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 $k=0$.

The empty and doubly occupied states of a paired momentum block form a two-level space. Define $S_k^{+}=a_{k\uparrow }^{†}{a}_{-k\downarrow }^{†}$, $S_k^{-}=a_{-k\downarrow }{a}_{k\uparrow }$ and $S_k^z=(n_{k\uparrow }+n_{-k\downarrow }-1)/2$. They obey the spin algebra on that space. The normalized factor $u_k|0⟩+v_k|\mathrm{pair}⟩$ is a spin-one-half coherent state; the product over blocks is the BCS ground state. Pauli exclusion prevents interpreting a block as an unrestricted harmonic-oscillator pair mode.