The Modular Hamiltonian in Asymptotically Flat Spacetime Conformal to Minkowski

We consider a four-dimensional globally hyperbolic and asymptotically flat spacetime (M, g) conformal to Minkowski spacetime, together with a massless, conformally coupled scalar field. Using a bulk-to-boundary correspondence, one can establish the existence of an injective ∗-homomorphism ΥM between W(M), the Weyl algebra of observables on M and a counterpart which is defined intrinsically on future null infinity ℑ+≃R×S2, a component of the conformal boundary of (M, g). Using invariance under the asymptotic symmetry group of ℑ+, we can individuate thereon a distinguished two-point correlation function whose pull-back to M via ΥM identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider Vx+, a future light cone stemming from x∈M as well as W(Vx+)=W(M)|Vx+, its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in Kx, a positive half strip on ℑ+. To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to Kx. We extend such correspondence replacing Kx and Vx+ with deformed counterparts, denoted by SC and VC. In addition, since the one particle Hilbert space at the boundary decomposes as a direct integral on the sphere of U(1)-currents defined on the real line, we prove that also the generator of the modular group associated to the standard subspace of VC decomposes as a suitable direct integral. This result allows us to study the relative entropy between coherent states of the algebras associated to the deformed cones VC establishing the quantum null energy condition.