  We construct a state in the loop quantum gravity theory with zero
cosmological constant, which should correspond to the flat spacetime vacuum
solution. This is done by defining the loop transform coefficients of a flat
connection wavefunction in the holomorphic representation which satisfies all
the constraints of quantum General Relativity and it is peaked around the flat
space triads. The loop transform coefficients are defined as spin foam state
sum invariants of the spin networks embedded in the spatial manifold for the
SU(2) quantum group. We also obtain an expression for the vacuum wavefunction
in the triad represntation, by defining the corresponding spin networks
functional integrals as SU(2) quantum group state sums.
