Energy-momentum operators with eigenfunctions localized along a line

Abstract

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 ψ= {\tilde p^0} ψ$ 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.