S matrix and vanishing fields

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Hello! I am reading about the S matrix, and I see that one of the assumption that the derivations are based on is the fact that interacting particles are free at ##t=\pm \infty## and I am not sure I understand why. One of the given examples is the ##\phi^4## theory which contains an interaction term of the form ##\frac{\lambda}{4!}\phi(x)^4##. Why would this term vanish far away from our experiment. More concretely, if we would have an electron Dirac spinor field (assuming we collide electron and positrons), why would we assume that far from the experiment the electron is free? Isn't it still interacting with the vacuum QED i.e. the vacuum would be ##|\Omega>## and not ##|0>## no matter at which point we are (even at big distances from our collision point). Can someone explain this to me please? Thank you!
 

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The interaction with the vacuum is treated separately as "property of the free electron". It is not part of the collision process.
 
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The interaction with the vacuum is treated separately as "property of the free electron". It is not part of the collision process.
So from the point of view of the S matrix, an interacting particle with momentum p and a non-interacting particle with momentum p are identical at ##\pm \infty##? But the operators used to create a particle with momentum p i.e. ##a_p^\dagger## is different (they evolve differently in time) in the 2 theories as the vacuum is different?
 
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vanhees71
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Haag's theorem is irrelevant FAPP. See the excellent book

The conceptual framework of QFT, Oxford University Press

It's precisely filling the gaps Weinberg's three volumes leave.
 
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I recommend you read section 1 of this (available in the amazon preview) to see how insanely fundamental to relativistic quantum theory this issue is:

Page 3 said:
'the description of such a process as occurring in the course of time is therefore just as unreal as the classical paths are in non-relativistic quantum mechanics. The only observable quantities are the properties of free particles'
 

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