Modular Hamiltonians on the null plane and the Markov property of the vacuum state

TL;DR

This work derives the modular Hamiltonians for regions whose future horizon lies on a null plane (equivalently regions bounded on a null cone in a CFT) and shows they admit a local expression as integrals of the stress tensor along each null generator. It provides two complementary proofs and reveals an infinite-dimensional Lie (Virasoro-like) structure governing these modular flows, with positivity and a local action on the null surface. A striking outcome is the Markov property of the vacuum for these regions, which leads to saturation of strong subadditivity and strong super-additivity of relative entropy, with implications for entropic proofs of RG flow theorems. The results generalize to massive deformations and connect to algebraic QFT via half-sided modular inclusions, offering deep insights into the structure of entanglement in QFT and possible links to asymptotic symmetry algebras. Overall, the paper establishes a rich, local-null-geometry framework for modular actions with broad implications across QFT, holography, and quantum information theory.

Abstract

We compute the modular Hamiltonians of regions having the future horizon lying on a null plane. For a CFT this is equivalent to regions with boundary of arbitrary shape lying on the null cone. These Hamiltonians have a local expression on the horizon formed by integrals of the stress tensor. We prove this result in two different ways, and show that the modular Hamiltonians of these regions form an infinite dimensional Lie algebra. The corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities, but act non locally outside the null plane. We regain this result in greater generality using more abstract tools on algebraic quantum field theory. Finally, we show that modular Hamiltonians on the null surface satisfy a Markov property that leads to the saturation of the strong sub-additive inequality for the entropies and to the strong super-additivity of the relative entropy.

Figures (4)

• Figure 1: Setup of the work: null plane $\mathcal{P}$ parallel to $\xi = (1,1,0,\cdots)$, with an arbitrary curve $\gamma(\lambda)$. We will determine the modular Hamiltonians associated to regions $R_\gamma$.
• Figure 2: The past null cone in Minkowski space. The spheres in the cone passing though the point $i$ (blue) are mapped to planes on the null plane with the conformal transformation (\ref{['maa']}). All other spheres (green) are mapped to parabolic regions on the null plane. The point $i$ and the red null line are mapped to infinity.
• Figure 3: The cylinder where the two vertical edges are identified. Minkowski space is conformal to the diamond shown with dashed lines. The modular flow of a double cone $A$ inside Minkowski space moves regions inside $A$ towards the future tip of $A$ (for positive modular parameter) and regions inside the complement ($D$ in the figure) towards the past tip of the complement in the cylinder. For some finite modular parameter a point in the complement will cross the past null boundary of Minkowski space making the flow in Minkowski non local. However, the flow is still local in the cylinder.
• Figure 4: Family of curves that move among themselves by the modular translations corresponding to the two dashed curves. Here we have plotted (\ref{['standardtrans']}) for $\tau=l/4$, $l=0,1,\cdots 4$. The modular flows of these curves will also move the curves among themselves though with different parametrizations.