The mass shell in the semi-relativistic Pauli-Fierz model

TL;DR

This paper constructs and analyzes the ground state eigenprojection of a semi-relativistic quantum electrodynamics model, demonstrating convergence of a sequence of approximations and properties of the ground state energy.

Contribution

It develops a novel iterative perturbation approach to establish the existence and properties of the ground state in a semi-relativistic Pauli-Fierz model with infra-red cutoff.

Findings

Convergence of ground state projections as infra-red cutoff vanishes

Ground state energy is a two-fold degenerate eigenvalue of the renormalized Hamiltonian

Ground state energy is twice continuously differentiable and strictly convex in total momentum

Abstract

We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo's iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain Hamiltonian unitarily equivalent to a renormalized fiber Hamiltonian acting in a coherent state representation space. The ground state energy is an exactly two-fold degenerate eigenvalue of the renormalized Hamiltonian, while it is not an eigenvalue of the original fiber Hamiltonian unless the total momentum is zero. These results hold true, for total momenta inside a ball about zero of arbitrary radius p>0, provided that the coupling constant is sufficiently small depending on p and the ultra-violet cutoff. Along the way we prove twice continuous differentiability and strict convexity of the ground state energy as a function of the total momentum inside that ball.