The Gravity Duals of Modular Hamiltonians

TL;DR

This paper explores the gravitational duals of modular Hamiltonians in quantum field theories with semi-classical gravity duals, providing new prescriptions for their calculation and revealing non-commuting properties with certain bulk operators.

Contribution

It introduces gravity dual prescriptions for modular Hamiltonians and uncovers their non-commuting behavior with space-like bulk operators.

Findings

Derived gravity dual formulas for modular Hamiltonian action.

Showed modular Hamiltonian does not commute with certain bulk operators.

Connected modular Hamiltonians to covariant holographic entanglement entropy.

Abstract

In this work, we investigate modular Hamiltonians defined with respect to arbitrary spatial regions in quantum field theory states which have semi-classical gravity duals. We find prescriptions in the gravity dual for calculating the action of the modular Hamiltonian on its defining state, including its dual metric, and also on small excitations around the state. Curiously, use of the covariant holographic entanglement entropy formula leads us to the conclusion that the modular Hamiltonian, which in the quantum field theory acts only in the causal completion of the region, does not commute with bulk operators whose entire gauge-invariant description is space-like to the causal completion of the region.