# Calculation of a 4-point function by path integrals

In Srednicki's book in chapter 8 a four-point function is computed as a sum of products of propagators:

$$<0|T\phi(x_1)\phi(x_2) \phi(x_3)\phi(x_4)|0> = \frac{1}{i^2}[\Delta(x_1 -x_2)\Delta(x_3-x_4) + \Delta(x_1 -x_3)\Delta(x_2-x_4)+ \Delta(x_1 -x_4)\Delta(x_2-x_3)]\tag{8.16}$$

whereas in chapter 10 the result is different: ($$\delta_j= \frac{1}{i}\frac{\delta}{\delta J(x_j)}$$ and $$i W = \log Z(J)$$, only 1-order terms are considered)

$$<0|T\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)|0>= \delta_1\delta_2\delta_3\delta_4 iW|_{J=0} +\frac{1}{i^2}[\Delta(x_1 -x_2)\Delta(x_3-x_4) + \Delta(x_1 -x_3)\Delta(x_2-x_4)+ \Delta(x_1 -x_4)\Delta(x_2-x_3)]\tag{10.4}$$

In the following it is even argued that for the transition amplitude only the first term is important. It seems that in chapter 8 a non-interaction theory is considered whereas in chapter 10 an interacting theory.

How can the difference be explained? I still do not feel so familiar with the path integral formalism that I would easily guess it.

• It seems that OP has answered his own question by pointing out that chapter 8 is for a free theory. – Qmechanic Nov 17 '18 at 15:03