allows for a neater expression, in which the quantity in square brackets is to be regarded as a differential operator: Consider next the matrix element appearing in the integral. Extracting an out state creation operator and subtracting the corresponding in state operator, with the assumption that no incoming particle has the same momentum, we have Remembering that , where , we can replace the annihilation operators with in fields using the adjoint of the inverse relation. Applying the asymptotic relation, we find Note that a time-ordering symbol has appeared, since the first term requires on the left, while the second term requires it on the right. Following the same steps as before, this expression reduces to The rest of the in and out states can then be extracted and reduced in the same way, ultimately resulting in The same procedure can be done for the scattering of d-type particles, for which 's are replaced by 's, and 's and 's are swapped. Field strength normalization The reason of the normalization factor in the definition of in and out fields can be understood by taking that relation between the vacuum and a single particle state with four-moment on-shell: Remembering that both and are scalar fields with their Lorentz transform according to: where is the four-momentum operator, we can write: Applying the Klein–Gordon operator on both sides, remembering that the four-moment is on-shell and that is the Green's function of the operator, we obtain: So we arrive to the relation: which accounts for the need of the factor . The in field is a free field, so it can only connect one-particle states with the vacuum. That is, its expectation value between the vacuum and a many-particle state is null. On the other hand, the interacting field can also connect many-particle states to the vacuum, thanks to interaction, so the expectation values on the two sides of the last equation are different, and need a normalization factor in between. The right hand side can be computed explicitly, by expanding the in field in creation and annihilation operators: Using the commutation relation between and we obtain: leading to the relation: by which the value of may be computed, provided that one knows how to compute . References The original paper is: A pedagogical derivation of the LSZ reduction formula can be found in: Peskin and Schroeder, Section 7.2, also in Srednicki, Section I.5,