Averaged Null Energy Condition from Causality

TL;DR

This paper proves that causality in quantum field theories implies the averaged null energy condition, leading to new constraints on operator couplings and strengthening existing bounds from various methods.

Contribution

It establishes a causality-based proof of the averaged null energy condition and derives new inequalities for spinning operators in conformal field theories.

Findings

A positivity sum rule for the averaged null energy operator.

Infinite family of new constraints for higher spin operators.

Stronger bounds on operator couplings than previous methods.

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.