Spectral Analysis of the Semi-relativistic Pauli-Fierz Hamiltonian

TL;DR

This paper analyzes the spectral properties of a semi-relativistic quantum Hamiltonian describing a charged spin-1/2 particle coupled to a quantized electromagnetic field, revealing ground state degeneracy and spectral gaps.

Contribution

It provides a spectral analysis of the semi-relativistic Pauli-Fierz Hamiltonian, establishing ground state degeneracy and spectral gap uniformity for non-zero photon mass.

Findings

Ground state is exactly two-fold degenerate for non-zero photon mass.

A spectral gap exists uniformly in total momentum.

The spectrum structure is characterized via fiber decomposition.

Abstract

We consider a charged particle, spin 1/2, with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as $H(P)$, $P\in \BbbR^3$. We study the spectrum of $H(P)$. In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in $P$, from the rest of the spectrum.