# What is a contraction in QFT?

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!

• Reading which QFT texts? – Qmechanic Mar 17 '18 at 19:25
• I am following Ashoke Das, Peskin & Schroeder and sometimes Ryder, Srednicki. But my question originates from college lectures. – Samapan Bhadury Mar 17 '18 at 19:34

## 1 Answer

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.

• When drawing Feynman diagrams, we are supposed to connect to points if they appear with contraction right? If that understanding of mine is correct then I am failing to connect these two. Why contraction is represented as connection between two points in the diagram? I hope I could explain my difficulty. I am being unable to figure out why the mathematical terms of contraction result in connection in the diagram? – Samapan Bhadury Mar 17 '18 at 19:12
• Often a contraction has an interpretation as a propagator. – Qmechanic Mar 17 '18 at 19:52
• Yeah, but why is that represented as connection between two points in a Feynman diagram? – Samapan Bhadury Mar 17 '18 at 19:59