Uniqueness Theorem for BMS-Invariant States of Scalar QFT on the Null Boundary of Asymptotically Flat Spacetimes and Bulk-Boundary Observable Algebra Correspondence

Abstract

This work concerns some features of scalar QFT defined on the causal boundary $\mathfrak{J}^{+}$ of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra $\mathcal{W}(\mathfrak{J}^{+})$.

(a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on $\mathcal{W}(\mathfrak{J}^{+})$, recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame $(u,\zeta,\overline{\zeta})$ on $\mathfrak{J}^{+}$, ( $u\in \mathbb{R}$ being the affine parameter of the complete null geodesics forming $\mathfrak{J}^{+}$ and $\zeta,\overline{\zeta}$ 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 $\mathcal{W}(\mathfrak{J}^{+})$ 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 λ. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state ω (like λ but not necessarily quasifree) on $\mathcal{W}(\mathfrak{J}^{+}), \omega$ 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 ı 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 $\mathcal{W}(\mathfrak{J}^{+})$. Using ı a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state λ on $\mathcal{W}(\mathfrak{J}^{+})$.