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.

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