  The postulate that coordinate and momentum representations are related to
each other by the Fourier transform has been accepted from the beginning of
quantum theory by analogy with classical electrodynamics. As a consequence, 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. This leads to several paradoxes. The most striking of them is that
coordinate wave functions of photons emitted by stars have cosmic sizes and
strong arguments indicate that this contradicts observational data. We argue
that the above postulate is based neither on strong theoretical arguments nor
on experimental data and propose a new consistent definition of the position
operator. Then WPS in directions perpendicular to the particle momentum is
absent and the paradoxes are resolved. 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. Implications of the results for
entanglement, quantum locality and the problem of time in quantum theory are
discussed.
