for small $\alpha$. Theorem \[Pauli\] implies, as a corollary, that for small $\alpha$ the [*total energy*]{} (including Coulomb energies) is bounded below by $+\alpha \Lambda N$. In other words, among the three components of energy (kinetic, field and Coulomb), the first two overwhelm the third — for small $\alpha$, at least.

All of these statements are true without mass renormalization and the situation could conceivably be more dramatic when the mass is renormalized. In any case, the true physical questions concern energy differences, and this question remains to be addressed.

NON-RELATIVISTIC ENERGY BOUNDS
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[*Theorem \[theo:nonrel\]:*]{} We sketch a proof of Theorem \[theo:nonrel\]. It is clear by taking the state $V^{-1/2}/ \otimes |0\rangle$ that the ground state energy is bounded above by ${\rm{(const)}} \alpha \Lambda^2$, which is the same result one gets from perturbation theory. Since the field energy in this state vanishes, such a computation ignores the tradeoff between the kinetic energy of the electron and the field energy. Thus, it is important to quantify this tradeoff. The main idea is to estimate the field energy in terms of selected modes.

Consider the operators (field modes), parametrized by $y \in {\mathbb
R}^3$, $$L(y) = \frac{1}{\sqrt{2V}} \sum_{|k| < \Lambda, \lambda} 