= Asymptotic scalar field
{title2=$\phi_{\rm in},\phi_{\rm out}$}
An <asymptotic scalar field> is a free field describing stable incoming or outgoing particles of an interacting <scalar field theory>. With a field that overlaps the stable one-particle state, the scattering asymptotic condition is $\phi\to\sqrt Z\phi_{\rm in}$ in the remote past and $\phi\to\sqrt Z\phi_{\rm out}$ in the remote future. These are scattering limits of suitably smeared matrix elements, not unrestricted strong pointwise limits of local fields. With covariant state normalization $\langle p'|p\rangle=(2\pi)^3 2E_p\delta^3(\mathbf p-\mathbf p')$, the free-field vacuum-to-one-particle matrix element is one. <Translation invariance> and the asymptotic condition then give $\langle p|\phi(0)|0\rangle=\sqrt Z$ after choosing its phase positive. The <wave-function renormalization> residue $Z$ measures that overlap.
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