How are correlation functions defined for non-uniform operator theories?

[UndeadCat]

Hi all, this is a question about Green's functions (sometimes called correlation 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...x_n):= <$\Omega$|T{$\Phi$_1H$\Phi$_2H...$\Phi$_nH}|$\Omega$> = <0|T{$\Phi$₁$\Phi$₂...$\Phi$_n}S|0>/<0|S|0> = sum of all connected Feynman graphs (where |$\Omega$> is the true vacuum of the interacting theory, normalized to H|$\Omega$> = 0; $\Phi$_nH = $\Phi$(x_n) 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?

 

[booster]

Hello

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.

 

[alemsalem]

I'm not sure but I would guess you'd would have something like this:
G^(n,m)(x₁, x₂...x_n ,y₁, y₂...y_m) instead of
G⁽ⁿ⁾(x₁, x₂...x_n)

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