= Free scalar field with quartic spatial dispersion
{title2=$E_{\mathbf p}^2=\kappa^2|\mathbf p|^4+m^2$}
A local <real scalar field> with <Lagrangian density> $\tfrac12[(\partial_t\phi)^2-\kappa^2(\nabla^2\phi)^2-m^2\phi^2]$ has the <equation of motion> $\partial_t^2\phi+\kappa^2\nabla^4\phi+m^2\phi=0$. Two spatial <integrations by parts> produce the positive <biharmonic operator> in this equation. Its <Hamiltonian> is a sum of positive squares and each nonzero-frequency spatial <Fourier mode> quantizes as an oscillator with the displayed <dispersion relation>. Higher spatial derivatives do not require higher time derivatives. The theory has a preferred time and does not possess <Lorentz invariance>.
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