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The background: I am looking at Compton scattering in its most general form with $p+\gamma^*\rightarrow p'+\gamma'^*$ in the Breit-frame (which implies that in my case $\vec{p} = -\vec{p}'$).

The question: How can I calculate the total helicity of the system before and after the scattering when I only have the individual helicities? I think an answer on how to add two helicities in general would also help.

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You cannot meaningfully "add" helicities. Helicity is the projection of spin onto momentum. If by "total helicity" you mean the projection of the total spin onto the total momentum, then this is impossible to compute from the individual helicities (take the extreme case where each individual spin is perpendicular to its corresponding individual momentum, so individual helicities are zero, to see that this is true).

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