Local Modular Hamiltonians from the Quantum Null Energy Condition

TL;DR

This paper derives a universal expression for the modular Hamiltonian of regions defined by smooth cuts of a null plane by leveraging the quantum null energy condition (QNEC) and its vacuum saturation. Under these assumptions, the second derivative of the modular Hamiltonian with respect to the deformation parameter along the null direction is tied to the stress tensor, enabling a double integration that generalizes the Rindler result $K = (2\pi/\hbar) \int x T_{tt} dx$ to a wider class of null-cut geometries via $\Delta K(\lambda) = \frac{2\pi}{\hbar} \int d^{d-2}y \int_{V(y;\lambda)}^{\infty} (v - V(y;\lambda)) T_{vv} \, dv$. The holographic calculation confirms vacuum QNEC saturation to all orders in $1/N$ by invoking entanglement wedge nesting and the causal wedge condition, which forces the bulk extremal surface to lie on the horizon ($\bar U=0$). The results extend to general Killing horizons and suggest broader implications for the relationship between entropy, modular Hamiltonians, and energy conditions, with connections to relative entropy and prior first-order analyses. Together, these findings provide a concrete, locally defined expression for modular Hamiltonians in a new class of regions and strengthen the operational link between entanglement and energy conditions in quantum field theory.

Abstract

The vacuum modular Hamiltonian $K$ of the Rindler wedge in any relativistic quantum field theory is given by the boost generator. Here we investigate the modular Hamiltoninan for more general half-spaces which are bounded by an arbitrary smooth cut of a null plane. We derive a formula for the second derivative of the modular Hamiltonian with respect to the coordinates of the cut which schematically reads $K" = T_{vv}$. This formula can be integrated twice to obtain a simple expression for the modular Hamiltonian. The result naturally generalizes the standard expression for the Rindler modular Hamiltonian to this larger class of regions. Our primary assumptions are the quantum null energy condition --- an inequality between the second derivative of the von Neumann entropy of a region and the stress tensor --- and its saturation in the vacuum for these regions. We discuss the validity of these assumptions in free theories and holographic theories to all orders in $1/N$.

Figures (1)

• Figure 1: This image depicts a section of the plane $u = t-x =0$. The region $\mathcal{R}$ is defined to be one side of a Cauchy surface split by the codimension-two entangling surface $\partial \mathcal{R} = \lbrace{ (u = 0, v = V(y), y) \rbrace}$. The dashed line corresponds to a flat cut of the null plane.