Moretti, Valter (2006) Quantum ground states holographically induced by asymptotic flatness: Invariance under spacetime symmetries, energy positivity and Hadamard property. UNSPECIFIED. (Unpublished)
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 conformally coupled scalar field propagating in M. λM is "holographically" induced in the bulk by the universal BMS-invariant state λ at infinity J+ 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. This induction is possible when some requirements are fulfilled. This happens in particular whenever the spacetime M and the associated unphysical one, ~M , are globally hyperbolic and M admits future infinity i+. As is known λM coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case, 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 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 λM. The unitary group has positive self-adjoint generator without zero modes in the one- particle space. (iii) λM is (globally) Hadamard and thus it can be used as starting point for perturbative renormalization procedure.
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