Green's function of the Klein–Gordon operator. The weak asymptotic relation between out field and interacting field is: The reduction formula for scalars The asymptotic relations are all that is needed to obtain the LSZ reduction formula. For future convenience we start with the matrix element: which is slightly more general than an S-matrix element. Indeed, is the expectation value of the time-ordered product of a number of fields between an out state and an in state. The out state can contain anything from the vacuum to an undefined number of particles, whose momenta are summarized by the index . The in state contains at least a particle of momentum , and possibly many others, whose momenta are summarized by the index . If there are no fields in the time-ordered product, then is obviously an S-matrix element. The particle with momentum can be 'extracted' from the in state by use of a creation operator: where the prime on denotes that one particle has been taken out. With the assumption that no particle with momentum is present in the out state, that is, we are ignoring forward scattering, we can write: because acting on the left gives zero. Expressing the construction operators in terms of in and out fields, we have: Now we can use the asymptotic condition to write: Then we notice that the field can be brought inside the time-ordered product, since it appears on the right when and on the left when : In the following, dependence in the time-ordered product is what matters, so we set: It's easy to show by explicitly carrying out the time integration that: so that, by explicit time derivation, we have: By its definition we see that is a solution of the Klein–Gordon equation, which can be written as: Substituting into the expression for and integrating by parts, we arrive at: That is: Starting from this result, and following the same path another particle can be extracted from the in state, leading to the insertion of another field in the time-ordered product. A very similar routine can extract particles from the out state, and the two can be iterated to get vacuum both on right and on left of the time-ordered product, leading to the general formula: Which is the LSZ reduction formula for Klein–Gordon scalars. It gains a much better looking aspect if it is written using the Fourier transform of