Why are the renormalisation constants the same in LSZ and renormalisation?

Question

To keep it simple I'll phrase everything in terms of scalar fields. We seem to have three constants called $Z$:

1. When we do LSZ reduction we say, as $t\rightarrow-\infty$ then $\phi\rightarrow \sqrt{Z}\phi_{free}$.

2. Later we renormalised the theory and rescaled the bare fields $\phi_0$ as $\phi_0=\sqrt{Z}\phi_R$.

3. We also have a factor $Z$ that is the residue of the two-point correlation function $\frac{Z}{p^2-m^2_R}$.

Why are these all the same $Z$? (ok, i sort of see that for 2 we can choose to rescale the field however we want, but that doesn't explain why 1 and 3 are the same)

Answer 1

The number $Z$ is defined as being overlap between the desired one particle state $|k\rangle$ with $k^2=m_R^2$ and the actual (more complicated) state created by $\phi(x)$. In other words
$$ \langle k|\phi(x)|{\rm vac}\rangle= \sqrt Z e^{-ikx}. $$ This gives both the residue of the pole in the Lehman-Kallen expansion, and the need for $Z$ in LSZ. Note that $Z\le 1$, with equality if it's a free theory.