On the Ehrenfest theorem and centroids of relativistic particles

We consider relativistic versions of the Ehrenfest relation between the expectation values of the coordinate and momentum of a quantum particle in free space: $d⟨\mathbf{r}⟩\mathrm{/}dt=⟨\mathbf{p}⟩\mathrm{/}m$. We find that the simple proportionality between the mean velocity and momentum holds true only for the simplest quadratic dispersion (i.e., dependence of the energy on the momentum). For relativistic dispersion, the mean velocity is generally not collinear with the mean momentum, but velocity of the {\it energy centroid} is directed along the mean momentum. This is related to the conservation of the Lorentz-boost momentum and has implications in possible decomposition of the mean orbital angular momentum into intrinsic and extrinsic parts. Neglecting spin/polarization effects, these properties depend solely on the dispersion relation, and can be applied to any waves, including classical electromagnetic or acoustic fields.