Energy-momentum operators with eigenfunctions localized along a line

Abstract

The momentum operator $ {\bf p} = - i {\bx \nabla} $ has radial component $\tilde{\mathbf{p}}\equiv -i\hat{\mathbf{r}}(\frac{1}{r}{\partial }_{r}r).$ We show that $\tilde{\mathbf{p}}$ 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.

10 pages, errors corrected