Saturday, February 17, 2018

quantum field theory - Contact terms in Dyson-Schwinger equation can be ignored?


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(xy). More precisely


H0|ϕ(x)ϕ(y)|0=f(others)+δ(xy)


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(xy) 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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