product is taken.

The ground state energy, $E^{fermion}_0(N)$, of $N$ charged fermions interacting with the radiation field satisfies $$C_1 \alpha^{1/2} \Lambda^{3/2} N \ \leq \ E^{fermion}_0(N) \ \leq  \ C_2 
\alpha^{2/7} \Lambda^{12/7} N
\label{eq:nonrelfer}$$ \[theo:nonrelfer\]

The [**“relativistic” kinetic energy**]{} for an electron is $$T^{\rm rel} = | p + \sqrt{\alpha} A (x) | = \sqrt{[p + \sqrt{\alpha} A(x)]^2}
\label{eq:4.1}$$ with $p=-i\nabla$. (Really, we should take $\sqrt{[p+\sqrt{\alpha}A
(x)]^2 + 1}$, but since $x < \sqrt{x^2 + 1} < x + 1$, the difference is bounded by $N$.)

Consider, first, the $N=1$ body problem with the Hamiltonian $$H =T^{\rm rel} + H_f ~.
\label{eq:4.2}$$

By simple length scaling (with a simultaneous scaling of the volume $V$) we easily see that $E_0 = \inf {\rm{spec}}~ (H) = C \Lambda$. Our goal here is to show that the constant, $C$, is strictly positive and to give an effective lower bound for it. But we would like to do more, namely investigate the dependence of this constant on $\alpha$. We also want to show, later on, that for $N$ fermions the energy is bounded below by a positive constant times $N \Lambda$. Our proof will contain some novel — even bizarre — features.

For the 