Free theory time-ordered correlation function with two internal fields

[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?

 

[eljose79]

I would like to clarify that there are actually only two distinct Wick contractions for this particular time-ordered correlation function. The third option you mentioned is not a valid Wick contraction as it would result in a repeated field pairing, which is not allowed in Wick's theorem.

The first Wick contraction you mentioned corresponds to the following product of two-point correlation functions:

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

The second Wick contraction corresponds to the product of three two-point correlation functions:

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

It is important to note that Wick's theorem only applies to free theory time-ordered correlation functions. In the case of interacting theories, the Wick contractions become more complicated and involve additional terms.

In conclusion, Wick's theorem is a powerful tool that simplifies the calculation of correlation functions in free theories. However, it is important to carefully consider all possible Wick contractions and eliminate any redundant ones. I hope this clarifies any confusion you may have had about the application of Wick's theorem to this particular correlation function.