Rigorous steps towards holography in asymptotically flat spacetimes
C. Dappiaggi, V. Moretti, N. Pinamonti
TL;DR
This work develops a rigorous bulk–boundary holographic framework for scalar quantum fields in four-dimensional asymptotically flat spacetimes by constructing a boundary QFT on future null infinity $\Im^+$ using a Weyl $C^*$-algebra built from a $G_{BMS}$-invariant symplectic form. It shows that boundary fields correspond to bulk massless fields through unitary BMS representations induced from the $\Delta$ little group, and demonstrates that a natural bulk–boundary correspondence requires the nuclear topology on the BMS supertranslations; under symplectic preservation, bulk Weyl algebras embed into boundary Weyl algebras via injective $*$-homomorphisms, with Minkowski space providing an explicit realization where the Minkowski vacuum maps to the boundary vacuum. The analysis unifies boundary and bulk perspectives through an induced representation framework and clarifies how Poincaré invariance in the bulk translates into BMS structures on the boundary. The results lay a rigorous foundation for holography in asymptotically flat spacetimes and point to future extensions to massive fields, interactions, and higher dimensions, while highlighting the crucial role of nuclear topology for a canonical bulk–boundary dictionary.
Abstract
Scalar QFT on the boundary $\Im^+$ at null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory is invariant under a suitable unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to $\Im^+$ of massless minimally coupled fields propagating in the bulk. The analysis of the found unitary BMS representation proves that such a field on $\Im^+$ coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group $Δ$, the semidirect product between SO(2) and the two dimensional translational group. The result proposes a natural criterion to solve the long standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually some theorems towards a holographic description on $\Im^+$ of QFT in the bulk are established at level of $C^*$ algebras of fields for strongly asymptotically predictable spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective $*$-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary $\Im^+$. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincaré invariance in the bulk and BMS invariance on the boundary at $\Im^+$ is established at level of QFT. It arises that the $*$-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on $\Im^+$.