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### Abstract

This thesis, based on two papers by Satishchandran and Wald [1,2], investigates the behavior of classical and quantum fields in scattering theory in asymptotically flat spacetimes. It has been known that the presence of massless fields will give rise to a ``memory effect'' in four dimensions. At order 1/r a massless field generically will not return to the same value at late retarded times as it had at early retarded times. This memory effect is deeply connected to the asymptotic symmetry group of an asymptotically flat spacetime as well as the infrared divergences encountered in quantum field theory and quantum gravity. The full scope of this thesis is to fully understand the relationship between these seemingly disparate phenomena and develop an infrared finite scattering theory in QFT and quantum gravity. To understand the origin of these relationships we investigate the behavior of massless scalar, electromagnetic, and gravitational perturbations near null infinity in all dimensions greater than or equal to 4 assuming that they admit an expansion in 1/r. We consider the gravitational memory effect and show that in even dimensions, the memory effect is at Coulombic order and can be decomposed into null and ordinary memory. In odd dimensions, the memory effect vanishes. In 4 dimensions, there is a close relationship between memory and the supertranslations charge/flux relations. We then show that the vanishing of memory of at radiative order is responsible for the lack of IR divergences in higher than 4 dimensions but is directly responsible for IR divergences in 4 dimensions. IR divergences are artifacts of trying to represent states with memory in the standard Fock space. 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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