\sqrt{\alpha}A(x))]^2 + H_f ,$$ satisfies the bounds $$\begin{aligned}
E_0 &\leq & C_3 \sqrt{\alpha}\Lambda^{3/2}  \\
E_0 &\geq & C_1 \alpha \Lambda  \quad
{\rm{for~small}} ~ \alpha\nonumber \\
E_0 &\geq & C_2 \alpha^{1/3} \Lambda  \quad
{\rm for~large} ~ \alpha \nonumber \end{aligned}$$ For $N$ fermions, the bounds above are multiplied by $N$ (and the constants are changed). \[Pauli\]

For the detail of the proof, we refer the reader to \[LL\]. We believe that the upper bound is closer to the truth since the main contributions to the self energy should come from the fluctuations of the $A^2$ term.

Theorem \[Pauli\] has the following consequence for stability of matter interacting with quantized fields. It was shown in [@LLS] that a system of electrons and nuclei interacting with Coulomb forces, with the Pauli kinetic energy for the electrons and with a [*classical*]{} magnetic field energy is stable (i.e., the ground state energy is bounded below by $N$) if and only if $\alpha$ is small enough. In [@BFG],[@FFG] this result was extended to [*quantized*]{}, ultraviolet cutoff magnetic fields (as here). Among other things, it was shown in [@FFG] that the ground state energy, $E_0$, of the electrons and nuclei problem is bounded below by $-\alpha^2
\Lambda N$ 