Characterization of Infrared Catastrophe by The Carleman Operator and Its Singularity

TL;DR

This paper explores the mathematical structure of the infrared catastrophe in quantum field theory using the Carleman operator, analyzing ground state non-existence and applying the theory to models involving electrons and quantum fields.

Contribution

It introduces a novel operator-theoretic framework for IR catastrophe and applies it to physical models, linking mathematical analysis with solid state physics.

Findings

Characterization of IR catastrophe via the Carleman operator

Proof of non-existence of ground states under IR catastrophe conditions

Application of the theory to electron-phonon and electron-polariton models

Abstract

This paper addresses some mathematical problems arising from the infrared (IR) catastrophe in quantum field theory. IR catastrophe is formulated and studied in operator theory, characterized by the Carleman operator. Non-existence of ground state under IR catastrophe is also investigated with the help of the characterization. The theory presented in this paper is applied to the Hamiltonian of the model describing a non-relativistic electron coupled with a quantum field of phonons or polaritons in the light of mathematics as well as solid state physics.