In the section 5 of the book, it says
The LSZ formula is valid provided that the field obeys $$\langle 0|\phi(x)|0\rangle=0, \langle k|\phi(x)|0\rangle=1.$$
The second one is needed to ensure one-particle state is correctly normolized. But I cannot understand. My question is: in the derivation of LSZ formula, at which step do we use the condition?
I have also seen this post, but the 8th step confues me: $$\hat{a}^{\dagger}_1(\infty)- \hat{a}^{\dagger}_1(-\infty) =\sqrt{Z}[\hat{a}^{\dagger}_{1,{\rm out}}(\infty) - \hat{a}^{\dagger}_{1,{\rm in}}(-\infty)].$$
What are the $\hat{a}^{\dagger}_{1,{\rm out}}(\infty)$ and $\hat{a}^{\dagger}_{1,{\rm in}}(-\infty)$ here?
