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 $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> -homomorphism $$\Upsilon _M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Υ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> between $$\mathcal {W}(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , the Weyl algebra of observables on M and a counterpart which is defined intrinsically on future null infinity $$\Im ^+\simeq \mathbb {R}\times \mathbb {S}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>ℑ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>≃</mml:mo> <mml:mi>R</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , a component of the conformal boundary of ( M , g ). Using invariance under the asymptotic symmetry group of $$\Im ^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℑ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , we can individuate thereon a distinguished two-point correlation function whose pull-back to M via $$\Upsilon _M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Υ</mml:mi> <mml:mi>M</mml:mi> </mml:msub> </mml:math> identifies a quasi-free Hadamard state for the bulk algebra of observables. In this setting, if we consider $$\textsf{V}^+_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>V</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:math> , a future light cone stemming from $$x\in M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>M</mml:mi> </mml:mrow> </mml:math> as well as $$\mathcal {W}(\textsf{V}^+_x)=\mathcal {W}(M)|_{\textsf{V}^+_x}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>V</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>W</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>V</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> </mml:msub> </mml:mrow> </mml:math> , its counterpart at the boundary is the Weyl subalgebra generated by suitable functions localized in $$\textsf{K}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math> , a positive half strip on $$\Im ^+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℑ</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> . To each such cone, we associate a standard subspace of the boundary one-particle Hilbert space, which coincides with the one associated naturally to $$\textsf{K}_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:math>