---
abstract: 'Various models of charged particles interacting with a quantized, ultraviolet cutoff radiation field (but not with each other) are investigated. Upper and lower bounds are found for the self- or ground state-energies without mass renormalization. For $N$ fermions the bounds are proportional to $N$, but for bosons they are sublinear, which implies ‘binding’, and hence that ‘free’ bosons are never free. Both ‘relativistic’ and non-relativistic kinematics are considered. Our bounds are non-perturbative and differ significantly from the predictions of perturbation theory.'
author:
- |
    Elliott H. Lieb[^1]\
    Departments of Mathematics and Physics\
    Princeton University, Princeton, New Jersey 08544-0708\
    [*lieb@math.princeton.edu*]{}
- |
    Michael Loss[^2]\
    School of Mathematics\
    Georgia Institute of Technology, Atlanta, Georgia 30332-0160\
    [*loss@math.gatech.edu*]{}
date: 'September 15, 1999'
title: 'SELF-ENERGY OF ELECTRONS IN NON-PERTURBATIVE QED[^3][^4]'
---

Introduction {#intro}
============

Quantum electrodynamics (QED), the theory of electrons interacting with photons (at least for small energies) is one of the great successes of physics. Among its major achievements is the explanation of the Lamb shift and the anomalous magnetic moment of the electron. Nevertheless, its computations, which are entirely based on perturbation theory, created some uneasiness among the practitioners. The occurrence of infinities was and is especially vexing. Moreover, a 