# Wick theorem applying to partly ordered operator

I symbolize $T$ as the time-ordered operator and $::$ as normal order symbol. I know that in quantum field theory generally we have:

$$T\phi_1(x_1)\dots\phi_n(x_n)=:\phi_1(x_1)\dots\phi_n(x_n):+A$$

where $A$ roughly means all possible contractions. I have read a convincing proof.

However when I encounter terms like:

$$T:\phi_1(x_1)\phi_2(x_2)::\phi_3(x_3)\phi_4(x_4):\phi_5(x_5)$$

I don't know how to apply Wick theorem. I guess that $A$ now contains only the contractions of pairs not included in the same normal order symbol $::$ (for example the contraction of $\phi_3\phi_4$ is eliminated)

What is the general rules? How can I understand it intuitively?

EDIT: Actually I have found a referrence that states the rules:

http://www.fysik.su.se/~kardell/QFT/Tutorial-8.pdf

• Unless I miss something, it seems to me that there is an error in the formulae $(3),(4),(5)$. I don't see any double contraction $(AD)(BC)$ ? – Trimok Aug 8 '14 at 10:04