Abstract
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.
Highlights
Energy conditions play a distinguished role at the interface between classical and quantum physics
A slightly different phrasing of this inquiry is the following: “Under what conditions can we regard the null energy of an effective field theory as a genuine operator?" For one, we show that no amount of smearing along the transverse lightsheet coordinates provides such a definition
We construct an explicit class of squeezed states that realize this bound, at least parametrically with the UV cutoff. After this we propose the lower-boundedness of the DSNEC and argue for its validity through two methods: firstly we calculate its vacuum two-point function and show that it is bounded and secondly we reexamine the DSNE in the same class of smeared states saturating the Smeared Null Energy Condition" (SNEC)
Summary
Energy conditions play a distinguished role at the interface between classical and quantum physics. Two important examples in this direction are the Achronal Average Null Energy Condition (AANEC) [3–7] and the Quantum Null Energy Condition (QNEC) [8–11]; see [12] for a nice review While these results have varying degrees of applicability to semi-classical gravity, they illuminate the fact that energy inequalities are interesting objects in their own right for a quantum field theory, revealing an interesting interplay between null energy, causality [13], and quantum information [7, 10]. A slightly different phrasing of this inquiry is the following: “Under what conditions can we regard the null energy of an effective field theory as a genuine operator?" For one, we show that no amount of smearing along the transverse lightsheet coordinates provides such a definition. Let us pause to mention the following at the onset: while motivated by the role of null energy conditions in semi-classical gravity, the results of this article are purely field theoretical.
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