= Momentum-space contact-interaction Hamiltonian
{title2=$H_{\rm int}=\frac{u}{2V}\sum_{k,p,q}a_{k+q}^\dagger a_{p-q}^\dagger a_pa_k$}
For a periodic box of <volume> $V$, expanding $\psi(\mathbf r)=V^{-1/2}\sum_{\mathbf k}e^{i\mathbf k\cdot\mathbf r}a_{\mathbf k}$ in $H_{\rm int}=\tfrac u2\int d^3r\,\psi^\dagger\psi^\dagger\psi\psi$ gives the displayed <normal ordering>. The spatial <integral> is $V\delta_{\mathbf k_1+\mathbf k_2,\mathbf k_3+\mathbf k_4}$ by <Fourier orthogonality>, enforcing <momentum conservation> and leaving the factor $V^{-1}$. The coupling $u$ in three dimensions has units of <energy> times <volume>.
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