Rigorous Steps Towards Holography in Asymptotically Flat Spacetimes

Abstract

Scalar QFT on the boundary ℑ + at future null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMSinvariant symplectic form. The constructed theory turns out to be invariant under a suitable stronglycontinuous unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to ℑ + of massless minimally coupled fields propagating in the bulk. The group theoretical analysis of the found unitary BMS representation proves that such a field on ℑ + coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group ∆, the semidirect product between SO(2) and the two-dimensional translations group. This wave function is massless with respect to the notion of mass for BMS representation theory. The presented 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 ℑ + of QFT in the bulk are established at level of C * algebras of fields for asymptotically flat at null infinity 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 ℑ + . 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 null infinity is established at level of QFT. It arises that, in this case, the * -homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on ℑ + .

Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property

“…In this paper we continue the analysis of the states determined by the asymptotic structure of fourdimensional asymptotically flat spacetimes at null infinity started in [DMP06] and fully developed in [Mo06]. Part of those results will be summarised in Sec.2.…”

“…Part of those results will be summarised in Sec.2. In [DMP06] and [Mo06] it has been established that the null boundary at future infinity + of an asymptotically flat spacetime admits a natural formulation of bosonic linear QFT living therein. A preferred quasifree pure state λ has been picked out in the plethora of algebraic states defined on the algebra of Weyl observables W( + ) of the QFT on + .…”

“…Above Π is the well-known surjective covering homomorphism SL(2, C) → SO(3, 1)↑ (see [DMP06] for further details). Two Bondi frames are connected each other through the transformations (2),(3) with Λ ∈ SU (2).…”

“…There is a natural representation of G BM S acting on (S( + ), σ) discussed in [DMP06,Mo06]. Start from the representation A of G BM S made of transformations on functions…”

“…Here W(S( + )) := W(S( + ), σ) has the natural interpretation of the algebra of observables for a linear bosonic QFT defined on + as discussed in [DMP06,Mo06] (see also the appendix A of [Mo06]). …”