Pair Annihilation

Now consider pair annihilation. In this case we insert for $\Psi_i(y)$ a solution that reduces to $\Psi_i^{(+E)}(y)$ at $t\rightarrow-\infty$. This positive-energy solution represents an electron that propagates forward in time into the interaction volume, to be scattered backward in time and emerges into a negative energy state. The $n$'th order amplitude that the electron scatters into a given final state $\psi_f^{(-E)}$, labelled by the physical quantum numbers $\vec{p}_+$, $s_+$, $\epsilon_f=-1$, is given by

$$S_{fi}^{(n)} = ie^n\int d^4y_1\cdots \int d^4y_n \overline{\... ... S_F(y_n-y_{n-1}) \cdots \not{\!\!A}(y_1) \psi_i^{(+E)}(y_1) .$$ (6.99)

In the language of hole theory this is the $n$'th order amplitude that a positive-energy electron is scattered into an electron state of negative three-momentum $-\vec{p}_+$ and spin $-s_+$.