I Calculating E+e- -> μ+ μ- Cross Section

[Silviu]

Hello! When calculating the matrix element for (let's say) $e^+e^- \to \mu^+ \mu^-$we have to average over initial spins and sum over final spins. I understand the motivation of this, but when the calculation is done, the sum is done for 2 cases: spin up and spin down, so you have to add 8 terms (2 for each particle) and divide by 4 (the 2 initial incoming particles). Why is this summation enough? Shouldn't one integrate over all possible values of spin? It is not like the particle will come with either spin up or spin down on a give axis, they can be a linear combination of these.

 

[ChrisVer]

Ehm, the fact that you have a linear combination of both spin up and down is that you have to sum over them... If the particle was in a known spin state, you wouldn't need to make any summation.

 

[Silviu]

Ehm, the fact that you have a linear combination of both spin up and down is that you have to sum over them... If the particle was in a known spin state, you wouldn't need to make any summation.

I understand this. My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations? You don't know the spin so it can be anything, not just up or down along the z axis.

 

[Orodruin]

My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations?

What you are doing is equivalent to considering all possible combinations. You have to consider is the ensemble of the in-state, which corresponds to a density matrix that is proportional to unity.

 

[ChrisVer]

I understand this. My question is why you sum over just the spin up and spin down and divide by 2, and not integrate over all the possible linear combinations? You don't know the spin so it can be anything, not just up or down along the z axis.

is there any other possible state for spin 1/2 ? as far as I know the dim is 2s+1=2.

 

[Silviu]

is there any other possible state for spin 1/2 ? as far as I know the dim is 2s+1=2.

My point was that the spin doesn't have to be along the z axis, it can be along any other axis. But Orodruin made it clear to me.

 

[ChrisVer]

in any axis it may be though, when written in the basis of z you are still having 2. ok