# Interference of spin in scattering processes

I am unable to understand how do we determine interference of matrix elements in processes such as positron electron annihilation into muons, i.e. $$e^++e^-\rightarrow \mu^++\mu_-$$. Let us consider the case where we haven't measured the electrons spins before hand (quoting Peskin and Schroeder "In most experiments the electron and positron beams are unpolarized").

With that in mind let us label the spins of the incoming particles as $$s$$ and $$s'$$ and of the outcoming particles $$r$$ and $$r'$$, denoting $$M_{ss'\rightarrow rr'}$$ the associated matrix element.

Now, I understand that if we don't discriminate the spin of the outgoing muons, we should ADD the probabilities since the different outcomes cannot interfere, as we could in principle distinguish each outcome. In other words, do something like \$\sum_{r,r'} |M_{ss'\rightarrow rr'}|^2

Now, consider the initial spins. If we have as we said, an "unpolarized" electron and positron beam, shouldn't the matrix elements interfere ? In this case in my opinion they should, and the appropriate matrix element should be : $$\sum_{r,r'} |\sum_{ss'}M_{ss'\rightarrow rr'}|^2$$.

However, in Peskin and Schroeder (and any other book I have been able to check), they seem to imply that even though the beam is unpolarized, there is not interference. In my interpretation this means that using an unpolarised beam is the same experiment as measuring the spin of each incoming electron and then performing the experiment, in which case the amplitude should reasonably be :

$$\frac{1}{4}\sum_{s,s'}\sum_{r,r'}|M_{ss'\rightarrow rr'}|^2$$

Now I'm sure that this expression is right even in the unpolarized beam case, but I am unable to understand how the different event do not interfere. Could someone offer a detailed explanation of the reasoning we have when determining if 2 events should interfere ?

My understanding was that if we could technically distinguish the two events, then the processes should not interfere. But in this case it seems to me that going out and measuring the spins of the incoming particles WILL actually change the experiment, since it will collapse particles into definite spin states.