Problem about scattering amptitude

[knightq]

Take the scattering of photon and electron as example, we first figure out the amptitude for transition definte polarization,
say f(ϵα(k),ϵα′(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,1/2∑α∑α′f? the initial and final polarization should be all possible combination of the independent polarizatition.

 

[azntoon]

How to understand the average and sum?The average and sum over the two independent polarizations is actually what's known as a "partial wave expansion", which is a way of expanding the scattering amplitude into a sum of terms that each correspond to a particular angular momentum. The idea is that the total scattering amplitude can be expressed as a sum of these individual terms, and each term is determined by the initial and final polarizations. By averaging and summing over the two independent polarizations, we are essentially separating out the amplitude into individual components that each correspond to a particular angular momentum. This then allows us to calculate the total scattering amplitude more efficiently.