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

This work concerns some features of scalar QFT defined on the causal boundary F+ of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra W(F+). ( a) ( i) It is noticed that the natural BMS invariant pure quasifree state lambda on W(F+), 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, zeta (zeta) over bar) on F+, (u is an element of R being the affine parameter of the complete null geodesics forming F+ and zeta,(zeta) over bar complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of the 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 the cluster property under u-displacements is valid for every ( not necessarily quasifree) pure state on W(F+) which is invariant under u displacements. (iii) It is established that 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 lambda. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state omega (like lambda but not necessarily quasifree) on W(F+), omega is the only state invariant under u-displacement. (b) It is proved that the theory can be formulated for spacetimes asymptotically flat at null infinity which also admit future time completion i(+) ( and fulfill other requirements related with global hyperbolicity). In this case a *-isomorphism i exists - with a natural geometric meaning - which identifies the ( Weyl) algebra of observables of a linear field propagating in the bulk spacetime with a sub algebra of W(F+). Using i a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state. on W(F+).