A New Look at the Position Operator in Quantum Theory

In standard quantum theory the momentum and position operators are on equal footing and wave functions in momentum and coordinate representations are related to each other by the Fourier transform. However, while the momentum operator is unambiguously defined as one of the operators of the symmetry algebra, the position operator has a physical meaning only in semiclassical approximation and should be defined from additional considerations. We show that, as a consequence of the inconsistent definition of standard position operator, an inevitable effect in standard theory is the wave packet spreading (WPS) of the photon coordinate wave function in directions perpendicular to the photon momentum and an analogous effect takes place in classical electrodynamics. This leads to the fundamental paradox that we should see not separate stars but only an almost continuous background from all stars. We propose a new consistent definition of the position operator. In this approach WPS in directions perpendicular to the particle momentum is absent regardless of whether the particle is nonrelativistic or relativistic. Hence the above paradox is resolved. Moreover, for an ultrarelativistic particle the effect of WPS is absent at all. Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation. PACS: 11.30.Cp, 03.65.-w, 03.63.Sq, 03.65.Ta