Infrared (IR) divergences arise in scattering theory with massless fields and

are manifestations of the memory effect. There is nothing singular about states

with memory, but they do not lie in the standard Fock space. IR divergences are

artifacts of trying to represent states with memory in the standard Fock space.

For collider physics, one can impose an IR cutoff and calculate inclusive

quantities. But, this approach cannot treat memory as a quantum observable and

is highly unsatisfactory if one views the S-matrix as fundamental in QFT and

quantum gravity, since the S-matrix is undefined. For a well-defined S-matrix,

it is necessary to define in/out Hilbert spaces with memory. Such a

construction was given by Faddeev and Kulish (FK) for QED. Their construction

“dresses” momentum states of the charged particles by pairing them with memory

states of the electromagnetic field to produce states of vanishing large gauge

charges at spatial infinity. However, in massless QED, due to collinear

divergences, the “dressing” has an infinite energy flux so these states are

unphysical. In Yang-Mills theory the “soft particles” used for dressing also

contribute to the current flux, invalidating the FK procedure. In quantum

gravity, the analogous FK construction would attempt to produce a Hilbert space

of eigenstates of supertranslation charges at spatial infinity. However, we

prove that there are no eigenstates of supertranslation charges except the

vacuum. Thus, the FK construction fails in quantum gravity. We investigate some

alternatives to FK constructions but find that these also do not work. We

believe that to treat scattering at a fundamental level in quantum gravity – as

well as in massless QED and YM theory – it is necessary to take an algebraic

viewpoint rather than shoehorn the in/out states into some fixed Hilbert space.

We outline the framework of such an IR finite scattering theory.