# The n-point Green functions and Heisenberg picture

Let's have the S-matrix: $$S_{\beta \alpha} = \langle \beta | \hat{S} | \alpha\rangle .$$ Here $|\alpha \rangle , | \beta \rangle$ are $t \to \mp \infty$ limit of the free states, $\hat {S} = \hat{T}e^{-i\int \hat{L}_{\int}d^{4}x}$, $\hat{L}_{\int}$ refers to the operator in the interaction picture. When we decide to get the matrix element of some process we will get $$\int d^{4}x_{1}...d^{4}x_{n}\langle \beta |\hat{T}(\hat{\varphi}_{1_{int}}(x_{1})...\hat{\varphi}_{m_{int}}(x_{n})) | \alpha \rangle .$$ So it's convenient to introduce n-point Green function, $$\tag 1 G_{n}(x_{1},...x_{n}) = \langle 0| \hat{T}(\hat {\varphi}_{1_{int}}(x_{1})...\hat{\varphi}_{n_{int}}(x_{n}))| 0\rangle$$ and generation functionals for it.

But recently I have read anywhere that as n-point Green function people use expression $$\tag 2 G^{H}_{n}(x_{1},...x_{n}) = \langle 0| \hat{T}(\hat {\varphi}_{1}(x_{1})...\hat{\varphi}_{n}(x_{n}))| 0\rangle ,$$ where the operators of fields are in the Heisenberg picture. So they need to rewrite the operators into interaction picture: $$\tag 3 G^{H}_{n}(x_{1},...x_{n}) = \langle 0| \hat{T}\left( \hat {\varphi}_{1_{int}}(x_{1})...\hat{\varphi}_{n_{int}}(x_{n})\hat{S}\right)|0\rangle .$$ I don't understand why we need the Green function $(2)$ where the fields operators are in the Heisenberg picture if $S$-matrix "generates" rather Green functions with operators in interaction picture.

Can you explain it why we don't use $(1)$ when talk about the Green functions?

• Could you add the sources that use these functions? I've seen many weird definitions of n-point function, generating functionals and the like, and often they have tailored it to their specific context (sometimes they're just confused, though). Commented Aug 7, 2014 at 15:59
• @ACuriousMind : for example, here, en.wikipedia.org/wiki/…, is used generating functional for $(3)$. Commented Aug 7, 2014 at 16:49

The normal Green's ($n$-point) functions are supposed to include all the interactions given by Feynman's vertices etc., so they need to be evaluated from the operators in the normal picture, i.e. the Heisenberg picture. The interaction picture is just a "fudged" compromise between the Heisenberg picture and the Schrödinger picture – a compromise that is useful and convenient but in no way fundamental. It's the Heisenberg picture and correlators in it that reduce to classical physics in the $\hbar\to 0$ classical limit.
• Thank you. But where these normal Green functions arise naturally (where we must introduce $(2)$, not $(1)$, for applying to the theory)? In the S-matrix there are only the Green function like my $(1)$, and nonperturbative results (like LSZ-theorem, Ward equalities etc.) also don't use $(2)$, if I don't confuse. Commented Aug 7, 2014 at 16:26