  We scrutinize the diagrammatic perturbation theory of noninteracting
electrons in a random potential with the aim to accomplish a consistent
comprehensive theory of quantum diffusion. Ward identity between the
one-electron self-energy and the two-particle irreducible vertex is generally
not guaranteed in the perturbation theory with only elastic scatterings. We
show how the Ward identity can be established in practical approximations and
how the functions from the perturbation expansion should be used to obtain a
fully consistent conserving theory. We derive the low-energy asymptotics of the
conserving full two-particle vertex from which we find an exact representation
of the diffusion pole and of the static diffusion constant in terms of Green
functions of the perturbation expansion. We illustrate the construction on the
leading vertex corrections to the mean-field diffusion due to maximally-crossed
diagrams responsible for weak localization.
