Rigorous proof of the ANEC in Minkowski spacetime

Prove rigorously that the averaged null energy condition (ANEC) holds for quantum field theories in Minkowski spacetime; concretely, establish that for each transverse position the null-integrated component of the stress tensor along a complete null generator has nonnegative expectation value in all physical states of the theory.

Background

The ANEC asserts the nonnegativity of the null-integrated stress tensor and underpins many classical results in general relativity. Although quantum fields can violate pointwise energy conditions, substantial evidence indicates that the ANEC should still hold in Minkowski spacetime. In particular, bootstrap- and algebraic-method-based arguments provide compelling support for the ANEC’s validity in flat space.

Despite this evidence, a fully rigorous proof encompassing general quantum field theories in Minkowski spacetime has not yet been established. Clarifying this would close an important conceptual gap and solidify many modern applications of light-ray operators and energy conditions in quantum field theory.