Why do we only average and sum over two independent polarizations in scattering?

[knightq]

Take the scattering of photon and electron as example, we first figure out the amptitude for transition definte polarization,
say f(\epsilon_{\alpha}(\textbf{k}),\epsilon_{\alpha'} (\textbf{k}'))
If the initial photon is unpolarized and the final polarization is not observed, we need to average the initial polarization and sum over the possible final polarization.
But why we just average and sum over the two independent polarization,\frac{1}{2}\sum_{\alpha}\sum_{\alpha'}f? the initial and final polarization should be all possible combination of the independent polarizatition.

 

[azntoon]

The reason why we only average and sum over the two independent polarizations is because we are assuming that the initial photon is unpolarized. Unpolarized light contains all possible polarizations, so it is not necessary to consider all possible combinations of the two independent polarizations. Additionally, since we are not observing the final polarization, we can take advantage of the fact that all possible final polarizations will contribute the same amount of intensity, so it is not necessary to consider each combination individually. Instead, we can just average the initial polarization and sum over the possible final polarizations.