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