The LSZ reduction formula obtains an S-matrix element from a time-ordered correlation function by amputating external propagators, taking external momenta on shell, and multiplying by pole-residue factors and external wavefunctions.
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 in the remote past and 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 , the free-field vacuum-to-one-particle matrix element is one. Translation invariance and the asymptotic condition then give after choosing its phase positive. The wave-function renormalization residue measures that overlap.
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The LSZ reduction formula, named after Lüders, Steinweg, and Ziman, is a fundamental result in quantum field theory (QFT) that relates S-matrix elements to time-ordered correlation functions (or Green's functions). It provides a method for calculating the S-matrix (which describes the scattering processes) from the theoretical correlation functions computed in a given quantum field theory.