Quantum null energy condition

File:Simple light cone diagram.svg

A light cone in flat spacetime. Its boundary consists of null directions followed by light.

The quantum null energy condition (QNEC) is an inequality in quantum field theory that relates energy flowing along a null direction to changes in entanglement entropy. Its classical counterpart, the null energy condition, requires the corresponding component of the stress–energy tensor to be non-negative. Quantum fields can violate that pointwise restriction, so QNEC replaces simple positivity with a lower bound determined by the quantum state.

Formulation

To formulate the condition, a codimension-two surface $\mathrm{\Sigma }$ is chosen through a point $p$, together with a null vector $k^a$ orthogonal to it. A small part of the surface is then displaced along the null direction. On a stationary null surface in flat spacetime, QNEC may be written as

$⟨T_{kk}(p)⟩\ge \frac{\hslash }{2\pi }\underset{{𝒜}\to 0}{\lim }\frac{S_{{\mathrm{o}\mathrm{u}\mathrm{t}}^{″}}}{{𝒜}}.$

Here, $T_{kk}=T_{ab}{k}^a{k}^b$ is the stress–energy tensor projected along the null direction. The quantity $S_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the von Neumann entropy of the quantum fields on one side of $\mathrm{\Sigma }$, while the double prime denotes its second variation as the surface is moved. Dividing by the small transverse area ${𝒜}$ produces a local comparison between energy and entanglement. Negative null energy is therefore allowed, but its magnitude is constrained by the accompanying change in entropy.

Development

QNEC arose from the quantum focusing conjecture, which extends the focusing of light rays in general relativity to systems containing quantum matter. The conjecture is expressed through the generalized entropy

$S_{\mathrm{g}\mathrm{e}\mathrm{n}}=\frac{A}{4G\hslash }+S_{\mathrm{o}\mathrm{u}\mathrm{t}},$

where $A$ is the area of a surface and $S_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the entropy outside it. For locally parallel light rays, the conjecture leads to QNEC when gravitational effects are taken to zero. The resulting inequality contains no Newton constant and can be formulated entirely within quantum field theory.

File:RindlerObserversCartesian.png

World lines of uniformly accelerated observers in a Rindler wedge. The red null lines form a Rindler horizon.

The first proof applied to free and superrenormalizable bosonic field theories at points lying on stationary null surfaces, including Rindler horizons in flat spacetime. A separate holographic proof established the condition for certain conformal field theories and related theories with gravitational duals. A later proof used modular Hamiltonians, causality and null deformations to establish QNEC for broad classes of interacting relativistic quantum field theories.

Scope and interpretation

The flat-spacetime form of QNEC is the most direct because its energy and entropy terms can be defined without additional curvature contributions. In curved spacetime, renormalization may introduce geometric terms depending on the spacetime dimension and on the expansion or shear of the chosen null surface. Additional conditions are therefore needed in some dimensions to obtain a finite, scheme-independent inequality.

QNEC connects the local flow of energy with the information contained in a quantum state. Rather than imposing an absolute prohibition on negative energy, it limits that energy through the response of entanglement entropy to a lightlike deformation. This relationship places the condition at the intersection of quantum field theory, quantum information and semiclassical gravity.

See also

• Averaged null energy condition
• Energy condition
• Quantum field theory in curved spacetime
• Quantum inequality
• Stress–energy tensor
• Von Neumann entropy

References

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