Harmonic-chain normal-mode reality condition ($Q_k^{†}=Q_{-k},[Q_k,P_{k^{\prime }}]=i\hslash {\delta }_{k,-k^{\prime }}$)

The discrete Fourier transform of real displacement operators produces complex coordinates obeying $Q_k^{†}=Q_{-k}$. They are not independent real oscillators. The canonical expansion $Q_k=\sqrt{\hslash /(2m{\omega }_k)}(a_k+a_{-k}^{†})$ and $P_k=-i\sqrt{m\hslash {\omega }_k/2}(a_k-a_{-k}^{†})$ 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.