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In Quantum Field Theory, truncated N-Point functions (or truncated Green's functions) are the N-Point functions of diagrams with their external legs chopped off.

I was told that the truncated N-point function, in momentum space, is given by the following relation:

$$\prod_{j=1}^N G^{(2)}(p_j)^{-1}G^{(N)}(p_1,\ldots,p_N)$$

In other words, it is the original N-Point function but dividing the 2-point function for each external momentum.

May I ask why is this the case, or what is the derivation behind it?

(The notation is: those functions are not connected and the 2-point functions are exact inverses, but feel free to point out any problems with the above expression if you think it is wrong.)

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  • $\begingroup$ Are $G_{}^{(2)}$ and $G_{}^{(N)}$ connected 2-point and N-point functions or not? Is $G_{}^{(2)}$ in the above expression is the inverse of exact 2-point function or the inverse of free 2-point function? $\endgroup$
    – Sunyam
    Aug 26, 2018 at 20:27
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    $\begingroup$ The notation is: not connected and exact inverse, but feel free to point out any problems with the above expression if you think it is wrong. $\endgroup$
    – user148792
    Aug 26, 2018 at 20:47

1 Answer 1

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It is basically a matter of being consistent with definitions/notation.

  1. The $n$-point function, denoted by $$ G^{(n)}(x_1,\dots,x_n) $$ is given by the sum over all Feynman diagrams with $n$ external legs, and where every line corresponds to a propagator, and every vertex to a bare vertex factor (e.g., $ig$); see ref.1, §6-1-1 for some more details.

  2. The connected $n$-point function, denoted by $$ G^{(n)}_c(x_1,\dots,x_n) $$ is given by the sum over all connected Feynman diagrams with $n$ external legs, and where every line corresponds to a propagator, and every vertex to a bare vertex factor (e.g., $ig$). The functions $G$ and $G_c$ are related by $$ G^{(n)}(x_1,\dots,x_n)=\sum_{\cup I_\alpha=I}\prod_\alpha G^{(n)}_c(\{x_{I_\alpha}\})\tag{5-52} $$ where $I=(1,\dots,n)$. (This relation is much more easily expressed in terms of generating functions, cf. $Z(j)=\exp(Z_c(j))$; see ref.1, §5-1-5 for some more details).

  3. The truncated $n$-point function, denoted by $$ G^{(n)}_t(x_1,\dots,x_n) $$ is given by the sum over all Feynman diagrams with $n$ external legs, and where every line corresponds to a propagator except for the external legs (which carry no factor), and every vertex to a bare vertex factor (e.g., $ig$). The functions $G$ and $G_t$ are related by $$ G^{(n)}(x_1,\dots,x_n)=\int\Delta(x_1,y_1)\cdots \Delta(x_n,y_n) G^{(n)}_t(y_1,\dots,y_n)\ \mathrm dy_1\cdots\mathrm dy_n $$

    One may similarly define the truncated and connected $n$-point function: $$ G_c^{(n)}(x_1,\dots,x_n)=\int\Delta(x_1,y_1)\cdots \Delta(x_n,y_n) G^{(n)}_{c,t}(y_1,\dots,y_n)\ \mathrm dy_1\cdots\mathrm dy_n $$ whose diagrammatic meaning should be clear.

  4. There is another notion of truncated $n$-point function, which I do not find useful, that reads $$ G^{(n)}(x_1,\dots,x_n)=\int G^{(2)}\cdots G^{(2)}(x_n,y_n) \tilde G^{(n)}_t(y_1,\dots,y_n)\ \mathrm dy_1\cdots\mathrm dy_n\tag{6-70} $$ with the full two-point function $G^{(2)}$ instead of the propagator $\Delta$. The main use for truncated $n$-point functions is that they can be fed directly into the LSZ formula; but as in the latter we are taking $p^2\to m^2$, and in this limit we have $G^{(2)}\to \Delta$, there is no difference: either of them works fine when calculating $S$-matrix elements. From a conceptual point of view, the function $G_t$ as defined above seems much more natural than $\tilde G_t$, but the reader should note that both notions are used in the literature.

  5. The proper $n$-point function, denoted by $$ \Gamma^{(n)}(x_1,\dots,x_n) $$ is given by the sum over all one-particle-irreducible Feynman diagrams with $n$ external legs, and where every line corresponds to a full two-point function $G^{(2)}=G^{(2)}_c$ except for the external legs (which carry no factor), and every vertex to a full vertex factor. The functions $G_c$ and $\Gamma$ are related by $$ G_c^{(n)}(x_1,\dots,x_n)=\int G^{(2)}(x_1,y_1)\cdots G^{(2)}(x_n,y_n) \Gamma^{(n)}(y_1,\dots,y_n)\ \mathrm dy_1\cdots\mathrm dy_n+\cdots $$ where $+\cdots$ corresponds to lower-order terms (with $\Gamma^{(n-1)},\Gamma^{(n-2)},\dots$). (The relation is much more easily expressed in terms of generating functions, cf. $Z_c(j)$ and $\Gamma(j)$ are Legendre-dual; see ref.1, §6-2-2 for more details).

It bears mentioning that all the relations between pairs of functions $G,G'$ written above are invertible, so one may express, say, $G_c$ as a function of $G$ and vice-versa. But this "inverse" has to be understood in the functional sense, e.g., $$ \int A(x,y)A^{-1}(y,z)\ \mathrm dy\equiv \delta(x,z) $$

This means that, for example, the inverse of $$ G^{(n)}(x_1,\dots,x_n)=\int\Delta(x_1,y_1)\cdots \Delta(x_n,y_n) G^{(n)}_t(y_1,\dots,y_n)\ \mathrm dy_1\cdots\mathrm dy_n $$ is $$ G_t^{(n)}(x_1,\dots,x_n)=\mathscr D_{x_1}\cdots \mathscr D_{x_n} G^{(n)}(x_1,\dots,x_n) $$ where $\mathscr D_x\Delta(x,y)\equiv \delta(x,y)$ (modulo signs that I don't care about here).

In momentum space, convolutions become standard multiplication (cf. this PSE post), and therefore the functional inverse is basically the algebraic inverse. Therefore, the relations above become $$ \begin{aligned} G^{(n)}(p_1,\dots,p_n)&=\Delta(p_1)\cdots \Delta(p_n) G^{(n)}_t(p_1,\dots,p_n)\\ G_t^{(n)}(p_1,\dots,p_n)&=\Delta(p_1)^{-1}\cdots \Delta(p_n)^{-1} G^{(n)}_t(p_1,\dots,p_n) \end{aligned} $$ as in the OP (or replace $\Delta\to G^{(2)}$ if you want to use $\tilde G_t$ instead of $G_t$).

Note that all these definitions work for theories where the fields may have any spin or Grassmann parity; if the field is non-scalar, one just introduces more indices here and there, and if it is Grassmann odd, there are some missing signs that I have neglected. Filling out the details are left to the reader.

References.

  1. Itzykson & Zuber - Quantum field theory.
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  • $\begingroup$ To fix: there are a couple of wrong statements in 5., the proper function! $\endgroup$ Nov 10, 2018 at 15:06

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