Propagator and $n$-point Green Function meaning with Path-integral

Question

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:

1. Is everything I wrote correct?

2. 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?

3. 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 $\phi_2$:

$$ G^{(2)}_{1, 2}(x , y)= \left[\frac{1}{Z[J]} \frac{\delta^{2} Z}{\delta J_{1}(x)\delta J_{2}(y)}\right]\Bigg|_{J=0}. $$

4. 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.

5. 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.

Answer 1

You are bringing up a lot of issues in this question. I think perhaps you need to slow down and try to understand each piece of the story on its own more carefully, and build up to a complete picture. A good resource that will address your questions is Srednicki's book, an online version is available for free, especially chapters 8-10, see https://chaosbook.org/FieldTheory/extras/SrednickiQFT03.pdf.

However, I'll try to give brief answers to the questions you raised, without writing a textbook.

Is everything I wrote correct?

Everything before this point is essentially a definition, so not really correct or incorrect. I would say your definitions are standard, although I would probably call $D_i(x-y)$ the free propagator rather than the propagator.

What's the utility of this new Green Function? I thought that the external sources 𝐽𝑖 where just a tool to write the 𝐺(𝑛) in term of a Generating Functional 𝑍[𝐽], but aside from that the 𝐽𝑖 shouldn't have a physical meaning... right?

Typically we are interested in Green's functions in the vacuum $|0\rangle$ with no sources, so you are correct. However, there are cases where a Green's function in a state with a classical source could be relevant, for example in the effective theory of gravitational waves (https://arxiv.org/abs/hep-th/0409156). In that specific example, the sources would correspond to the black holes creating background spacetime curvature.

If the propagator is defined in the free theory then what does the 2-point function represent, what's the difference from the propagator?

The 2-point function $G^2_{a,b}(x,y)$, the way you defined it, is different from the free propagator, $D_{a,b}(x-y)$. You can see this if you just try to compute $G^2_{a,b}(x,y)$ perturbatively, you'll see that $D_{a,b}(x-y)$ is the first order contribution, but there are also higher order corrections. This is computed explicitly in many QFT books. You could view the former as the amplitude for an excitation to travel from $x$ to $y$ in the full theory, and the latter as the amplitude in the free theory.

what's the difference from the 2-point function evaluated with different fields

If you want a physical interpretation, you can think of it in terms of particle oscillation. The amplitude $G^{2}_{1,2}(x-y)$ tells you the amplitude for a particle of type $\phi_1$ to start at $x$, and arrive at $y$ as particle type $\phi_2$.

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.

The leading order contribution to $G^4$ is a sum of tree level diagrams with 4 external legs. This will include the 4-point vertex, as well as a diagram with two three-point vertices plus a propagator connecting them. There are also higher order loop corrections you will find if you expand out the expression for $G^{4}$ in terms of derivatives of $Z_0[J]$ -- again, this calculation is done in many QFT books.

There's also a deeper construction using the quantum effective action (which is a Legendre transformation of the generating functional $Z[J]$, and which should not be confused with the Wilson effective action), where the exact four-point function is given by the quartic vertex computed from the full quantum action.

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.)

If you've understood everything up to this point, the generalization to arbitrary $n$ should be straightforward. You write the exact expression for $G^n$ in terms of derivatives of $Z_0[J]$, and compute the derivatives to any desired order. The result will be a series of Feynman diagrams with $n$ external legs, built out of the vertices of the original theory.