# Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity

**Authors:** Kartik Prabhu, Gautam Satishchandran, Robert M. Wald  
**arXiv:** [2203.14334](https://arxiv.org/abs/2203.14334)  
**Date:** 2022-03-27  
**Summary page:** https://www.summarizepaper.com/en/paper/2203.14334v1/

## Key points

- Infrared (IR) divergences in scattering theory involving massless fields are linked to the memory effect, where states with memory do not fit into the standard Fock space representation.
- Attempting to fit states with memory into the conventional Fock space framework leads to IR divergences.
- Imposing an IR cutoff in collider physics to address IR divergences falls short in treating memory as a quantum observable and becomes inadequate when considering the S-matrix as fundamental in Quantum Field Theory (QFT) and quantum gravity.
- To establish a well-defined S-matrix, it is crucial to define in/out Hilbert spaces incorporating memory.
- Faddeev and Kulish (FK) proposed a construction for Quantum Electrodynamics (QED) pairing momentum states of charged particles with memory states of the electromagnetic field, but collinear divergences render these dressed states unphysical in massless QED.
- The application of FK construction encounters challenges in Yang-Mills theory due to contributions from "soft particles" undermining its validity.
- An analogous FK construction in quantum gravity aiming to produce a Hilbert space comprising eigenstates of supertranslation charges at spatial infinity fails due to the absence of eigenstates aside from the vacuum state.
- Exploring alternatives to FK constructions reveals their inefficacy, leading authors to advocate for an algebraic viewpoint over forcing in/out states into fixed Hilbert spaces when addressing scattering at a fundamental level.

## Summary

The paper "Infrared Finite Scattering Theory in Quantum Field Theory and Quantum Gravity" by Kartik Prabhu, Gautam Satishchandran, and Robert M. Wald delves into the issue of infrared (IR) divergences that arise in scattering theory involving massless fields. These divergences are linked to the memory effect, where states with memory do not conform to the standard Fock space representation. The authors highlight that IR divergences stem from attempting to fit states with memory into the conventional Fock space framework. In collider physics, one approach to address IR divergences is by imposing an IR cutoff to compute inclusive quantities. However, this method falls short in treating memory as a quantum observable and becomes inadequate when considering the S-matrix as fundamental in Quantum Field Theory (QFT) and quantum gravity due to its undefined nature. To establish a well-defined S-matrix, it is crucial to define in/out Hilbert spaces incorporating memory. Faddeev and Kulish (FK) proposed a construction for Quantum Electrodynamics (QED) where momentum states of charged particles are paired with memory states of the electromagnetic field to generate states with vanishing large gauge charges at spatial infinity. Yet, in massless QED, collinear divergences result in infinite energy flux within these dressed states, rendering them unphysical. The application of FK construction in Yang-Mills theory encounters challenges as "soft particles" used for dressing contribute to current flux, undermining the FK procedure's validity. In quantum gravity, an analogous FK construction aims to produce a Hilbert space comprising eigenstates of supertranslation charges at spatial infinity. However, the authors demonstrate that aside from vacuum state, there are no eigenstates of supertranslation charges, leading to the failure of FK construction in quantum gravity. Exploring alternatives to FK constructions reveals their inefficacy as well. The authors advocate for an algebraic viewpoint over forcing in/out states into fixed Hilbert spaces when addressing scattering at a fundamental level in quantum gravity, massless QED, and Yang-Mills theory. They propose an IR finite scattering theory framework as a more promising avenue for tackling these issues comprehensively. This comprehensive analysis sheds light on the intricate challenges posed by IR divergences and offers insights into potential avenues for advancing our understanding of scattering phenomena within the realms of quantum field theory and quantum gravity.