BCS ground state (${\prod }_k(u_k+v_k{c}_{k\uparrow }^{†}{c}_{-k\downarrow }^{†})|0⟩$)

The BCS ground state is the normalized product of empty and occupied pair states ${\prod }_k(u_k+v_k{b}_k^{†})|0⟩$. 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 $k=0$.