The S-matrix and Infrared Divergences in Quantum Gravity

We study the construction of an ``infrared finite'' S-matrix in Quantum Gravity. Infrared divergences in Quantum Gravity (and QFT) are a direct consequence of a classical observable known as the Memory Effect. The memory effect implies that ``out'' scattering states live in an uncountably infinite set of unitarily inequivalent Hilbert spaces (one for each memory effect). In order to construct an ``IR finite'' S-matrix we seek a ``in'' and ``out'' Hilbert space of scattering states which is (1) separable, (2) invariant under the asymptotic symmetry group and (3) preserved under scattering. The analogous problem in QED is solved by building a Hilbert space of ``dressed states''. We clarify that this procedure fails in Quantum Gravity and we argue that, in contrast to QED, there is no natural Hilbert space of ``in'' and ``out'' scattering states in any IR finite description of Quantum Gravity. Nevertheless, the ``in'' and ``out'' Algebra of Observables are well defined and, in the absence of a preferred Hilbert space, an IR finite description of Q.G. requires the formulation of an ``S matrix'' as a map on algebraic states. We present some progress towards this construction.