Scalar QFT on the boundary ℑ + at future null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMSinvariant symplectic form. The constructed theory turns out to be invariant under a suitable stronglycontinuous unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to ℑ + of massless minimally coupled fields propagating in the bulk. The group theoretical analysis of the found unitary BMS representation proves that such a field on ℑ + coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group ∆, the semidirect product between SO(2) and the two-dimensional translations group. This wave function is massless with respect to the notion of mass for BMS representation theory. The presented result proposes a natural criterion to solve the long standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually some theorems towards a holographic description on ℑ + of QFT in the bulk are established at level of C * algebras of fields for asymptotically flat at null infinity spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective * -homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary ℑ + . Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincaré invariance in the bulk and BMS invariance on the boundary at null infinity is established at level of QFT. It arises that, in this case, the * -homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on ℑ + .

The discovery of the radiation properties of black holes prompted the search for a natural candidate quantum ground state for a massless scalar field theory on Schwarzschild spacetime, here considered in the Eddington-Finkelstein representation. Among the several available proposals in the literature, an important physical role is played by the so-called Unruh state, which is supposed to be appropriate to capture the physics of a black hole formed by spherically symmetric collapsing matter. Within this respect, we shall consider a massless Klein-Gordon e-print archive: http://lanl.arXiv.org/abs/0907.1034 CLAUDIO DAPPIAGGI ET AL.field and we shall rigorously and globally construct such state, that is on the algebra of Weyl observables localised in the union of the static external region, the future event horizon and the non-static black hole region. Eventually, out of a careful use of microlocal techniques, we prove that the built state fulfils, where defined, the so-called Hadamard condition; hence, it is perturbatively stable, in other words realizing the natural candidate with which one could study purely quantum phenomena such as the role of the back reaction of Hawking's radiation.From a geometrical point of view, we shall make a profitable use of a bulk-to-boundary reconstruction technique which carefully exploits the Killing horizon structure as well as the conformal asymptotic behaviour of the underlying background. From an analytical point of view, our tools will range from Hörmander's theorem on propagation of singularities, results on the role of passive states, and a detailed use of the recently discovered peeling behaviour of the solutions of the wave equation in Schwarzschild spacetime.

Abstract. As a starting point, we state some relevant geometrical properties enjoyed by the cosmological horizon of a certain class of Friedmann-Robertson-Walker backgrounds. Those properties are generalised to a larger class of expanding spacetimes M admitting a geodesically complete cosmological horizon ℑ − common to all co-moving observers. This structure is later exploited in order to recast, in a cosmological background, some recent results for a linear scalar quantum field theory in spacetimes asymptotically flat at null infinity. Under suitable hypotheses on M , encompassing both the cosmological de Sitter background and a large class of other FRW spacetimes, the algebra of observables for a Klein-Gordon field is mapped into a subalgebra of the algebra of observables W(ℑ − ) constructed on the cosmological horizon. There is exactly one pure quasifree state λ on W(ℑ − ) which fulfils a suitable energy-positivity condition with respect to a generator related with the cosmological time displacements. Furthermore λ induces a preferred physically meaningful quantum state λM for the quantum theory in the bulk. If M admits a timelike Killing generator preserving ℑ − , then the associated self-adjoint generator in the GNS representation of λM has positive spectrum (i.e. energy). Moreover λM turns out to be invariant under every symmetry of the bulk metric which preserves the cosmological horizon. In the case of an expanding de Sitter spacetime, λM coincides with the Euclidean (Bunch-Davies) vacuum state, hence being Hadamard in this case. Remarks on the validity of the Hadamard property for λM in more general spacetimes are presented.