be binding for all large enough $N$, irrespective of the cutoff $\Lambda$. It also has to be remarked that the Coulomb repulsion will, in all likelihood, not alter this result since it has an effect on energy scales of the order of $\Lambda$ and not $\Lambda^{12/7}$ or $\Lambda^{3/2}$.

FERMIONS
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The real issue for physics is what happens with fermions. We cannot show that there is no binding but we can show that the energy is extensive as in Theorem \[theo:nonrelfer\]. The Hamiltonian is the same as (\[manyham\]) but it acts on antisymmetric functions tensored with ${\mathcal F}$. (Spin can be ignored for present purposes.)

[*Rough sketch of the proof of Theorem \[manyham\]*]{}.

The difficulty in proving this theorem stems from the fact that the field energy is not extensive in any obvious way.

Define $\underbar{X} = (x_1, \cdots , x_N)$ and define the function $$n_j (\underbar{X}, R) = \# \{ x_i \neq x_j ~:~ |x_i - x_j| < R \} ~ .$$ This function counts the number of electrons that are within a distance $R$ of the $j^{\rm{th}}$ electron. Note that this function is not smooth, so that all the following computations have to be modified. (See [@LL].) We 