  The fermion propagator is derived in detail from the model of fermion coupled
to loop quantum gravity. As an ingredient of the propagator, the vacuum state
is defined as the ground state of some effective fermion Hamiltonian under the
background geometry given by a coherent state resembling the classical
Minkowski spacetime. Moreover, as a critical feature of loop quantum gravity,
the superposition over graphs is employed to define the vacuum state. It turns
out that the graph superposition leads to the propagator being the average of
the propagators of the lattice field theory over various graphs so that all
fermion doubler modes are suppressed in the propagator. This resolves the
doubling problem in loop quantum gravity. Our result suggests that the
superposition nature of quantum geometry should, on the one hand, resolve the
tension between fermion and the fundamental discreteness and, on the other
hand, relate to the continuum limit of quantum gravity.
