Following section 7.1 and 7.2 in Peskin and Schroeder (P&S), I've tried to consider what the derivation of the LSZ formula looks like for (spin $1/2$) fermions (in the text, they explicitly consider a scalar field instead). On page 216, they briefly mention some of the essential equations for fermions. For instance, eq 7.12 gives
$$\langle\Omega|\psi(0)|p,s\rangle = \sqrt{Z_2}u^s(p)$$
where $|\Omega\rangle$ is the vacuum of the interacting theory, $\psi$ is a Dirac field, $|p,s\rangle$ is a single-particle state of the interacting theory with 4-momentum $p$ and spin $s$ and $u^s(p)$ is the usual 4-spinor (with 4-momentum $p$ and spin $s$). $Z_2$ is called the field strength renormalization.
P&S briefly mention that for antiparticles, we must use $\bar{v}$ instead of $u$. I assume that the similar equation for antiparticles must be
$$\langle\Omega|\bar{\psi}(0)|p,s\rangle' = \sqrt{Z_2}\bar{v}^{-s}(p)$$
where $|p,s\rangle'$ is now a single-antiparticle state. Here we use the spinor with spin $-s$ (essentially just the opposite spin of $s$), since antiparticle states and the corresponding spinors have opposite spin.
Using these equations, I was able to derive the LSZ formula for fermions. The problem is that both equations can be used to define $Z_2$, and it's not clear to me why they have to be consistent. I tried to see if I could argue that, for instance, they can be made consistent by choosing a suitable phase factor in the definition of the single-particle states (we have already chosen a normalization for the states (the relativistic normalization), so the only freedom we have left is in the phase factors), but I don't see how that can be done. In particular, I don't see how to argue that there wouldn't have to be some extra multiplicative factor (with norm different from 1) in one of the equations to make them consistent. Of course, in the free theory, one can simply use the Fourier mode expansion of $\psi$ to show that $Z_2$ is the same in both equations (they're both 1 in that case), but here we're of course considering the interacting theory. Any tips?
