  A charge conserving approximation scheme determining the excitations of
crystalline solids is proposed. Like other such approximations, it relies on
"downfolding" of the original microscopic model to a simpler electronic model
on the lattice with pairwise interactions. A systematic truncation of the set
of Dyson - Schwinger equations for correlators of the low energy (downfolded)
model of a material, supplemented by a "covariant" calculation of correlators
lead to a converging series of approximates. The covariance preserves all the
Ward identities among correlators describing various condensed matter probes.
It is shown that the third order approximant of this kind beyond classical and
gaussian (Hartree - Fock) is precise enough and due to several fortunate
features the complexity of calculation is surprisingly low so that a realistic
material computation is feasible. Focus here is on the electron field
correlator describing the electron (hole) excitations measured in photoemission
and other probes. The scheme is tested on several solvable benchmark models
