# How to find the number of distinct contraction cases in Wick's Theorem?

Let $\mathcal{G}^8_{un}:=(t_1,t_2,t_1'^3,t_2'^3)=\langle 0 \mid T[Q_{un}(t_1)Q_{un}(t_2)Q(t_1')^3Q(t_2')^3] \mid 0 \rangle_{un}$

We want to use Wicks theorem to write this function as the sum of 2-point functions. Apparently there are eight different types of contraction. Some examples are such as: (and there are 5 other cases).

Can you determine the number of distinct contractions for a general n point function before you go through Wicks theorem by brute force? This would provide a good of checking that you have covered all possible cases when using Wick's Theorem.

For reference see page 14 of these notes http://www.maths.bris.ac.uk/~dc13950/lecturenotes.pdf

## 1 Answer

Why not try to draw the Feynman diagrams first and characterize them by different topologies; then count the possible contractions for each cases?

• In my course it goes from the contractions to the Feynman diagrams. I dont see how you would go straight to the diagrams? – Permian Apr 23 '15 at 9:52
• Well, I'm not 100 percent sure if I understand your question correctly. But according to my understanding, in your problem, I think there are two external legs(t1,t2) and two vertex, right? So just need to pairly connect them I would say. – Chuan Chen Apr 23 '15 at 10:48