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masteralien
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- TL;DR Summary
- In QFT one computes Scattering Amplitudes between Initial and Final States however sometimes the system doesn’t evolve into a single final state but a superposition how does one compute the S Matrix or Green Functions for this case.
In QFT one important calculation is the Scattering amplitude the overlap between an initial state and a final state due to an interaction. For example the S Matrix element for electron electron scattering is the overlap between an initial 2 electron state in the far past and a 2 electron state in the far future.
However the system may not evolve into a single final state rather a superposition. For example during Compton Scattering a recoiling electron may also emit photons due to Bremsstrahlung or during an electron positron interaction the final state could either be a photon or an electron and a positron either Bhabha Scattering or Pair Annihilation can happen. How would one compute scattering amplitudes and Correlation Functions in this case when there is more than a single final state.
Presumably one would simply add up the amplitudes for electron positron interaction as
S_fi=S1+S2
And Green Functions as the sum of the 2 possibilities
G=G4(x1,x2,x3,x4)+G3(x5,x3,x4)
=<Ω|Tψ(x1)ψ(x2)ψ^+(x3)ψ^+(x4)|Ω>+<Ω|TA_μ(x5)ψ^+(x3)ψ^+(x4)|Ω>
Where x3 and x4 are the initial spacetime positions of the electron and positron x1 and x2 final spacetime positions and x5 the Photon’s spacetime position.
Presumably it would follow the S Matrix is simply a sum of the 2.
Disclaimer the electron positron scattering was an example there might be more than 2 possible final states this was simply an example
However the system may not evolve into a single final state rather a superposition. For example during Compton Scattering a recoiling electron may also emit photons due to Bremsstrahlung or during an electron positron interaction the final state could either be a photon or an electron and a positron either Bhabha Scattering or Pair Annihilation can happen. How would one compute scattering amplitudes and Correlation Functions in this case when there is more than a single final state.
Presumably one would simply add up the amplitudes for electron positron interaction as
S_fi=S1+S2
And Green Functions as the sum of the 2 possibilities
G=G4(x1,x2,x3,x4)+G3(x5,x3,x4)
=<Ω|Tψ(x1)ψ(x2)ψ^+(x3)ψ^+(x4)|Ω>+<Ω|TA_μ(x5)ψ^+(x3)ψ^+(x4)|Ω>
Where x3 and x4 are the initial spacetime positions of the electron and positron x1 and x2 final spacetime positions and x5 the Photon’s spacetime position.
Presumably it would follow the S Matrix is simply a sum of the 2.
Disclaimer the electron positron scattering was an example there might be more than 2 possible final states this was simply an example