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

This paper continues the analysis of the quantum states introduced in previous works and 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 λM, of a massless conformally coupled scalar field φ propagating in M. λM is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity + of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in particular whenever the spacetime M and the associated unphysical one, ˜M , are globally hyperbolic and M admits future time infinity i+. λM 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 λM enjoys the following further remarkable properties: (i) λM is invariant under the (unit component of the Lie) group of isometries of the bulk spacetime M. (ii) λM 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 time-like Killing vector, the associated one-parameter group of isometries is represented by a strongly-continuous unitary group in the GNS representation of λM. The unitary group has positive self-adjoint generator without zero modes in the one-particle space. In this case λM is a so-called regular ground state. (iii) λM is (globally) Hadamard in M and thus it can be used as the starting point for the perturbative renormalisation procedure of QFT of φ in M.