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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)

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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.

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