# A Free theory time-ordered correlation function with two internal fields

1. Nov 19, 2016

### spaghetti3451

Wick's theorem allows one to write a free theory time-ordered $n$-point correlation function as a product of free theory time-ordered $2$-point correlation function.

The procedure involves the pairwise Wick contraction of fields such that external fields are not paired up each other.

Consider the following time-ordered correlation function:

$$\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x) \phi(x) \phi(x) \phi(y) \phi(y) \phi(y) \} | 0 \rangle.$$

This time-ordered $8$-point correlation function can be Wick contracted in multiple ways.

1. One possible Wick contraction pairs $\phi(x_{1})$ with one $\phi(x)$ and pairs $\phi(x_{2})$ with another $\phi(x)$.

2. Another possible Wick contraction pairs $\phi(x_{1})$ with one $\phi(x)$ and pairs $\phi(x_{2})$ with one $\phi(y)$.

3. A third possible Wick contraction pairs $\phi(x_{1})$ with one $\phi(y)$ and pairs $\phi(x_{2})$ with another $\phi(y)$. However, this appears to be redundant as it is already covered in the first choice of Wick contractions. Am I wrong?

2. Nov 24, 2016