  We study the quantum fermions+gravity system, that is, the gravitational
counterpart of QED. We start from the standard Einstein-Weyl theory,
reformulated in terms of Ashtekar variables; and we construct its non-
perturbative quantum theory by extending the loop representation of general
relativity. We construct the fermion equivalent to the loop variables. Not
surprisingly, fermions can be incorporated in the loop representation simply by
including open curves into ''Loop space''. We explicitely construct the
diffeomorphism and hamiltonain operators. The first can be fully solved as in
pure gravity. The second is constructed by using a background-independent
regularization technique. The theory retains the clean geometrical features of
the pure quantum gravity. In particular, the hamiltonian constraint admits the
same simple geometrical interpretation as its pure gravity counterpart. Quite
surprisingly, a simple action codes the full dynamics of the interacting
theory. To unravel the dynamics of the theory we study the evolution of the
fermion-gravity system in the physical-time defined by an additional coupled
(''clock''-) scalar field. We construct the Hamiltonian operator that evolves
the system in this physical time. We show that this Hamiltonian is finite,
diffeomorphism invariant, and has a simple geometrical action. The theory
fermions+gravity evolving in the clock time is finally given by the
combinatorial and geometrical action of this Hamiltonian on a set of graphs
with a finite number of end points. This action defines the "topological
Feynman rules" of the theory.
