ABSTRACT
Using the reduction formulae, we can translate the rules for the calculation of Feynman diagrams for the Green’s functions, discussed in Section 8.2, into rules for the calculation of S-matrix elements. The starting point is the Fourier transform which appears in the reduction formula, equation (7.51). To simplify the notation we consider all particles as if they belonged to the initial state. For a particle in the final state, it is sufficient to change the sign of the momentum and the energy. We must therefore calculate the following expression ∏ k = 1 n ( p k 2 - m 2 ) i Z ∫ ∏ k = 1 n d 4 x k ∏ k = 1 n e - i p k x k 0 | T ϕ ( x 1 ) … ϕ ( x n ) | 0 . $$ \begin{aligned} \prod _{k=1}^{n} \left( \frac{(p_k^2-m^2)}{i\sqrt{Z}} \right) \int \prod _{k=1}^{n} d^{4}x_{k} \prod _{k=1}^{n}e^{-ip_{k}x_{k}}\; \left\langle {0|T\left(\phi ( x_{1})\ldots \phi ( x_{n})\right)|0}\right\rangle \ . \end{aligned} $$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315369723/48d43afe-85b0-4e2b-a932-df2bd1af07be/content/math9_437.tif"/>
