# Difference between positron and electron scattering in Coulomb field

In first order of perturbation theory the S-matrix amplitude for electron scattering in the Coulomb field will be (up to normalization factors) $$S_{fi} = \frac{iZ q^2}{\sqrt{2E_{f}2E_{i}}}\bar {u}(p_{f}, s_{f})\gamma_{0}u (p_{i}, s_{i}) \int \frac{d^{4}xe^{i(p_{f} - p_{i})x}}{|\mathbf x|},$$ where $f, i$-indices mark correspondingly final and initial waves, $s_{i, f}$ marks the polarization.

What is the difference for the case of positron scattering (the amplitude of scattering has different expression)?

• A difference in the Feynman rules for positrons is that they carry a $\bar{v}$ spinor instead of $u$ if they are incoming. They satisfy $\sum_{\mathrm{spins}} v(p)\bar{v}(p)=\gamma^{\mu}p_{\mu} -m$ rather than $\sum_{\mathrm{spins}} u(p)\bar{u}(p) = \gamma^{\mu}p_{\mu} +m$, neglecting spin indices. – user32361 Mar 13 '14 at 21:55