Free scalar field with quartic spatial dispersion ($E_{\mathbf{p}}^2={\kappa }^2|\mathbf{p}{|}^4+m^2$)

A local real scalar field with Lagrangian density $\frac{1}{2}[({\partial }_t\varphi {)}^2-{\kappa }^2({\nabla }^2\varphi {)}^2-m^2{\varphi }^2]$ has the equation of motion ${\partial }_t^2\varphi +{\kappa }^2{\nabla }^4\varphi +m^2\varphi =0$. 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.