Wave function renormalization at finite temperature

We present a derivation of the medium dependent wave function renormalization for a spinor field in the presence of a thermal bath. We show that, as already pointed out in the literature, projector operators are not multiplicatively renormalized and the effect involves a nontrivial spinor dependence, which disappears in the zero temperature covariant limit. The results, which differ from what is already found in literature, are then applied to the decay of a massive scalar boson into two fermions and to the $\beta$-decay and crossed related processes relevant for primordial nucleosynthesis.