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 $1/N$.
Highlights
AND SUMMARYThe reduced density operator ρ for a region in quantum field theory encodes all of the information about observables localized to that region
We conclude by discussing the generality of our analysis, some implications and future directions, and connections with previous work
Though we restricted to cuts of Rindler horizons in flat space for simplicity, all of our results continue to hold for cuts of bifurcate Killing horizons for QFTs defined in arbitrary spacetimes, assuming the quantum null energy condition (QNEC) is true and saturated in the vacuum in this context
Summary
The reduced density operator ρ for a region in quantum field theory encodes all of the information about observables localized to that region. Above an arbitrary cut v 1⁄4 VðyÞ of a null plane is given by dd−2y ðv − VðyÞÞTvvdv: VðyÞ ð4Þ This equation has been previously derived by Wall for free field theories [6] building on [7,8] and to linear order in the. II, the only remaining question is whether the QNEC is saturated in the vacuum state for entangling surfaces which are cuts of a null plane This has been shown for free theories in [11]. IV we will conclude with a discussion of possible extensions to curved backgrounds and more general regions, connections between the relative entropy and the QNEC, and relations to other work
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