  Young's classic double-slit experiment demonstrates the reality of
interference when waves and particles travel simultaneously along two different
spatial paths. Here, we propose a double-slit experiment in momentum space,
realized in the free-space elastic scattering of vortex electrons. We show that
this process proceeds along two paths in momentum space, which are well
localized and well separated from each other. For such vortex beams, the
(plane-wave) amplitudes along the two paths acquire adjustable phase shifts and
produce interference fringes in the final angular distribution. We argue that
this experiment can be realized with the present day technology. We show that
it gives experimental access to the Coulomb phase, a quantity which plays an
important role in all charged particle scattering but which usual scattering
experiments are insensitive to.
