Crystal momentum conservation ($k_1+k_2=k_3+k_4(\mathrm{mod}G)$)

A lattice-translation-invariant interaction conserves total crystal momentum modulo a reciprocal lattice vector $G$. On a periodic chain of $M$ sites with unit spacing, the discrete Fourier transform supplies ${\sum }_{j=1}^M{e}^{i(k_1+k_2-k_3-k_4)j}=M{\delta }_{k_1+k_2,k_3+k_4(\mathrm{mod}2\pi )}$. This Fourier orthogonality proves the selection rule directly; it permits transfer by a reciprocal lattice vector rather than requiring equality of unrestricted momenta.