Electron Scattering

Consider ordinary scattering of electrons. In this case

$$\Psi_i(y) \stackrel{y_0\rightarrow-\infty}{\Longrightarrow} ... ...c{m}{E_-}} \frac{1}{(2\pi)^{3/2}} u(p_-,s_-) e^{-ip_-\cdot y}$$ (6.95)

reduces to an incoming electron wave with positive energy $E_-$, momentum $\vec{p}_-$ and spin $s_-$. The minus sign designates the negative charge of the electron.

The $n$'th order contribution to the $S$-matrix element is

$$S_{fi}^{(n)} = -ie^n \int d^4y_1 \cdots d^4y_n \overline{\Psi}_f^{(+E)}(y_n) \not{\!\!A}(y_n) S_F(y_n-y_{n-1}) \not{\!\!A}(y_{n-1})$$ (6.96)

$$\cdots S_F(y_2-y_1) \not{\!\!A}(y_1) \psi_i^{(+E)}(y_1) .$$

This expression contains both types of graphs as shown in figure 6.8. That is, in addition to ordinary scattering intermediate pair creation and pair annihilation are included in the series, since the various $d^4y_i$ integrations also allow for a reverse time ordering, $y_{n+1}^0<y_n^0$.

Figure 6.8: Two 3'rd order graphs for electron scattering.