Harmonic chain ($H={\sum }_j[p_j^2/(2m_0)+\kappa (u_{j+1}-u_j{)}^2/2]$)

A uniform harmonic chain couples neighbouring displacements by quadratic spring energies. With periodic boundaries and spacing $a$, a Fourier transform gives ${\omega }_k=2\sqrt{\kappa /m_0}|\sin (ka/2)|$. Quantization makes each nonzero normal mode a phonon. The zero-frequency uniform displacement is a free centre-of-mass coordinate unless pinned, not an ordinary oscillator.^[ref]

A quantum harmonic chain has quadratic couplings between canonical coordinates and momenta. A normal mode transformation turns it into independent quantum harmonic oscillators. A continuum elastic action with time coefficient $\chi$ and spatial stiffness $\kappa$ has long-wave frequency ${\omega }^2=(\kappa /\chi )q^2$, the structure of a smooth condensate phase action.

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.

The uniform displacement $X=N^{-1}{\sum }_j{u}_j$ and total momentum $P={\sum }_j{p}_j$ satisfy $[X,P]=i\hslash$. The nonzero-frequency phonon expansion alone yields the site commutator $i\hslash ({\delta }_{jl}-1/N)$; including $X,P$ restores the missing uniform part. The thermal free-coordinate factor requires a specified volume, pinning or a fixed centre of mass. Substituting zero frequency into an oscillator partition function instead produces a spurious divergence without defining the physical zero mode.

At wavelengths much longer than the lattice spacing, $u_{j+1}-u_j\simeq a{\partial }_{x}u$ gives $L=\int dx[\rho ({\partial }_{t}u{)}^2-K({\partial }_{x}u{)}^2]/2$. The field obeys a wave equation with sound speed $v=\sqrt{K/\rho }$. Its conjugate momentum density is $\Pi =\rho {\partial }_{t}u$ and canonical quantization gives $[u(x),\Pi (x^{\prime })]=i\hslash \delta (x-x^{\prime })$. The lattice remains the ultraviolet cutoff of this long-wave theory.