# Infrared problem in perturbative quantum field theory

## TL;DR

This paper develops a rigorous construction of the scattering matrix and interacting fields in quantum field theories with massless particles and long-range interactions, addressing the infrared problem.

## Contribution

It introduces a modified scattering matrix and interacting fields using the ideas of Dollard, Kulish, and Faddeev, applicable to models like QED with infrared issues.

## Key findings

- Constructed the modified scattering matrix in first and second order.
- Proved existence of the adiabatic limit in perturbation theory.
- Described non-standard particle interpretation with photon clouds.

## Abstract

We propose a mathematically rigorous construction of the scattering matrix and the interacting fields in models of relativistic perturbative quantum field theory with massless fields and long-range interactions. We consider quantum electrodynamics and a certain model of interacting scalar fields in which the standard definition of the scattering matrix is not applicable because of the infrared problem. We modify the Bogoliubov construction using the ideas of Dollard, Kulish and Faddeev. Our modified scattering matrix and modified interacting fields are constructed with the use of the adiabatic limit which is expected to exist in arbitrary order of perturbation theory. In the paper we prove this assertion in the case of the first- and the second-order corrections to the modified scattering matrix and the first-order corrections to the modified interacting fields. Our modified scattering matrix and modified interacting fields are defined in the standard Fock space. However, the particle interpretation of states in this space is non-standard. In particular, the electrons and positrons are always surrounded by irremovable clouds of photons. Moreover, the physical energy-momentum operators do not coincide with the standard ones and their joint spectrum does not contain the mass hyperboloid.