consistent description of a system of stable charged relativistic or non-relativistic bosons interacting with the radiation field, while the Pauli exclusion principle, on the other hand, is able to prevent the above mentioned pathology.

In the remainder of the section we explain our notation and state the results. In the subsequent sections we sketch the proof of some of them but for details we refer the reader to \[LL\].

We measure the energy in units of $mc^2$ where $m$ is the bare mass of the electron, the length in units of the Compton wave length $ \ell_C = \hbar/mc$ of the bare electron. We further choose $\ell_C^{-1} \sqrt{\hbar c}$ as the unit for the vector potential $A$ and $\ell_C^{-2} \sqrt{\hbar c}$ as the unit for the magnetic field $B$. The argument is the dimensionless quantity $ \ell_C^{-1} x$. As usual, $\alpha = e^2 / \hbar c \approx 1/137.04$ is the fine structure constant.

In the expression below, $A(x)$ denotes an ultraviolet cutoff radiation field localised in a box $L\times L\times L$ with volume $V=L^3$,

$$A(x)=\frac {1 }{ \sqrt{2V}} \sum_{|k| < \Lambda} \sum_ {\lambda =1,2} \frac{1}{
\sqrt{|k|}}
\varepsilon_{\lambda}(k)\bigl[ a_{\lambda}(k) e^{ i x\cdot k} +
 a_{\lambda}^*(k) e^{- i x\cdot k} \bigr] \ 