The modular Hamiltonian in asymptotically flat spacetime conformal to Minkowski

Abstract

We consider a four-dimensional globally hyperbolic 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 $\Upsilon_M$ between $\mathcal{W}(M)$, the Weyl algebra of observables on $M$ and a counterpart which is defined intrinsically on future null infinity $\Im^+\simeq\mathbb{R}\times\mathbb{S}^2$, a component of the conformal boundary of $(M,g)$. Using invariance under the asymptotic symmetry group of $\Im^+$, we can individuate thereon a distinguished two-point correlation function whose pull-back to $M$ via $\Upsilon_M$ identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider $\mathsf{V}^+_x$, a future light cone stemming from $x\in M$ as well as $\mathcal{W}(\mathsf{V}^+_x)=\mathcal{W}(M)|_{\mathsf{V}^+_x}$, its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in $\mathsf{K}_x$, a positive half strip on $\Im^+$. To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to $\mathsf{K}_x$. We extend such correspondence replacing $\mathsf{K}_x$ and $\mathsf{V}^+_x$ with deformed counterparts, denoted by $\mathsf{S}_C$ and $\mathsf{V}_C$. 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 $\mathsf{V}_C$ 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 $\mathsf{V}_C$ establishing the quantum null energy condition.