The Substitution Rules

One of the most powerful consequences of the structure of the $S$-matrix is the substitution rule. If two diagrams $S^\prime$ and $S$ differ only in one external line, such that this line is an outgoing photon, electron, or positron line in $S^\prime$ and an ingoing photon, positron, or electron line in $S$, respectively, then the $S$-matrix elements associated with $S^\prime$ and $S$ are related as shown in table 7.1. In case of circular polarization: right-circular $\leftrightarrow$ left-circular polarization. The double arrow indicates that the substitution necessary to obtain $S$ from $S^\prime$ is reversible, so that one can also obtain $S^\prime$ from $S$.

Table 7.1: Substitution rules for photons, electrons and positron.

$S^\prime$		$S$		Kinematics			Spin
photon	$k^\prime$ out	photon	$k$ in	$k^\prime$	$\leftrightarrow$	$k$	${\epsilon^*}^\prime$	$\leftrightarrow$	$\epsilon$
electron	$p^\prime$ out	positron	$q$ in	$p^\prime$	$\leftrightarrow$	$-q$	$\overline{u}(p^\prime)$	$\leftrightarrow$	$v(q)$
positron	$q^\prime$ out	electron	$p$ in	$q^\prime$	$\leftrightarrow$	$-p$	$\overline{v}(q^\prime)$	$\leftrightarrow$	$u(p)$

This substitution rule can also be applied to the square of the matrix elements after a spin summation has been carried out. In that case, the wave functions have disappeared and projection operators take their place. The substitution can then be carried out directly in these, as follows

Table 7.2: Substitution rules for spin sums.

$\overline{\mathcal{M}}$		Projection Operator
electron	$p^\prime\leftrightarrow q$	$\Lambda_-(p^\prime)$	$\leftrightarrow$	$-\Lambda_+(q) = -\Lambda_-(-q)$
positron	$q^\prime\leftrightarrow p$	$\Lambda_+(q)$	$\leftrightarrow$	$-\Lambda_-(p) = -\Lambda_+(-p)$

This simply means that there is also an overall sign change of the trace in addition to the momentum substitution.

Repeated application of the substitution rule permits ``bending'' of the external lines of a diagram in all possible ways. Some of the resulting diagrams will be excluded, of course, by conservation of energy and momentum.

As an example, the three processes of Compton scattering, two-photon free-pair annihilation, and pair production by two photons. When any one of these processes is computed the others follow by substitution. However, the original matrix elements (or traces) must be known in complete generality (without restriction to a special coordinate system) in order to make this procedure work.