Diffraction in time in terms of Wigner distributions and tomographic probabilities

Long ago appeared a discussion in quantum mechanics of the problem of opening a completely absorbing shutter on which were impinging a stream of particles of definite velocity. The solution of the problem was obtained in a form entirely analogous to the optical one of diffraction by a straight edge. The argument of the Fresnel integrals was though time dependent and thus the first part in the title of this article. In section 1 we briefly review the original formulation of the problem of diffraction in time. In section 2 and 3 we reformulate respectively this problem in Wigner distributions and tomographical probabilities. In the former case the probability in phase space is very simple but, as it takes positive and negative values, the interpretation is ambiguous, but it gives a classical limit that agrees entirely with our intuition. In the latter case we can start with our initial conditions in a given reference frame but obtain our final solution in an arbitrary frame of reference.