Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence

Scalar BMS-invariant QFT defined on the causal boundary $\text{scri}$ of an asymptotically flat spacetime is discussed. (a)(i) It is noticed that the natural $BMS$ invariant pure quasifree state $\lambda$ on $\text{cW}(\text{scri})$, recently introduced by Dappiaggi, Moretti an Pinamonti, enjoys positivity of the self-adjoint generator of $u$-translations with respect to {\em every} Bondi coordinate frame $(u,\text{z},\text{bz})$ on $\text{scri}$, $u\in \text{bR}$ being the affine parameter of the null geodesics forming $\text{scri}$. This fact may be interpreted as a remnant of spectral condition inherited from Minkowski spacetime. (ii) It is proved cluster property under $u$-displacements holds for $u$-invariant pure state on $\text{cW}(\text{scri})$. (iii) It is proved that there is a unique algebraic pure quasifree state invariant under $u$-displacements (of a fixed Bondi frame) having positive self-adjoint generator of $u$-displacements. It coincides with the GNS-invariant state $\lambda$.(iv) It is showed that in the folium of a pure $u$-invariant state $\omega$ (not necessarily quasifree) on $\text{cW}(\text{scri})$, $\omega$ is the only state invariant under $u$-displacement. (b) It is proved that the theory can formulated for spacetimes asymptotically flat at null infinity which admit future time completion. In this case a $\ast$-isomorphism $\mathrm{ı}$ exists which identifies the (Weyl) algebra of observables of linear fields in the bulk with a sub algebra of $\text{cW}(\text{scri})$. A preferred state on the field algebra in the bulk is induced by the $BMS$-invariant state $\lambda$.