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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.

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In the beams we have in the laboratory, the particles do not superpose or interact with each other, due to the large space time distances for the individual particles in the beam. Within our experimental accuracies it is as if each electron is all alone in the universe when interacting with a positron. Whether polarized or not each incoming particle can be considered non interacting with other beam particles.

It might help if you think of a light beam quantum mechanically: the number of photons superposing to form the beam is enormous, way beyond anything that could be attained in the laboratory for particles.

In any case, in order to study the scattering of e+e- one does not want interactions in the incoming beam at the level of superpositions.

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