Quantum out-states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property

This paper continues the analysis of the quantum states determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state lambda_M, of a massles conformally coupled scalar field phi propagating in M. lambda_M is ``holographically'' induced in the bulk by the universal BMS-invariant state lambda at infinity scri of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at null infinity discussed in previous papers. The induction is possible when some requirements are fulfilled, in particular the spacetime M and the associated unphysical one are globally hyperbolic and M admits future time infinity i^+. lambda_M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state lambda_M enjoys the following further properties. (1) lambda_M is invariant under the group of isometries of the bulk spacetime M. (2) lambda_M fulfills a natural energy-positivity condition with respect to every notion of Killing time (if any) in the bulk spacetime M: If M admits a complete time-like Killing vector, the associated one-parameter group of isometries is represented by a strongly-continuous unitary group in the GNS representation of lambda_M. The unitary group has positive self-adjoint generator without zero modes in the one-particle space. (3) lambda_M is (globally) Hadamard in M and thus lambda_M can be used as starting point for perturbative renormalization procedure of QFT of phi in M.