On the Infrared Problem for the Dressed Non-Relativistic Electron in a Magnetic Field

TL;DR

This paper investigates the existence of ground states for a non-relativistic electron coupled with a magnetic and quantized electromagnetic field, revealing conditions under which ground states exist in Fock and non-Fock representations.

Contribution

It establishes a precise criterion for the existence of ground states based on the derivative of the energy-momentum relation, extending understanding of infrared problems in quantum electrodynamics.

Findings

Ground state exists in Fock space if and only if the energy derivative is zero.

Non-Fock ground states exist when the energy derivative is non-zero.

Results hold for small coupling constants.

Abstract

We consider a non-relativistic electron interacting with a classical magnetic field pointing along the $x_3$-axis and with a quantized electromagnetic field. The system is translation invariant in the $x_3$-direction and we consider the reduced Hamiltonian $H(P_3)$ associated with the total momentum $P_3$ along the $x_3$-axis. For a fixed momentum $P_3$ sufficiently small, we prove that $H(P_3)$ has a ground state in the Fock representation if and only if $E'(P_3)=0$, where $P_3 \mapsto E'(P_3)$ is the derivative of the map $P_3 \mapsto E(P_3) = \inf \sigma (H(P_3))$. If $E'(P_3) \neq 0$, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant.