I am studying QFT from Peskin & Schroeder. There I found a physical interpretation of a 2-point correlation function. According to Peskin & Schroeder, the 2-point correlation function is nothing but a propagator between two points and I am happy with that. But the question that arises immediately is, what are the n-point correlation functions?

e.g.- does the 3-point correlation function given by, $\langle\omega|T[\phi(x)\phi(y)\phi(z)]|\omega\rangle$ imply the amplitude of propagation from $x$ to $z$ via $y$?

Is this understanding correct?


In a typical QFT, Wicks theorem tells you that the expectation value of all higher point correlation functions can be expressed as two point correlation functions. Essentially, a $2n$-point correlation function tells you about $n$ particles propagating from point $x_i$ to $x_j$, where $i,j = 1...n$.

For example, a 4 points look like, $$\langle\Omega|T\left\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\right\}|\Omega\rangle = \langle\Omega|T\left\{\phi(x_1)\phi(x_2)\right\}|\Omega\rangle\langle\Omega|T\left\{\phi(x_3)\phi(x_4)\right\}|\Omega\rangle + \langle\Omega|T\left\{\phi(x_1)\phi(x_3)\right\}|\Omega\rangle\langle\Omega|T\left\{\phi(x_2)\phi(x_4)\right\}|\Omega\rangle + \langle\Omega|T\left\{\phi(x_1)\phi(x_4)\right\}|\Omega\rangle\langle\Omega|T\left\{\phi(x_2)\phi(x_3)\right\}|\Omega\rangle $$

Which corresponds to one particle going from $x_1$ to $x_2$ and one going from $x_3$ to $x_4$, plus all the other possibilities. Diagramatically, this looks like

Free scalar Feynman Diagrams

You will notice that your example of a three point correlation function is zero according to this theorem.

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  • $\begingroup$ Yes, I have encountered the fact that odd point correlation functions turn out to be zero. But I think that was for free field. Do odd point correlation functions vanish even for interacting fields? $\endgroup$ – Samapan Bhadury Mar 19 '18 at 17:16
  • $\begingroup$ By the way, that was a very helpful answer. You pointed out the fact that a 2n point correlation function is about n particles moving from one point to another. :) $\endgroup$ – Samapan Bhadury Mar 19 '18 at 17:18

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