 Asymptotic scalar field

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 $\varphi \to \sqrt{Z}{\varphi }_{\mathrm{in}}$ in the remote past and $\varphi \to \sqrt{Z}{\varphi }_{\mathrm{out}}$ 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 $⟨p^{\prime }|p⟩=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-{\mathbf{p}}^{\prime })$, the free-field vacuum-to-one-particle matrix element is one. Translation invariance and the asymptotic condition then give $⟨p|\varphi (0)|0⟩=\sqrt{Z}$ after choosing its phase positive. The wave-function renormalization residue $Z$ measures that overlap.