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.
Categories: quant-ph math-ph math.MP