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Corrolation/Green's functions

  1. Jul 17, 2012 #1
    Hi all, this is a question about Green's functions (sometimes called corrolation functions), used in the LSZ reduction formula. They are defined in section 3.7 of http://www.damtp.cam.ac.uk/user/tong/qft/three.pdf in two different (but equivalent) ways:
    G(n)(x1, x2...xn):= <[itex]\Omega[/itex]|T{[itex]\Phi[/itex]1H[itex]\Phi[/itex]2H...[itex]\Phi[/itex]nH}|[itex]\Omega[/itex]> = <0|T{[itex]\Phi[/itex]1[itex]\Phi[/itex]2...[itex]\Phi[/itex]n}S|0>/<0|S|0> = sum of all connected Feynman graphs (where |[itex]\Omega[/itex]> is the true vacuum of the interacting theory, normalized to H|[itex]\Omega[/itex]> = 0; [itex]\Phi[/itex]nH = [itex]\Phi[/itex](xn) in the Heisenberg picture; T is the time-ordering operator and S is the scattering matrix). The link above has a very nice proof that these are all equivalent, but my question is: how, then, does one define the correlation functions for a theory where NOT all the operators are the same? At a guess, it would be defined as the above with a different choice of field operators as each combination for the LSZ formula requires...can anyone verify this or else tell me how such objects are calculated or where I can find out more?
    Also, if anyone can point me in the direction of some resources where some of the phenomena mentioned related to Green's functions are calculated?
  2. jcsd
  3. Jul 17, 2012 #2

    I am not totally sure I understood your question, and I couldn't access the link you posted (although I did take that course by David Tong, once upon a time!)

    In general the greens / correlation functions used in the LSZ formula can involve different fields operators. I think Srednicki chapters 5-10 give a nice walkthrough from the LSZ formula to scattering amplitudes ( don't panic, very short chapters and available online : http://web.physics.ucsb.edu/~mark/qft.html ).

    You may find problem 9.5 particularly illuminating. It doesn't need path integrals, so may be more in line with Tong's treatment.
    Last edited: Jul 17, 2012
  4. Jul 17, 2012 #3
    I'm not sure but I would guess you'd would have something like this:
    G(n,m)(x1, x2...xn ,y1, y2...ym) instead of
    G(n)(x1, x2...xn)

    in the single field case you have to specify the number of fields and in the two field case you specify the number of both fields. not sure if I answered your question
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