= Solution
A <real scalar field> suffices to describe one neutral species of bosonic <quasiparticle>. Let $\pi(\mathbf x)$ be its <canonical momentum>. In three spatial dimensions, choose the positive classical <Hamiltonian>
$$
\boxed{H[\phi,\pi]=\frac12\int d^3x\left[\pi^2+\kappa^2(\nabla^2\phi)^2+m^2\phi^2\right].}
$$
The same construction works in any number of spatial dimensions by replacing $d^3x$ appropriately. Under a spatial <Fourier transform>, $\nabla^2\phi$ becomes $-|\mathbf p|^2\widetilde\phi$, so the <Hamiltonian> is a collection of independent <quantum harmonic oscillators> with squared <angular frequency> $\omega_{\mathbf p}^2=\kappa^2|\mathbf p|^4+m^2$. The usual reality condition pairs opposite <momenta>; it does not introduce a second species.
The canonical <action> is $S=\int dt\,[\int d^3x\,\pi\dot\phi-H]$. Varying $\pi$ gives $\pi=\dot\phi$; eliminating it by the <Legendre transform> gives
$$
\boxed{S[\phi]=\frac12\int dt\,d^3x\left[(\partial_t\phi)^2-\kappa^2(\nabla^2\phi)^2-m^2\phi^2\right].}
$$
This <Lagrangian density> is local: only the field and finitely many derivatives at the same point occur. It contains second spatial derivatives but only first time derivatives, so no extra canonical time data are needed. A square root of a differential operator in the classical <Hamiltonian> is unnecessary. The fourth-order spatial term breaks <Lorentz invariance> while preserving spatial rotations and translations. This is a <free scalar field with quartic spatial dispersion>.
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