  We construct the propagator for a free fermionic unparticle field from basic
considerations of scale and Lorentz invariance. The propagator is fixed up to a
normalization factor which is required to recover the result of a free massless
fermion field in the canonical limit of the scaling dimension. Two new features
appear compared to the bosonic case. The propagator contains both \gamma and
non-\gamma terms, and there is a relative phase of \pi/2 between the two in the
time-like regime for arbitrary scaling dimension. This should result in
additional interference effects on top of the one known in the bosonic case.
The non-\gamma term can mediate chirality flipped transitions that are not
suppressed by a light fermion mass but are enhanced by a large bosonic mass in
loops, compared to the pure particle case. We employ this last feature to set
stringent bounds on the Yukawa couplings between a fermionic unparticle and an
ordinary fermion through electromagnetic dipole moments and radiative decays of
light fermions.
