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

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Discussion Overview

The discussion revolves around the calculation of the matrix element for the process ##e^+e^- \to \mu^+ \mu^-##, specifically focusing on the treatment of spin states in the context of averaging and summing over initial and final spins. The scope includes theoretical considerations related to particle physics and quantum mechanics.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions why the summation over spins is limited to just spin up and spin down, suggesting that integration over all possible spin states should be considered instead.
  • Another participant argues that summing over spin up and down is sufficient because if the particle were in a known spin state, no summation would be necessary.
  • There is a reiteration of the initial question regarding the adequacy of summing only two spin states and not integrating over linear combinations of spins.
  • A participant mentions that considering the ensemble of the in-state corresponds to a density matrix that is proportional to unity, implying this encompasses all possible combinations.
  • There is a discussion about the dimensionality of spin states, with one participant affirming that for spin 1/2, the dimension is indeed 2, while another emphasizes that the spin can be oriented along any axis, not just the z-axis.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of spin states in the calculation, with no consensus reached on whether summing over just spin up and down is sufficient or if a more comprehensive integration approach is warranted.

Contextual Notes

The discussion highlights potential limitations in the treatment of spin states, particularly regarding the assumptions made about the orientation of spins and the representation of quantum states.

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

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