Proof of the quantum null energy condition
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Abstract
We prove the quantum null energy condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the quantum focusing conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and super-renormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point $p$ and null vector ${k}^{a}$ define a null plane $N$ (a Rindler horizon). Given any codimension-2 surface $\mathrm{\ensuremath{\Sigma}}$ that contains $p$ and lies on $N$, one can consider the von Neumann entropy ${S}_{\mathrm{out}}$ of the quantum state restricted to one side of $\mathrm{\ensuremath{\Sigma}}$. A second variation ${S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}}$ can be defined by deforming $\mathrm{\ensuremath{\Sigma}}$ along $N$, in a small neighborhood of $p$ with area $\mathcal{A}$. The QNEC states that $⟨{T}_{kk}(p)⟩\ensuremath{\ge}\frac{\ensuremath{\hbar}}{2\ensuremath{\pi}}\underset{\mathcal{A}\ensuremath{\rightarrow}0}{\mathrm{lim}}{S}_{\mathrm{out}}^{\ensuremath{'}\ensuremath{'}}/\mathcal{A}$.