Bounds on negative energy densities in flat spacetime

We generalize results of Ford and Roman which place lower bounds—known as quantum inequalities—on the renormalized 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>~2) for the free real scalar field of mass m>~0. We discuss various features of our bounds in 2 and 4 dimensions. In particular, for massless field theory in two-dimensional Minkowski space, we show that our quantum inequality is weaker than Flanagan’s optimal bound by a factor of 3/2.