Modular Time Evolution and the QNEC
Stefan Hollands
TL;DR
This work establishes a rigorous bound on the large-s evolution of states under modular flow for a wedge in quantum field theory, linking chaos-like growth to the quantum null energy condition via measured relative entropy in half-sided modular inclusions. By formulating and proving inequalities involving S_{meas}(ϕ_s || ϕ_s ∘ T ∘ R) and leveraging non-commutative L_p interpolation, it connects dilations in Rindler time to an upper bound on Lyapunov-type growth with exponent 2π, and situates these results within the CF18 QNEC framework. The analysis generalizes beyond ergodic modular flows, using abstract modular-theoretic constructs and providing a mathematical bridge between chaos bounds and energy conditions, with potential implications for holography and semi-classical gravity.
Abstract
We establish an inequality restricting the evolution of states in quantum field theory with respect to the modular flow of a wedge, $Δ^{is}$, for large $|s|$. Our bound is related to the quantum null energy condition, QNEC. In one interpretation, it can be seen as providing a ``chaos-bound'' $\le 2π$ on the Lyapunov exponent with respect to Rindler time, $s$. Mathematically, our inequality is a statement about half-sided modular inclusions of von Neumann algebras.