= BCS ground state
{c}
{title2=$\prod_k(u_k+v_kc_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger)|0\rangle$}
The BCS ground state is the normalized product of empty and occupied pair states $\prod_k(u_k+v_kb_k^\dagger)|0\rangle$. For $u_k^2+v_k^2=1$, each pair factor is normalized. The inverse <Bogoliubov transformation> annihilates it, so it is the vacuum of positive-energy <quasiparticles>. Although individual factors use opposite momenta, $k$ runs over the relative pair momenta; it is not restricted to $k=0$.
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