(out)}$ are introduced to create states $|k_1,k_2,\cdots k_N; {\rm in(out)}\rangle$ from the vacuum. The in- and out- fields $\phi_{\rm in(out)}$ are then defined using superposition. They look like free fields, but are not, since for the [*total*]{} four-momentum $P_\mu$, we have $$P_\mu |k_1,k_2,\cdots k_N; {\rm in(out)}\rangle = (\sum_i k_{i \mu})
|k_1,k_2,\cdots k_N; {\rm in(out)}\rangle .$$

It is also assumed that

\(a) The vacuum and single particle states are unique up to a phase. Then after a phase choice, there is only one vacuum $|0\rangle$, $\langle 0 | 0\rangle =1$, and $$S|0\rangle = |0\rangle.$$

\(b) There exists an interpolating field $\phi$ in the Heisenberg representation such that matrix elements of $\phi(x,t)$ between in- and out- states go to those of $\phi_{\rm in, out}(x,t)$ in the infinite past and future, $$\phi(x,t)~ -~ \phi_{\rm in, out}(x,t) \rightarrow 0 \quad {\rm as} \quad t
\rightarrow \mp \infty$$ in weak topology. (We treat the case of just one scalar field for simplicity.)

Then LSZ formalism shows that $$\langle k'_1,k'_2,\cdots k'_N; {\rm out}|k_1,k_2,\cdots k_M; {\rm
  in}\rangle = \int {\cal{I}}~~G_{N+M}(x_1',\cdots,x_N';x_1\cdots x_M)$$ where $${\cal{I}}~=~\prod d^4x'_i\prod d^4x_j~e^{i(k_i'\cdot
x_i'~-~k_j\cdot x_j)} i(\partial_i'^2+m^2)\cdot i(\partial_j^2+m^2)
\label{I}$$ and $$G_{N+M} (x_1',\cdots,x_N';x_1\cdots x_M)\equiv \langle
0|T(\phi(x_1')\cdots \phi(x_N')\phi(x_1)\cdots \phi(x_M))|0\rangle$$

It is now convenient to regard all the momenta 