= Crystal momentum conservation
{title2=$k_1+k_2=k_3+k_4\pmod{G}$}
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 $\sum_{j=1}^Me^{i(k_1+k_2-k_3-k_4)j}=M\delta_{k_1+k_2,k_3+k_4\pmod{2\pi}}$. This <Fourier orthogonality> proves the selection rule directly; it permits transfer by a <reciprocal lattice> <vector> rather than requiring equality of unrestricted momenta.
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