= Harmonic-chain normal-mode reality condition
{title2=$Q_k^\dagger=Q_{-k},\quad [Q_k,P_{k'}]=i\hbar\delta_{k,-k'}$}
The <discrete Fourier transform> of real displacement operators produces complex coordinates obeying $Q_k^\dagger=Q_{-k}$. They are not independent real oscillators. The canonical expansion $Q_k=\sqrt{\hbar/(2m\omega_k)}(a_k+a_{-k}^\dagger)$ and $P_k=-i\sqrt{m\hbar\omega_k/2}(a_k-a_{-k}^\dagger)$ has independent occupation operators for all nonzero wavevectors, with the uniform free coordinate handled separately. At self-inverse zone-boundary wavevectors the same formulas reduce to one ordinary real oscillator.
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