Momentum-space contact-interaction Hamiltonian ($H_{\mathrm{int}}=\frac{u}{2V}{\sum }_{k,p,q}{a}_{k+q}^{†}{a}_{p-q}^{†}{a}_p{a}_k$)

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 $H_{\mathrm{int}}=\frac{u}{2}\int d^{3}r{\psi }^{†}{\psi }^{†}\psi \psi$ 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.