Averaged Null Energy Condition from Causality

TL;DR

The paper proves an averaged null energy condition (ANEC) in any unitary, Lorentz-invariant QFT with an interacting UV fixed point in spacetime dimensions $d>2$ by deriving a causality-based lightcone OPE that isolates a universal, positive contribution from the integrated null energy $\mathcal{E}=\int du\,T_{uu}$. It provides a manifestly positive sum rule connecting $\mathcal{E}$ to four-point functions and extends the construction to an infinite family of higher-spin operators $\mathcal{E}_s$, yielding new sign constraints on three-point couplings and coupling signs that must be consistent across probes. The approach reproduces, unifies, and strengthens known bounds such as Hofman-Maldacena conformal collider constraints, deep inelastic scattering, and lightcone bootstrap results, while clarifying their information-theoretic origins. It also discusses extensions to non-conformal theories and outlines a deeper link between causality and quantum information inequalities in QFT. The results provide a broad, positivity-based framework for constraining operator couplings across spinning probes and offer a path toward a more general connection between causality and information theory in quantum field theory.

Abstract

Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int du T_{uu}$, must be positive. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

Figures (4)

• Figure 1: Kinematics for the derivation of the ANEC. The leading correction to the $\psi\psi$ OPE is the null energy integrated over the red line, which in the limit of large $u$ takes the form $\langle \overline{O} \int du T_{uu} O\rangle$. This is then related to an expectation value by a Euclidean rotation.
• Figure 2: Operator insertions in Minkowski spacetime. In the limit where the two $\psi$'s become null, but are widely separated in $u$, the leading non-identity term in the $\psi\psi$ OPE is $\int du T_{uu}$, integrated over the red null line.
• Figure 3: Operator insertions on the $\tau y$-plane defining the smeared 4-point function $\langle \overline{\Theta}_0 \Theta_0 \rangle$.
• Figure 4: Two different ways to interpret the same Euclidean theory. The Euclidean $R^d$ (horizontal orange plane, parameterized by $(\tau,y,\vec{x})$) is the same in both pictures, but the definition of states and corresponding notion of Minkowski spacetime (vertical blue planes) is different in the two cases. On the left, the continuation to Lorentzian is $\tau \to i t$, states of the theory are defined on the plane $\tau = 0$, and $y$ is a space direction. On the right, the continuation to Lorentzian is $y \to i t'$, states are defined at $y = 0$, and $\tau=y'$ is a space direction. The two theories are identical, since they are determined by the same set of Euclidean correlators, but the map of observables and matrix elements from one description to the other is nontrivial.