensional Minkowski spacetime. This constraint has the form of an uncertainty principle-type limitation on the magnitude and duration of the negative energy density. That result was obtained after a somewhat complicated analysis. The goal of the current paper is to present a much simpler method for obtaining such constraints. Similar ``quantum inequality'' bounds on negative energy density are derived for the electromagnetic field, and for the massive scalar field in both two and four-dimensional Minkowski spacetime.