Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence
The paper originates from results previously reported by the same author and his collaborators in Ref. [\textit{C. Dappiaggi, V. Moretti} and \textit{N. Pinamonti}, Rev. Math. Phys. 18, No. 4, 349--415 (2006; Zbl 1107.81040)], thus aiming at providing a better understanding of features of scalar QFT defined on the causal boundary \({\mathfrak I}^+\) of an asymptotically flat at null infinity spacetime and based on a certain Weyl algebra \(\mathcal{W} ({\mathfrak I}^+)\) of observables associated to a symplectic space invariant under the action of the Bondi-Metzner-Sachs (BMS) group. The author basically addresses two questions. The first one concerns peculiarities of BMS-invariant states. In this respect the author shows that that: 1) the natural BMS-invariant state on \(\mathcal{W} ({\mathfrak I}^+)\) (the Gelfand-Naimark-Segal invariant state also considered in [Dappiaggi, Moretti and N. Pinamonti (loc. cit.)]) enjoys positivity of the self-adjoint generator of the affine parameter displacements w.r.t. every admissible frame (i.e. w.r.t. every global frame on the boundary \({\mathfrak I}^+\) related with a Bondi frame on \({\mathfrak I}^+\) by means of a BMS transformation); 2) the validity of positivity for the self-adjoint generator of affine parameter translations in a fixed admissible frame individuates such state uniquely; 3) in the folium of a pure state on \(\mathcal{W} ({\mathfrak I}^+)\) which is invariant under affine parameter displacement, the state itself is the only state possessing the invariance. The author next moves to the second issue, which regards the possibility to isometrically map the Weyl algebra of linear QFT in the bulk onto a sub-algebra of the BMS-invariant Weyl algebra \(\mathcal{W} ({\mathfrak I}^+)\) and, consequently, to induce a preferred state in the bulk by the natural BMS-invariant state. He proves, in fact, that the required isometric \(\ast\)-homomorphism do exist for the class of space-times that are flat at null infinity and also admit future time completion. Such extension of the analysis performed in [Dappiaggi, Moretti and N. Pinamonti (loc. cit.)] (which was restricted to Minkowski space-time) relies on a proper enlargment of the symplectic space. The article is mathematically self-consistent (with the three useful appendices: (a) Quasifree states on Weyl algebras, (b) Spacetime infinities, and (c) Fourier-Plancherel Transform on \(\mathbb{R}\times \mathbb{S}^{2}\)), but the reader would benefit of consulting the aforementioned earlier work [Dappiaggi, Moretti and N. Pinamonti (loc. cit.)] to better realize the general physical motivations underlying the study.
scientific article