Bounds on negative energy densities in flat spacetime

TL;DR

This paper extends quantum inequalities on negative energy densities for scalar fields in flat spacetime, providing bounds for a broader class of sampling functions and analyzing their features in various dimensions.

Contribution

It introduces new quantum inequalities applicable to smooth, non-negative sampling functions that are either compactly supported or decay rapidly, generalizing previous results.

Findings

Bounds are established for $d$-dimensional Minkowski space.

In 2D massless case, the bounds are weaker than Flanagan's optimal bound by 3/2.

Results apply to free real scalar fields with mass $m \\ge 0$.

Abstract

We generalise results of Ford and Roman which place lower bounds -- known as quantum inequalities -- on the renormalised energy density of a quantum field averaged against a choice of sampling function. Ford and Roman derived their results for a specific non-compactly supported sampling function; here we use a different argument to obtain quantum inequalities for a class of smooth, even and non-negative sampling functions which are either compactly supported or decay rapidly at infinity. Our results hold in $d$-dimensional Minkowski space ($d\ge 2$) for the free real scalar field of mass $m\ge 0$. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in 2-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan's optimal bound by a factor of 3/2.