What is a contraction in QFT?

Question

I have been reading QFT and I am stumbling upon the idea of Wick's theorem. The correlation functions have something to do with "contractions". I want to understand what the physical meaning of a contraction is!

Answer 1

I) Definition. Given two operator ordering prescriptions, denoted by, say, $T$ and $::$, the corresponding (generalized) contraction $$C(\hat{A},\hat{B})~=~ T(\hat{A}\hat{B})~-~:\hat{A}\hat{B}:$$ of two operators $\hat{A}$ and $\hat{B}$ is the difference in ordering prescriptions.

II) In applications, a contraction $C(\hat{A},\hat{B})~\propto~{\bf 1}$ is typically proportional to the identity operator ${\bf 1}$. Then $$C(\hat{A},\hat{B})~=~\langle \Omega|C(\hat{A},\hat{B}) |\Omega\rangle ~{\bf 1}.$$ With a slight abuse of language, the corresponding correlator $\langle \Omega|C(\hat{A},\hat{B}) |\Omega\rangle$ is often referred to as the contraction.

III) To see how contractions are important in Wick's theorem, see e.g. this & this Phys.SE posts.