Is it possible that the incoming particles, in a scattering experiment, have a superposition state of up and down spin? In this case, aren't we supposed, after we calculate the matrix elements for the different spin states, to add these amplitudes rather than their squares?
In other words, why do we always do this $\sum_{ss'}\left|\mathcal{M}\right|^2$ rather than $\left|\sum_{ss'}\mathcal{M}\right|^2$?
There are two types of uncertainty. The first is the classical kind where the system is in a particular state and we don't know what that state is. In that case one should add the probabilities which are given by the magnitude of the amplitude squared. The second type of uncertainty is the quantum mechanical one. In this case the system is not in one particular state but instead has different amplitudes for more than one state. Then, if one wants to work out the probability, one first adds over the amplitudes and then takes the magnitude squared of the result. The case you discuss is of the first kind as the incoming particles are in a particular spin state, we just don't know what that spin state is.
