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

TL;DR

This paper proves the uniqueness of a BMS-invariant pure quasifree state on the null boundary of asymptotically flat spacetimes, establishing a bulk-boundary algebra correspondence and analyzing state invariance and positivity properties.

Contribution

It demonstrates the uniqueness of the BMS-invariant pure quasifree state with positive generator and establishes a bulk-boundary algebra isomorphism for scalar QFT on null infinity.

Findings

The BMS-invariant state enjoys positivity of the generator in all Bondi frames.

Cluster property holds for u-invariant pure states.

Unique invariant state coincides with the GNS state.

Abstract

Scalar BMS-invariant QFT defined on the causal boundary $\scri$ of an asymptotically flat spacetime is discussed. (a)(i) It is noticed that the natural $BMS$ invariant pure quasifree state $\lambda$ on $\cW(\scri)$, 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 $(u,\z,\bz)$ on $\scri$, $u\in \bR$ being the affine parameter of the null geodesics forming $\scri$. 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 $\cW(\scri)$. (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 $\cW(\scri)$, $\omega$ 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 $*$-isomorphism $\imath$ exists which identifies the (Weyl) algebra of observables of linear fields in the bulk with a sub algebra of $\cW(\scri)$. A preferred state on the field algebra in the bulk is induced by the $BMS$-invariant state $\lambda$.