# Rigorous steps towards holography in asymptotically flat spacetimes

June 11, 2005

Scalar QFT on the boundary $\Im^+$ at null infinity of a general
asymptotically flat 4D spacetime is constructed using the algebraic approach
based on Weyl algebra associated to a BMS-invariant symplectic form. The
constructed theory is invariant under a suitable unitary representation of the
BMS group with manifest meaning when the fields are interpreted as suitable
extensions to $\Im^+$ of massless minimally coupled fields propagating in the
bulk. The analysis of the found unitary BMS representation proves that such a
field on $\Im^+$ coincides with the natural wave function constructed out of
the unitary BMS irreducible representation induced from the little group
$\Delta$, the semidirect product between SO(2) and the two dimensional
translational group. The 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 $\Im^+$ of QFT in
the bulk are established at level of $C^*$ algebras of fields for strongly
asymptotically predictable 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 $\Im^+$. Those results are, in particular, applied to 4D Minkowski
spacetime where a nice interplay between Poincar\'e invariance in the bulk and
BMS invariance on the boundary at $\Im^+$ is established at level of QFT. It
arises that the $*$-homomorphism admits unitary implementation and Minkowski
vacuum is mapped into the BMS invariant vacuum on $\Im^+$.

Keywords:

QFT on curved spacetimes, holographic principle