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

## TL;DR

This paper derives explicit local expressions for modular Hamiltonians on null planes in conformal field theories, demonstrating their algebraic structure and Markov property, which has implications for entropy inequalities in quantum field theory.

## Contribution

It provides a novel derivation of modular Hamiltonians on null surfaces, showing their local form, algebraic structure, and Markov property in a unified framework.

## Key findings

- Modular Hamiltonians have a local expression on the null horizon.
- The modular Hamiltonians form an infinite-dimensional Lie algebra.
- They satisfy a Markov property leading to entropy inequality saturation.

## 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.