  In this paper, we formulate the Lorentz-breaking extension for the spin-3/2
field theory and couple it to the Abelian gauge field in a Lorentz violating
(LV) manner. Next, we calculate the lower LV quantum corrections, that is, the
Carroll-Field-Jackiw (CFJ) term, which, being superficially divergent, turns
out to be finite but ambiguous, and also the higher-derivative CFJ term.
Besides, we compute the aether term, being the lowest CPT-even LV term,
involving the second order in the LV vector.
