= Continuum limit of a harmonic chain
{title2=$\rho=m_0/a,\quad K=\kappa a,\quad v=a\sqrt{\kappa/m_0}$}
At wavelengths much longer than the lattice spacing, $u_{j+1}-u_j\simeq a\partial_xu$ gives $L=\int dx[\rho(\partial_tu)^2-K(\partial_xu)^2]/2$. The field obeys a <wave equation> with <sound speed> $v=\sqrt{K/\rho}$. Its conjugate <momentum> density is $\Pi=\rho\partial_tu$ and <canonical quantization> gives $[u(x),\Pi(x')]=i\hbar\delta(x-x')$. The lattice remains the <ultraviolet cutoff> of this long-wave theory.
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