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Is there a deep reason for this Green's function co-incidences in Quantum Field Theory:

When you compute the time ordered vaccuum expectation value of a quantum field, it turns out to be the same as the Green's function of the classical field.

Ryder Rude
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  • Write down the LSZ formula for $\langle p\vert q\rangle$ (a single particle propagating). 2. I don't understand what you mean here - how does the path integral of a free particle "derive interaction physics"? Also, that the path integral is a kernel for the differential operator in the evolution equation of the wave function (I'm not sure why you say "a classical field" here, either) is true pretty much by construction. | Finally, please ask only a single question per post - these two questions look unrelated except for both being about Green's functions.
    • – ACuriousMind Aug 31 '22 at 10:48
  • @ACuriousMind 1. I haven't studied the LSZ formula. Does it resolve the first co-incidence? I have only seen the first co-incidence explicitly computed. 2. I meant that that path integral derives the Green's function, which has the Coulomb potential built into it. So this co-incidence relates interaction physics to free-particle physics. Can you please write an answer about your resolution of this co-incidence? I really thought these two belonged in the same question. I apologise for breaking site rules – Ryder Rude Aug 31 '22 at 10:53
  • @ACuriousMind Okay I'll ask a separate question about the second co-incidence – Ryder Rude Aug 31 '22 at 10:57