= Odd-momentum fermionic Bogoliubov angle
{title2=$\theta_{-p}=-\theta_p$}
For a spinless <Bogoliubov transformation> $c_p=\cos\theta_p\,\alpha_p+i\sin\theta_p\,\alpha_{-p}^\dagger$, consistency of the doubled <Nambu spinor> requires $\cos\theta_{-p}=\cos\theta_p$ and $\sin\theta_{-p}=-\sin\theta_p$. Thus choose $\theta_{-p}=-\theta_p$ modulo $2\pi$. The cross <anticommutator> from the first-row formulas is $\{c_p,c_{-p}\}=i\sin(\theta_p+\theta_{-p})$; a nontrivial even angle would violate the <canonical anticommutation relations>. A self-paired zero mode must instead have vanishing mixing in this convention. Real-angle <unitarity> within each two-by-two block alone does not enforce consistency between duplicated momentum labels.
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