Proof of the Quantum Null Energy Condition (1509.02542v2)

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 $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$.

• The paper rigorously proves the QNEC, establishing a bound between the stress-energy tensor and the second variation of von Neumann entropy.
• It employs null quantization and the replica trick on stationary null surfaces in Minkowski space to simplify the complex analysis of quantum fields.
• The findings reinforce the significance of entropic measures in constraining quantum energy conditions, independent of gravitational influences.

Overview of the Proof of the Quantum Null Energy Condition

The paper "Proof of the Quantum Null Energy Condition" authored by Raphael Bousso, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall, offers a rigorous proof of the Quantum Null Energy Condition (QNEC) within quantum field theory (QFT). This paper delves deeply into the fundamental constraints of stress-energy tensors in quantum systems and explores the intricate link between quantum information theory and physical phenomena in quantum field theory.

The QNEC provides a lower bound on the expectation value of the stress-energy tensor component associated with a null vector, expressed in terms of the second variation of the von Neumann entropy of a quantum region. Originally, the QNEC emerged as a consequence of the Quantum Focusing Conjecture (QFC) related to quantum gravity. However, the present focus on proving it within QFT highlights its independence from gravitational considerations.

Structure of the Paper

The authors methodically structure their proof within the framework of free and superrenormalizable bosonic field theories, focusing on applications to points residing on stationary null surfaces. Their approach primarily leverages Minkowski space as an exemplar backdrop where a null plane is defined, facilitating the derivation of the QNEC. This methodological focus enhances the theoretical clarity of the proof by isolating the effect of curvature and gravitational influences, thereby providing a more generalized assertion within QFT.

Key conceptual innovations in the proof include:

• Utilizing the properties of null surfaces where expansion and shear vanish, thereby simplifying the mathematical complexity.
• Leveraging null quantization techniques, which effectively reduce the problem to analyzing a large number of one-dimensional chiral scalar fields, thus exploiting the dimensionality to clarify the entropy-stress tensor relationship.
• Employing the replica trick for computing entropy via analytic continuation of Renyi entropies, ensuring rigorous treatment of nontrivial state variations.

Numerical and Theoretical Implications

The results from this paper solidify the theoretical foundation for employing entropy as a fundamental quantity in bounding energetic properties of quantum fields. The formal relationship established between stress tensors and von Neumann entropy highlights a novel physical constraint imposing an "information-theoretic" boundary to quantum energy inequalities.

Furthermore, the authors make significant progress in aligning concepts from quantum gravity with those in quantum field theory, emphasizing how entropic bounds can emerge naturally within nongravatational settings by extending intuitive geometrical intuition into the quantum regime.

Future Research Directions

The paper implicitly suggests multiple avenues for further inquiry. One particularly promising direction is examining the QNEC in the context of interacting quantum field theories, exploring its validity across non-Gaussian states or fields with higher spins beyond simple bosonic theories.

Additionally, investigating the conjectural extensions of the QNEC to fermionic fields will provide a broader applicability and understanding of how such quantum information bounds operate across more comprehensive spectra of the standard model fields and beyond.

Conclusion

This paper contributes a meticulous and insightful proof of the QNEC, reinforcing its role as a cornerstone in understanding energy conditions in quantum contexts. By disentangling gravity from quantum energy constraints, the authors pave the way for a deeper exploration of the quantum structure of spacetime and the fundamental interplay between energy, entropy, and information in quantum field theories.