Scale anomalies imply violation of the averaged null energy condition

It is important to know whether or not quantum field theory on a fixed but curved background spacetime satisfies the so-called “averaged null energy condition” (ANEC). This result has a bearing on various important issues in semiclassical quantum gravity; such as the singularity theorems, the topological censorship theorem, and the positive mass theorem. A comment by Wald and Yurtsever indicates that the ANEC is often violated in curved spacetimes. In this letter I explore this issue in more detail, and succeed in characterizing a broad class of spacetimes in which the ANEC is guaranteed to be violated. I shall show that, modulo appropriate technical caveats, the occurrence of a particular quantum field theoretic anomaly — the scale anomaly — implies generic violations of the ANEC. Implications of this result are then briefly discussed.