Hamiltonian in \[eq:4.2\] there are positive constants, $C, C',
C''$ such that $$\begin{aligned}
E_0 &\leq& C \sqrt{\alpha} \Lambda\\
E_0 &\geq& C' \sqrt{\alpha} \Lambda~{\rm{for~ small}}~ \alpha\\
E_0 &\geq& C'' \Lambda~{\rm{for ~ large}}~ \alpha ~ .\end{aligned}$$ \[theo:4.1\]

The generalization of this to $N$ fermions is similar to the nonrelativistic generalization, except that the power of $\Lambda$ is the same on both sides of the inequalities.

For $N$ fermions with Hamiltonian $$H_N = \sum_{i=1}^N T^{\rm rel}(x_i) + H_f$$ there are positive constants $C, C', C'',$ independent of $\alpha$ and $N$, such that $$\begin{aligned}
E_0 &\leq& C N \sqrt{\alpha} \Lambda\nonumber   \\ E_0 &\geq& C' N \sqrt{\alpha} \Lambda \quad {\rm{for~small}} ~ \alpha\nonumber 
 \\E_0 &\geq& C'' N \Lambda  \quad {\rm{for~large}} ~ \alpha
\label{eq:4.23}\end{aligned}$$ \[theo:4.3\]

We close this introduction by mentioning one last result about the Pauli–operator. The kinetic energy expression is given by $$T^{\rm Pauli}=[\sigma \cdot (p+ \sqrt{\alpha}A(x))]^2 = (p+
\sqrt{\alpha}A(x))^2 + \sqrt{\alpha}\ \sigma \cdot B(x) \ .
\label{eq:4.4}$$ where $\sigma$ denotes the vector consisting of the Pauli matrices. Observe that this term automatically accounts for the spin–field interaction. Our result for the self energy of a Pauli electron is the following.

The ground state energy $E_0$ of the Hamiltonian with Pauli energy, $$\frac{1}{2}[\sigma \cdot (p+ 