Infrared Finite Scattering Theory in QFT and Quantum Gravity*

The "infrared problem" is the generic emission of an infinite number of low-frequency quanta in any scattering process with massless degrees of freedom. That the ``out'' state contains an infinite number of such quanta implies that it does not lie in the standard Fock representation. Consequently, the standard S-matrix is undefined as a map between "in" and "out" states in the standard Fock space. This fact is due to the existence of a low-frequency tail of the radiation field (i.e. the memory effect) as well as the existence of an infinite number of conserved charges at spatial infinity. In massive QED, the scattering representations known as "Kulish-Faddeev" representations have been argued to yield an I.R. finite S-matrix. We clarify the "preferred status" of such representations as eigenstates of the conserved ``large gauge charge''. We prove a "No-Go" theorem for the existence of a suitable Hilbert space analogously constructed scattering states in massless QED, QCD, linearized quantum gravity with massive/massless sources, and in full quantum gravity. We then develop an "infrared-finite" formulation of scattering theory without any a priori choice of "in/out" Hilbert space.