 Solution

The canonical anticommutation relations are preserved precisely when the real matrix acting on $({\gamma }_{k\uparrow },{\gamma }_{-k\downarrow }^{†})$ is orthogonal. Directly,The other relations are identical or vanish between disjoint pairs. Hence $u_k^2+v_k^2=1$, and one may write $u_k=\cos {\theta }_k$, $v_k=\sin {\theta }_k$. This fermionic Bogoliubov transformation uses an ordinary rotation; bosonic mixing instead preserves a difference of squared coefficients.