Semi-local Bounds on Null Energy in QFT

We investigate whether the null energy, averaged over some region of spacetime, is bounded below in QFT. First, we use light-sheet quantization to prove a version ``Smeared Null Energy Condition" (SNEC) proposed [1], applicable for free and super-renormalizable QFT's equipped with UV cutoff. Through an explicit construction squeezed states, show that SNEC bound cannot be improved by smearing on alone. propose energy two directions defines operator independent cutoff, what call ``Double-Smeared Condition," or DSNEC. indicate schematically how this behaves respect lengths argue DSNEC displays transition when are comparable correlation length.