Energy-momentum operators with eigenfunctions localized along a line

The momentum operator ${\bf p} = - i {\bx \nabla}$ has radial component ${\bf \tilde p} \equiv - i {\bf \hat{r}} ({1 \over r} \partial_r r).$ We show that ${\bf \tilde p}$ is the space part of a 4-vector operator, the zero component of which is a positive operator. Their eigenfunctions are localized along an axis through the origin. The solutions of the evolution equation $i \partial_t \psi = {\tilde p^0} \psi$ are waves along the propagation axis. Lorentz transformations of these waves yield the aberration and Doppler shift. We briefly consider spin-half and spin-one representations.