On the positivity of light-ray operators

TL;DR

This paper investigates the positivity properties of light-ray operators in quantum field theory, providing proofs in certain cases, identifying contradictions, and proposing a refined conjecture for their positivity in physically realizable states.

Contribution

It proves positivity of light-ray operators in specific free scalar theories and two-dimensional CFTs, and introduces a conjecture restricting positivity to physically preparable states.

Findings

Light-ray operators are positive in free scalar theories.

In 2D CFTs, $ L_2$ is positive semi-definite.

Counterexamples exist exploiting infrared loopholes.

Abstract

We consider light-ray operators $\mathcal{L}_{2n} = \int\mathrm{d} x^+ (x^+)^{2n}T_{++}$, where $x^+$ is a null coordinate and $n$ a positive integer, in QFT in Minkowski spacetime in arbitrary dimensions. These operators are generalizations of the average null energy operator, which is positive. We give a proof that the light-ray operators are positive in a non-minimally coupled but otherwise free scalar field theory, and we present various arguments that show that $\mathcal{L}_2$ is positive semi-definite in two-dimensional conformal field theories. However, we are also able to construct reasonable states which contradict these results by exploiting an infrared loophole in our proof. To resolve the resulting tension, we conjecture that the light-ray operators are positive in a more restrictive set of states. These states satisfy stronger conditions than the Hadamard condition, and have the interpretation of states that can be physically prepared. Our proposal is nontrivial even in two-dimensional CFT.