According to this text here
http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-4-4p.pdf
contact terms do not affect the scattering amplitude. But These contact Terms are there; the question is: When contact Terms are relevant for scattering Amplitude computation?
My idea:
By starting with the connected Partition function G[J]:=logZ[J] where Z[J] is ordinary Partition function corresponding to the Action
S=Stheory+∫d4x Jϕ
for some fields ϕ and the source J one can derive cumulants belonging to Stheory by multiple Derivation of G[J] by J and Setting J=0. Only the equation for quadratic cumulants ⟨0|ϕ(x)ϕ(y)|0⟩ will contain an equation with the contact term δ(x−y). More precisely
H⟨0|ϕ(x)ϕ(y)|0⟩=f(others)+δ(x−y)
for an Operator H that I assume to be linear and nonlinear corrections f(others).
Neglecting nonlinearities I see that ⟨0|ϕ(x)ϕ(y)|0⟩ is exactly the Green function generated by H. This Green function Δ(x−y) vanishes if the Observation time t is set to ∞. And infinitely Observation times are assumed in the LSZ formula for scattering amplitudes.
Will contact Terms be relevant for finite Observation times? Why on scattering Amplitude/ cross section computation infinitely Long Observation times are assumed?
No real process has infinitely Long Observation times. But maybe uncertainty in energy is cancelled if Δt↦∞ is assumed.
Help would be greatly appreciated.
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