The Modular Hamiltonian in Asymptotically Flat Spacetime Conformal to Minkowski

Abstract

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 \(\Upsilon _M\) 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 \(\textsf{V}^+_x\), a future light cone stemming from \(x\in M\) as well as \(\mathcal {W}(\textsf{V}^+_x)=\mathcal {W}(M)|_{\textsf{V}^+_x}\), its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in \(\textsf{K}_x\), 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 \(\textsf{K}_x\). We extend such correspondence replacing \(\textsf{K}_x\) and \(\textsf{V}^+_x\) with deformed counterparts, denoted by \(\textsf{S}_C\) and \(\textsf{V}_C\). 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 \(\textsf{V}_C\) 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 \(\textsf{V}_C\) establishing the quantum null energy condition.