Proof of the Quantum Null Energy Condition

TL;DR

This paper proves the Quantum Null Energy Condition (QNEC), establishing a fundamental lower bound on the stress tensor in quantum field theories without gravity, applicable to free and stationary null surface points.

Contribution

The paper provides the first proof of the QNEC within quantum field theory, independent of gravity, for free and superrenormalizable bosonic theories on stationary null surfaces.

Findings

QNEC proven for free and superrenormalizable bosonic theories.

QNEC applies at points on stationary null surfaces like Rindler horizons.

Establishes a lower bound on stress tensor via second variation of entropy.

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 Focussing 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 superrenormalizable 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 $\Sigma$ that contains $p$ and lies on $N$, one can consider the von Neumann entropy $S_\text{out}$ of the quantum state restricted to one side of $\Sigma$. A second variation $S_\text{out}^{\prime\prime}$ can be defined by deforming $\Sigma$ along $N$, in a small neighborhood of $p$ with area $\cal A$. The QNEC states that $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$.