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

Features of scalar QFT defined on the causal boundary J+ of an asymptotically flat at infinity spacetime and based on the BMS-invariant Weyl algebra W (J+) are discussed. (a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on W (J+), recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame (u; ζ, ζ) on J+, (u ∈ R being the affine parameter of the complete geodesics forming J+ and ζ, ζ complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of spectral condition inherited from QFT in Minkowski spacetime (and it is the spectral condition for free fields when the bulk is the very Minkowski space). (ii) It is also proved that cluster property under u-displacements is valid for every (not necessarily quasifree) pure state on W(=+) which is invariant under u displacements. (iii) It is established that positivity of the self-adjoint generator of u-translations in a fixed Bondi frame individuates the BMS-invariant state λ uniquely (without requiring BMS invariance) in the class of pure algebraic quasifree states on W (J+): there is exactly one algebraic pure quasifree state which is invariant under u-displacements (of a fixed Bondi frame) and has positive self-adjoint generator of u-displacements. It coincides with the GNS-invariant state λ. (iv) Finally it is showed that in the folium of a pure u-displacement invariant state (like λ but not necessarily quasifree) on W (J+) the state itself is the only state invariant under u-displacement. (b) It is proved that all the theory can rested for spacetimes asymptotically flat at infinity which also admit future time completion i+ (and fulfills some other requirements related with global hyperbolicity). In this case a *-isomorphism exists automatically which identifies the (Weyl) algebra of observables of a linear fields propagating in the bulk spacetime with a sub algebra of W (J+). Moreover the preferred BMS-invariant state λ on W (J+) induces a preferred state on the field algebra in the bulk spacetime.