As an example we will use a Lagrangian of $k$ scalar fields $\phi_i$ with a generic quartic interaction between these fields:
$$ \mathcal{L} = \left(\sum_{i}\mathcal{L}_{i, free} \right) + \left(\sum_{i} \lambda_i \phi_i^{4} \right) + g_1 (\phi_1)^{2} (\phi_2)^2 + g_2 (\phi_1)^{2} (\phi_2)(\phi_3) + etc. $$
From what I understand the generic $n$-point Green function is:
$$ G^{(n)}_{{a_1} ... {a_n}}(x_1, ..., x_n) = \frac{\langle 0| \mathcal{T}\left[ \phi_{a_1} (x_1) ... \phi_{a_n}(x_n) \right]|0\rangle}{\langle 0|0\rangle} $$
With $\phi_{a_1} ... \phi_{a_n}$ a generic combination of $n$-fields of the $k$ scalar fields.
We can derive generic $n$-point Green function also using the Functional Generator $Z[J]$:
$$ Z[J]= \int \mathcal D \phi_1 ... D \phi_k \, e^{i \int d^4 x \left(\mathcal{L} + J^{i}(x)\phi_{i}(x) \right)} $$
$$ Z[J=0] = \langle 0 | 0 \rangle. $$
And we have:
$$ G^{(n)}_{{a_1} ... {a_n}}(x_1, ..., x_n)= \left[\frac{1}{Z[J]} \frac{\delta^{n} Z}{\delta J_{a_1}(x_1)...\delta J_{a_n}(x_n)}\right]\Bigg|_{J=0}. $$
Finally for each $\phi_i$ the Propagator $D_{i}(x-y)$ is the 2-point Green Function in the FREE theory, i.e. if we define:
$$ Z_{0}[J]= \int \mathcal D \phi_1 ... D \phi_k \, e^{i \int d^4 x \left(\sum_i \mathcal{L_{i,free}} + J^{i}(x)\phi_{i}(x) \right)}. $$
Then:
$$ D_{i}(x-y) = \left[\frac{1}{Z_{0}[J]} \frac{\delta^{2} Z_{0}}{\delta J_{i}(x)\delta J_{i}(y)}\right]\Bigg|_{J=0} $$
So my questions are:
Is everything I wrote correct?
I also saw we can define a $n$-point function without putting thee sources $J_i$ at zero after deriving Z[J]:
$$ \tilde{G}^{(n)}_{{a_1} ... {a_n}}(x_1, ..., x_n)= \frac{1}{Z[J]} \frac{\delta^{n} Z}{\delta J_{a_1}(x_1)...\delta J_{a_n}(x_n)} = \frac{_{J}\langle 0| \mathcal{T}\left[ \phi_{a_1} (x_1) ... \phi_{a_n}(x_n) \right]|0\rangle_{J}}{_{J}\langle 0|0\rangle_{J} } $$ Where $|0\rangle_{J}$ is the "new" vacuum in the sense that it is the vacuum of the new Lagrangian given by $\mathcal{L}$ plus an extra interaction term between the fields and the sources $J^i\phi_{i}$.
What's the utility of this new Green Function? I thought that the external sources $J_i$ where just a tool to write the $G^{(n)}$ in term of a Generating Functional $Z[J]$, but aside from that the $J_{i}$ shouldn't have a physical meaning... right?
- If the propagator is defined in the free theory then what does the 2-point function represent, what's the difference from the propagator?
In particular what is the 2-point function evaluated with the same field, for example $\phi_1$:
$$ G^{(2)}_{1, 1}(x , y)= \left[\frac{1}{Z[J]} \frac{\delta^{2} Z}{\delta J_{1}(x)\delta J_{1}(y)}\right]\Bigg|_{J=0} $$
And what's the difference from the 2-point function evaluated with different fields, for example $\phi_1$ and $\phi_2$:
$$ G^{(2)}_{1, 2}(x , y)= \left[\frac{1}{Z[J]} \frac{\delta^{2} Z}{\delta J_{1}(x)\delta J_{2}(y)}\right]\Bigg|_{J=0}. $$
What about for the $4$-point function? I think it is related in some way to the vertex, but I don't know precisely how.
What about a generic $n$-point function? In particular what is it if $n$ is different from $2$ or $4$ (or in general if we had a cubic interaction also from $3$, etc.)
I think the $1$ point function should be related to the (normalized) vacuum expectation value of the field:
$$ G^{(1)}_{i}(x) = \left[\frac{1}{Z[J]} \frac{\delta Z}{\delta J_{i}(x)}\right]\Bigg|_{J=0} = \frac{\langle 0|\phi_{i}|0\rangle}{\langle 0|0\rangle}. $$
But I read in a book that the $n$-point Green function is zero if $n$ is odd (is this true?) Edit: Apparently this is true only for the free theory, due to parity.
I'm sorry if these question have been asked before, but I searched deeply in this forum and I didn't find a satisfying answer.