= Solution
The <canonical anticommutation relations> are preserved precisely when the real matrix acting on $(\gamma_{k\uparrow},\gamma_{-k\downarrow}^\dagger)$ is orthogonal. Directly,
$$
\{c_{k\uparrow},c_{k\uparrow}^\dagger\}=u_k^2+v_k^2,\qquad \{c_{k\uparrow},c_{-k\downarrow}\}=-u_kv_k+v_ku_k=0.
$$
The other relations are identical or vanish between disjoint pairs. Hence \b[$u_k^2+v_k^2=1$], 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.
Back to article page
